Quantum Algorithm Zoo

Algebraic and Number Theoretic Algorithms

Algorithm: Factoring
Speedup: Superpolynomial
Implementation: Classiq, Cirq, PennyLane, Qrisp
Description: Given an n-bit integer, find the prime factorization. The quantum algorithm of Peter Shor solves this in $\tilde{O}(n^3)$ time [82,125]. The fastest known classical algorithm for integer factorization is the general number field sieve, which is believed to run in time $2^{\tilde{O}(n^{1/3})}$. The best rigorously proven upper bound on the classical complexity of factoring is $O(2^{n/4+o(1)})$ via the Pollard-Strassen algorithm [252, 362]. Shor's factoring algorithm breaks RSA public-key encryption and the closely related quantum algorithms for discrete logarithms break the DSA and ECDSA digital signature schemes and the Diffie-Hellman key-exchange protocol. A quantum algorithm even faster than Shor's for the special case of factoring “semiprimes”, which are widely used in cryptography, is given in [271]. If small factors exist, Shor's algorithm can be beaten by a quantum algorithm using Grover search to speed up the elliptic curve factorization method [366]. Additional optimized versions of Shor's algorithm are given in [384, 386, 431]. There are proposed classical public-key cryptosystems not believed to be broken by quantum algorithms, cf. [248]. At the core of Shor's factoring algorithm is order finding, which can be reduced to the Abelian hidden subgroup problem, which is solved using the quantum Fourier transform. A number of other problems are known to reduce to integer factorization including the membership problem for matrix groups over fields of odd order [253], and certain diophantine problems relevant to the synthesis of quantum circuits [254].

Algorithm: Discrete-log
Speedup: Superpolynomial
Implementation: Classiq, Qrisp
Description: We are given three n-bit numbers a, b, and N, with the promise that $b=a^s\mathrm{mod}N$ for some s. The task is to find s. As shown by Shor [82], this can be achieved on a quantum computer in poly(n) time. The fastest known classical algorithm requires time superpolynomial in n. By similar techniques to those in [82], quantum computers can solve the discrete logarithm problem on elliptic curves, thereby breaking elliptic curve cryptography [109, 14]. Further optimizations to Shor's algorithm are given in [385, 432]. The superpolynomial quantum speedup has also been extended to the discrete logarithm problem on semigroups [203, 204]. See also Abelian hidden subgroup.

Algorithm: Pell's Equation
Speedup: Superpolynomial
Description: Given a positive nonsquare integer d, Pell's equation is $x^2-d{y}^2=1$. For any such d there are infinitely many pairs of integers (x,y) solving this equation. Let $(x_1,y_1)$ be the pair that minimizes $x+y\sqrt{d}$. If d is an n-bit integer (i.e. $0\le d<2^n$ ), $(x_1,y_1)$ may in general require exponentially many bits to write down. Thus it is in general impossible to find $(x_1,y_1)$ in polynomial time. Let $R=\log (x_1+y_1\sqrt{d})$. $\lfloor R\rceil$ uniquely identifies $(x_1,y_1)$. As shown by Hallgren [49], given a n-bit number d, a quantum computer can find $\lfloor R\rceil$ in poly(n) time. No polynomial time classical algorithm for this problem is known. Factoring reduces to this problem. This algorithm breaks the Buchman-Williams cryptosystem. See also Abelian hidden subgroup.

Algorithm: Principal Ideal
Speedup: Superpolynomial
Description: We are given an n-bit integer d and an invertible ideal I of the ring $\mathbb{Z}[\sqrt{d}]$. I is a principal ideal if there exists $\alpha \in \mathbb{Q}(\sqrt{d})$ such that $I=\alpha \mathbb{Z}[\sqrt{d}]$. $\alpha$ may be exponentially large in d. Therefore $\alpha$ cannot in general even be written down in polynomial time. However, $\lfloor \log \alpha \rceil$ uniquely identifies $\alpha$. The task is to determine whether I is principal and if so find $\lfloor \log \alpha \rceil$. As shown by Hallgren, this can be done in polynomial time on a quantum computer [49]. A modified quantum algorithm for this problem using fewer qubits was given in [131]. A quantum algorithm solving the principal ideal problem in number fields of arbitrary degree (i.e. scaling polynomially in the degree) was subsequently given in [329]. Factoring reduces to solving Pell's equation, which reduces to the principal ideal problem. Thus the principal ideal problem is at least as hard as factoring and therefore is probably not in P. See also Abelian hidden subgroup.

Algorithm: Unit Group
Speedup: Superpolynomial
Description: The number field $\mathbb{Q}(\theta )$ is said to be of degree d if the lowest degree polynomial of which $\theta$ is a root has degree d. The set $\mathcal{O}$ of elements of $\mathbb{Q}(\theta )$ which are roots of monic polynomials in $\mathbb{Z}[x]$ forms a ring, called the ring of integers of $\mathbb{Q}(\theta )$. The set of units (invertible elements) of the ring $\mathcal{O}$ form a group denoted ${\mathcal{O}}^{\ast }$. As shown by Hallgren [50], and independently by Schmidt and Vollmer [116], for any $\mathbb{Q}(\theta )$ of fixed degree, a quantum computer can find in polynomial time a set of generators for ${\mathcal{O}}^{\ast }$ given a description of $\theta$. No polynomial time classical algorithm for this problem is known. Hallgren and collaborators subsequently discovered how to achieve polynomial scaling in the degree [213]. See also [329]. The algorithms rely on solving Abelian hidden subgroup problems over the additive group of real numbers.

Algorithm: Class Group
Speedup: Superpolynomial
Description: The number field $\mathbb{Q}(\theta )$ is said to be of degree d if the lowest degree polynomial of which $\theta$ is a root has degree d. The set $\mathcal{O}$ of elements of $\mathbb{Q}(\theta )$ which are roots of monic polynomials in $\mathbb{Z}[x]$ forms a ring, called the ring of integers of $\mathbb{Q}(\theta )$, which is a Dedekind domain. For a Dedekind domain, the nonzero fractional ideals modulo the nonzero principal ideals form a group called the class group. As shown by Hallgren [50], a quantum computer can find a set of generators for the class group of the ring of integers of any constant degree number field, given a description of $\theta$, in time poly(log($|\mathcal{O}|$)). An improved quantum algorithm, whose runtime is also polynomial in d was subsequently given in [329]. No polynomial time classical algorithm for these problems are known. See also Abelian hidden subgroup.

Algorithm: Gauss Sums
Speedup: Superpolynomial
Description: Let ${\mathbb{F}}_q$ be a finite field. The elements other than zero of ${\mathbb{F}}_q$ form a group ${\mathbb{F}}_q^{\times }$ under multiplication, and the elements of ${\mathbb{F}}_q$ form an (Abelian but not necessarily cyclic) group ${\mathbb{F}}_q^{+}$ under addition. We can choose some character ${\chi }^{\times }$ of ${\mathbb{F}}_q^{\times }$ and some character ${\chi }^{+}$ of ${\mathbb{F}}_q^{+}$. The corresponding Gauss sum is the inner product of these characters: $\sum _{x\ne 0\in {\mathbb{F}}_q}{\chi }^{+}(x){\chi }^{\times }(x)$ As shown by van Dam and Seroussi [90], Gauss sums can be estimated to polynomial precision on a quantum computer in polynomial time. Although a finite ring does not form a group under multiplication, its set of units does. Choosing a representation for the additive group of the ring, and choosing a representation for the multiplicative group of its units, one can obtain a Gauss sum over the units of a finite ring. These can also be estimated to polynomial precision on a quantum computer in polynomial time [90]. No polynomial time classical algorithm for estimating Gauss sums is known. Discrete log reduces to Gauss sum estimation [90]. Certain partition functions of the Potts model can be computed by a polynomial-time quantum algorithm related to Gauss sum estimation [47].

Algorithm:Primality Proving
Speedup:Polynomial
Description: Given an n-bit number, return a proof of its primality. The fastest classical algorithms are AKS, the best versions of which [393, 394] have essentially-quartic complexity, and ECPP, where the heuristic complexity of the fastest version [395] is also essentially quartic. The fastest known quantum algorithm for this problem is the method of Donis-Vela and Garcia-Escartin [396], with complexity $O(n^2(\log n{)}^3\log \log n)$. This improves upon a prior factoring-based quantum algorithm for primality proving [397] that has complexity $O(n^3\log n\log \log n)$. A recent result of Harvey and Van Der Hoeven [398] can be used to improve the complexity of the factoring-based quantum algorithm for primality proving to $O(n^3\log n)$ and it may be possible to similarly reduce the complexity of the Donis-Vela-Garcia-Escartin algorithm to $O(n^2(\log n{)}^3)$ [399].

Algorithm:Solving Exponential Congruences
Speedup:Polynomial
Description: We are given $a,b,c,f,g\in {\mathbb{F}}_q$. We must find integers $x,y$ such that $a{f}^x+b{g}^y=c$. As shown in [111], quantum computers can solve this problem in $\tilde{O}(q^{3/8})$ time whereas the best classical algorithm requires $\tilde{O}(q^{9/8})$ time. The quantum algorithm of [111] is based on the quantum algorithms for discrete logarithms and searching.

Algorithm: Matrix Elements and Multiplicity Coefficients of Group Representations
Speedup: Superpolynomial
Description: All representations of finite groups and compact linear groups can be expressed as unitary matrices given an appropriate choice of basis. Conjugating the regular representation of a group by the quantum Fourier transform circuit over that group yields a direct sum of the group's irreducible representations. Thus, the efficient quantum Fourier transform over the symmetric group [196], together with the Hadamard test, yields a fast quantum algorithm for additively approximating individual matrix elements of the arbitrary irreducible representations of $S_n$. Similarly, using the quantum Schur transform [197], one can efficiently approximate matrix elements of the irreducible representations of SU(n) that have polynomial weight. Direct implementations of individual irreducible representations for the groups U(n), SU(n), SO(n), and $A_n$ by efficient quantum circuits are given in [106]. Instances that appear to be exponentially hard for known classical algorithms are also identified in [106]. Kronecker coefficients count the multiplicity of a given irreducible representation in the tensor product of a given pair of irreducible representations. In [460] it was shown that normalized Kronecker coefficients of the symmetric group can be approximated to within an additive inverse polynomial error by a polynomial time quantum algorithm, whereas no polynomial time classical algorithm achieving this was known at the time. These quantum speedups were generalized to other multiplicity coefficients of the symmetric group in [516]. Prompted by these results, the state of the art in classical algorithms for computing Kronecker coefficients and more general multiplicity coefficients was improved in [515]. Nevertheless, as discussed in [516], these advances do not eliminate all quantum speedups for approximating multiplicity coefficients.

Algorithm: Verifying Matrix Products
Speedup: Polynomial
Description: Given three $n\times n$ matrices, A,B, and C, the matrix product verification problem is to decide whether AB=C. Classically, the best known (randomized) algorithm achieves this in time $O(n^2)$, whereas the best known classical algorithm for matrix multiplication runs in time $O(n^{2.373})$. Ambainis et al. discovered a quantum algorithm for this problem with runtime $O(n^{7/4})$ [6]. Subsequently, Buhrman and Špalek improved upon this, obtaining a quantum algorithm for this problem with runtime $O(n^{5/3})$ [19]. This latter algorithm is based on results regarding quantum walks that were proven in [85].

Algorithm: Subset-sum
Speedup: Polynomial
Description: Given a list of integers $x_1,\dots ,x_n$, and a target integer s, the subset-sum problem is to determine whether the sum of any subset of the given integers adds up to s. This problem is NP-complete, and therefore is unlikely to be solvable by classical or quantum algorithms with polynomial worst-case complexity. In the hard instances the given integers are of order $2^n$ and much research on subset sum focuses on average case instances in this regime. In [178], a quantum algorithm is given that solves such instances in time $2^{0.241n}$, up to polynomial factors. This quantum algorithm works by applying a variant of Ambainis's quantum walk algorithm for element-distinctness [7] to speed up a sophisticated classical algorithm for this problem due to Howgrave-Graham and Joux. The fastest known classical algorithm for such instances of subset-sum runs in time $2^{0.291n}$, up to polynomial factors [404].

Algorithm: Decoding
Speedup: Varies
Description: Classical error correcting codes allow the detection and correction of bit-flips by storing data reduntantly. Maximum-likelihood decoding for arbitrary linear codes is NP-complete in the worst case, but for structured codes or bounded error efficient decoding algorithms are known. Quantum algorithms have been formulated to speed up the decoding of convolutional codes [238] and simplex codes [239].

Algorithm: Quantum Cryptanalysis
Speedup: Various
Description: It is well-known that Shor's algorithms for factoring and discrete logarithms [82,125] completely break the RSA and Diffie-Hellman cryptosystems, as well as their elliptic-curve-based variants [109, 14]. (A number of "post-quantum" public-key cryptosystems have been proposed to replace these primitives, which are not known to be broken by quantum attacks.) Beyond Shor's algorithm, there is a growing body of work on quantum algorithms specifically designed to attack cryptosystems. These generally fall into three categories. The first is quantum algorithms providing polynomial or sub-exponential time attacks on cryptosystems under standard assumptions. In particular, the algorithm of Childs, Jao, and Soukharev for finding isogenies of elliptic curves breaks certain elliptic curve based cryptosystems in subexponential time that were not already broken by Shor's algorithm [283]. The second category is quantum algorithms achieving polynomial improvement over known classical cryptanalytic attacks by speeding up parts of these classical algorithms using Grover search, quantum collision finding, etc. Such attacks on private-key [284, 285, 288, 315, 316] and public-key [262, 287] primitives, do not preclude the use of the associated cryptosystems but may influence choice of key size. The third category is attacks that make use of quantum superposition queries to block ciphers. These attacks in many cases completely break the cryptographic primitives [286, 289, 290, 291, 292]. However, in most practical situations such superposition queries are unlikely to be feasible.

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Oracular Algorithms

Algorithm: Searching
Speedup: Polynomial
Implementation: Classiq, Cirq, PennyLane, Cirq, Qrisp (Grover), Qrisp (Quantum Counting), Qrisp (Amplitude Amplification)
Description: We are given an oracle with N allowed inputs. For one input w ("the winner") the corresponding output is 1, and for all other inputs the corresponding output is 0. The task is to find w. On a classical computer this requires $\mathrm{\Omega }(N)$ queries. The quantum algorithm of Lov Grover achieves this using $O(\sqrt{N})$ queries [48], which is optimal [216]. This algorithm has subsequently been generalized to search in the presence of multiple "winners" [15], evaluate the sum of an arbitrary function [15,16,73], find the mean, median, and global minimum of an arbitrary function [35,75, 255,465,472], take advantage of alternative initial states [100] or nonuniform probabilistic priors [123], work with oracles whose runtime varies between inputs [138], approximate definite integrals [77], and converge to a fixed-point [208, 209, 433]. Considerations on optimizing the depth of quantum search circuits are given in [405]. The generalization of Grover's algorithm known as amplitude estimation [17] is now an important primitive in quantum algorithms. Amplitude estimation forms the core of most known quantum algorithms related to collision finding and graph properties. One of the natural applications for Grover search is speeding up the solution to NP-complete problems such as 3-SAT. Doing so is nontrivial, because the best classical algorithm for 3-SAT is not quite a brute force search. Nevertheless, amplitude amplification enables a quadratic quantum speedup over the best classical 3-SAT algorithm, as shown in [133]. Quadratic speedups for other constraint satisfaction problems are obtained in [134]. (Slightly superquadratic speedups relative to brute search are obtained using means beyond amplitude amplification in [493,492].) For further examples of application of Grover search and amplitude amplification see [261, 262]. A problem closely related to, but harder than, Grover search, is spatial search, in which database queries are limited by some graph structure. On sufficiently well-connected graphs, $O(\sqrt{n})$ quantum query complexity is still achievable [274,275,303, 304, 305, 306, 330].

Algorithm: Abelian Hidden Subgroup
Speedup: Superpolynomial
Implementation: Classiq, Cirq
Description: Let G be a finitely generated Abelian group, and let H be some subgroup of G such that G/H is finite. Let f be a function on G such that for any $g_1,g_2\in G$, $f(g_1)=f(g_2)$ if and only if $g_1$ and $g_2$ are in the same coset of H. The task is to find H (i.e. find a set of generators for H) by making queries to f. This is solvable on a quantum computer using $O(\log |G|)$ queries, whereas classically $\mathrm{\Omega }(|G|)$ are required. This algorithm was first formulated in full generality by Boneh and Lipton in [14]. However, proper attribution of this algorithm is difficult because, as described in chapter 5 of [76], it subsumes many historically important quantum algorithms as special cases, including Simon's algorithm [108], which was the inspiration for Shor's period finding algorithm, which forms the core of his factoring and discrete-log algorithms. The Abelian hidden subgroup algorithm is also at the core of the Pell's equation, principal ideal, unit group, and class group algorithms. In certain instances, the Abelian hidden subgroup problem can be solved using a single query rather than order $\log (|G|)$, as shown in [30]. It is normally assumed in period finding that the function $f(x)\ne f(y)$ unless $x-y=s$, where $s$ is the period. A quantum algorithm which applies even when this restiction is relaxed is given in [388]. Period finding has been generalized to apply to oracles which provide only the few most significant bits about the underlying function in [389].

Algorithm: Non-Abelian Hidden Subgroup
Speedup: Superpolynomial
Description: Let G be a finitely generated group, and let H be some subgroup of G that has finitely many left cosets. Let f be a function on G such that for any $g_1,g_2$, $f(g_1)=f(g_2)$ if and only if $g_1$ and $g_2$ are in the same left coset of H. The task is to find H (i.e. find a set of generators for H) by making queries to f. This is solvable on a quantum computer using $O(\log (|G|)$ queries, whereas classically $\mathrm{\Omega }(|G|)$ are required [37,51]. However, this does not qualify as an efficient quantum algorithm because in general, it may take exponential time to process the quantum states obtained from these queries. Efficient quantum algorithms for the hidden subgroup problem are known for certain specific non-Abelian groups [81,55,72,53,9,22,56,71,57,43,44,28,126,207,273]. A slightly outdated survey is given in [69]. Of particular interest are the symmetric group and the dihedral group. A solution for the symmetric group would solve graph isomorphism. A solution for the dihedral group would solve certain lattice problems [78]. Despite much effort, no polynomial-time solution for these groups is known, except in special cases [312]. However, Kuperberg [66] found a time $2^{O(\sqrt{\log N})})$ algorithm for finding a hidden subgroup of the dihedral group $D_N$. Regev subsequently improved this algorithm so that it uses not only subexponential time but also polynomial space [79]. A further improvement in the asymptotic scaling of the required number of qubits is obtained in [218]. Quantum query speedups (though not necessarily efficient quantum algorithms in terms of gate count) for somewhat more general problems of testing for isomorphisms of functions under sets of permutations are given in [311]

Algorithm: Bernstein-Vazirani
Speedup: Polynomial Directly, Superpolynomial Recursively
Implementation: Classiq, Cirq, PennyLane
Description: We are given an oracle whose input is n bits and whose output is one bit. Given input $x\in \{0,1{\}}^n$, the output is $x\odot h$, where h is the "hidden" string of n bits, and $\odot$ denotes the bitwise inner product modulo 2. The task is to find h. On a classical computer this requires n queries. As shown by Bernstein and Vazirani [11], this can be achieved on a quantum computer using a single query. Furthermore, one can construct recursive versions of this problem, called recursive Fourier sampling, such that quantum computers require exponentially fewer queries than classical computers [11]. See [256, 257] for related work on the ubiquity of quantum speedups from generic quantum circuits and [258, 270] for related work on a quantum query speedup for detecting correlations between the an oracle function and the Fourier transform of another.

Algorithm: Deutsch-Jozsa
Speedup: Exponential over P, none over BPP
Implementation: Classiq, PennyLane
Description: We are given an oracle whose input is n bits and whose output is one bit. We are promised that out of the $2^n$ possible inputs, either all of them, none of them, or half of them yield output 1. The task is to distinguish the balanced case (half of all inputs yield output 1) from the constant case (all or none of the inputs yield output 1). It was shown by Deutsch [32] that for n=1, this can be solved on a quantum computer using one query, whereas any deterministic classical algorithm requires two. This was historically the first well-defined quantum algorithm achieving a speedup over classical computation. (A related, more recent, pedagogical example is given in [259].) A single-query quantum algorithm for arbitrary n was developed by Deutsch and Jozsa in [33]. Although probabilistically easy to solve with O(1) queries, the Deutsch-Jozsa problem has exponential worst case deterministic query complexity classically.

Algorithm: Formula Evaluation
Speedup: Polynomial
Description: A Boolean expression is called a formula if each variable is used only once. A formula corresponds to a circuit with no fanout, which consequently has the topology of a tree. By Reichardt's span-program formalism, it is now known [158] that the quantum query complexity of any formula of O(1) fanin on N variables is $\mathrm{\Theta }(\sqrt{N})$. This result culminates from a long line of work [27,8,80,159,160], which started with the discovery by Farhi et al. [38] that NAND trees on $2^n$ variables can be evaluated on quantum computers in time $O(2^{0.5n})$ using a continuous-time quantum walk, whereas classical computers require $\mathrm{\Omega }(2^{0.753n})$ queries. In many cases, the quantum formula-evaluation algorithms are efficient not only in query complexity but also in time-complexity. The span-program formalism also yields quantum query complexity lower bounds [149]. Although originally discovered from a different point of view, Grover's algorithm can be regarded as a special case of formula evaluation in which every gate is OR. The quantum complexity of evaluating non-boolean formulas has also been studied [29], but is not as fully understood. Childs et al. have generalized to the case in which input variables may be repeated (i.e. the first layer of the circuit may include fanout) [101]. They obtained a quantum algorithm using $O(min\{N,\sqrt{S},N^{1/2}{G}^{1/4}\})$ queries, where N is the number of input variables not including multiplicities, S is the number of inputs counting multiplicities, and G is the number of gates in the formula. References [164], [165], and [269] consider special cases of the NAND tree problem in which the number of NAND gates taking unequal inputs is limited. Some of these cases yield superpolynomial separation between quantum and classical query complexity.

Algorithm: Hidden Shift
Speedup: Superpolynomial
Implementation: Classiq, Cirq
Description: We are given oracle access to some function f on ${\mathbb{Z}}_N$. We know that f(x) = g(x+s) where g is a known function and s is an unknown shift. The hidden shift problem is to find s. By reduction from Grover's problem it is clear that at least $\sqrt{N}$ queries are necessary to solve hidden shift in general. However, certain special cases of the hidden shift problem are solvable on quantum computers using O(1) queries. In particular, van Dam et al. showed that this can be done if f is a multiplicative character of a finite ring or field [89]. The previously discovered shifted Legendre symbol algorithm [88,86] is subsumed as a special case of this, because the Legendre symbol $\left(\frac{x}{p}\right)$ is a multiplicative character of ${\mathbb{F}}_p$. No classical algorithm running in time O(polylog(N)) is known for these problems. Furthermore, the quantum algorithm for the shifted Legendre symbol problem would break a certain cryptographic pseudorandom generator given the ability to make quantum queries to the generator [89]. A quantum speedup for hidden shift problems of difference sets is given in [312], and this also subsumes the Legendre symbol problem as a special case. Roetteler has found exponential quantum speedups for finding hidden shifts of certain nonlinear Boolean functions [105,130]. Building on this work, Gavinsky, Roetteler, and Roland have shown [142] that the hidden shift problem on random boolean functions $f:{\mathbb{Z}}_2^n\to {\mathbb{Z}}_2$ has O(n) average case quantum complexity, whereas the classical query complexity is $\mathrm{\Omega }(2^{n/2})$. The results in [143], though they are phrased in terms of the hidden subgroup problem for the dihedral group, imply that the quantum query complexity of the hidden shift problem for an injective function on ${\mathbb{Z}}_N$ is O(log n), whereas the classical query complexity is $\mathrm{\Theta }(\sqrt{N})$. However, the best known quantum circuit complexity for injective hidden shift on ${\mathbb{Z}}_N$ is $O(2^{C\sqrt{\log N}})$, achieved by Kuperberg's sieve algorithm [66]. A recent result, building upon [408, 43], achieves exponential quantum speedups for some generalizations of the Hidden shift problem including the hidden multiple shift problem, in which one has query access to $f_s(x)=f(x-hs)$ over some allowed range of s and one wishes to infer h [407].

Algorithm: Polynomial interpolation
Speedup: Varies
Description: Let $p(x)=a_d{x}^d+\dots +a_{1}x+a_0$ be a polynomial over the finite field $\mathrm{GF}(q)$. One is given access to an oracle that, given $x\in \mathrm{GF}(q)$, returns $p(x)$. The polynomial reconstruction problem is, by making queries to the oracle, to determine the coefficients $a_d,\dots ,a_0$. Classically, $d+1$ queries are necessary and sufficient. (In some sources use the term reconstruction instead of interpolation for this problem.) Quantumly, $d/2+1/2$ queries are necessary and $d/2+1$ queries are sufficient [360,361]. For multivariate polynomials of degree d in n variables the interpolation problem has classical query complexity $(\genfrac{}{}{0ex}{}{n+d}{d})$. As shown in [387], the quantum query complexity is $O\left(\frac{1}{n+1}(\genfrac{}{}{0ex}{}{n+d}{d})\right)$ over $\mathbb{R}$ and $\mathbb{C}$ and it is $O\left(\frac{d}{n+d}(\genfrac{}{}{0ex}{}{n+d}{d})\right)$ over ${\mathbb{F}}_q$ for sufficiently large q. Quantum algorithms have also been discovered for the case that the oracle returns $\chi (f(x))$ where $\chi$ is a quadratic character of $\mathrm{GF}(q)$ [390], and the case where the oracle returns $f(x{)}^e$ [392]. These generalize the hidden shift algorithm of [89] and achieve an exponential speedup over classical computation. A quantum algorithm for reconstructing rational functions over finite fields given noisy and incomplete oracle access to the function values is given in [391].

Algorithm: Pattern matching
Speedup: Superpolynomial
Description: Given strings T of length n and P of length m < n, both from some finite alphabet, the pattern matching problem is to find an occurrence of P as a substring of T or to report that P is not a substring of T. More generally, T and P could be d-dimensional arrays rather than one-dimensional arrays (strings). Then, the pattern matching problem is to return the location of P as a $m\times m\times \dots \times m$ block within the $n\times n\times \dots \times n$ array T or report that no such location exists. The $\mathrm{\Omega }(\sqrt{N})$ query lower bound for unstructured search [216] implies that the worst-case quantum query complexity of this problem is $\mathrm{\Omega }(\sqrt{n}+\sqrt{m})$. A quantum algorithm achieving this, up to logarithmic factors, was obtained in [217]. This quantum algorithm works through the use of Grover's algorithm together with a classical method called deterministic sampling. More recently, Montanaro showed that superpolynomial quantum speedup can be achieved on average case instances of pattern matching, provided that m is greater than logarithmic in n. Specifically, the quantum algorithm given in [215] solves average case pattern matching in $\tilde{O}((n/m{)}^{d/2}{2}^{O(d^{3/2}\sqrt{\log m})})$ time. This quantum algorithm is constructed by generalizing Kuperberg's quantum sieve algorithm [66] for dihedral hidden subgroup and hidden shift problems so that it can operate in d dimensions and accomodate small amounts of noise, and then classically reducing the pattern matching problem to this noisy d-dimensional version of hidden shift. A quantum algorithm for string matching with $\tilde{O}(\sqrt{n})$ complexity is given in [435] in a different input model, where the strings are written out in their entirety using $n+m$ qubits rather than through quantum queries to an oracle providing individual bits.

Algorithm: Ordered Search
Speedup: Constant factor
Description: We are given oracle access to a list of N numbers in order from least to greatest. Given a number x, the task is to find out where in the list it would fit. Classically, the best possible algorithm is binary search which takes ${\log }_{2}N$ queries. Farhi et al. showed that a quantum computer can achieve this using 0.53 ${\log }_{2}N$ queries [39]. Currently, the best known deterministic quantum algorithm for this problem uses 0.433 ${\log }_{2}N$ queries [103]. A lower bound of $\frac{\ln 2}{\pi }{\log }_{2}N$ quantum queries has been proven for this problem [219, 24]. In [10], a randomized quantum algorithm is given whose expected query complexity is less than $\frac{1}{3}{\log }_{2}N$.

Algorithm: Graph Properties in the Adjacency Matrix Model
Speedup: Polynomial
Description: Let G be a graph of n vertices. We are given access to an oracle, which given a pair of integers in {1,2,...,n} tells us whether the corresponding vertices are connected by an edge. Building on previous work [35,52,36], Dürr et al. [34] show that the quantum query complexity of finding a minimum spanning tree of weighted graphs, and deciding connectivity for directed and undirected graphs have $\mathrm{\Theta }(n^{3/2})$ quantum query complexity, and that finding lowest weight paths has $O(n^{3/2}{\log }^{2}n)$ quantum query complexity. Deciding whether a graph is bipartite, detecting cycles, and deciding whether a given vertex can be reached from another (st-connectivity) can all be achieved using a number of queries and quantum gates that both scale as $\tilde{O}(n^{3/2})$, and only logarithmically many qubits, as shown in [317], building upon [13, 272, 318]. A span-program-based quantum algorithm for detecting trees of a given size as minors in $\tilde{O}(n)$ time is given in [240]. A graph property is sparse if there exists a constant c such that every graph with the property has a ratio of edges to vertices at most c. Childs and Kothari have shown that all sparse graph properties have query complexity $\mathrm{\Theta }(n^{2/3})$ if they cannot be characterized by a list of forbidden subgraphs and $o(n^{2/3})$ (little-o) if they can [140]. The former algorithm is based on Grover search, the latter on the quantum walk formalism of [141]. By Mader's theorem, sparse graph properties include all nontrivial minor-closed properties. These include planarity, being a forest, and not containing a path of given length. According to the widely-believed Aanderaa-Karp-Rosenberg conjecture, all of the above problems have $\mathrm{\Omega }(n^2)$ classical query complexity. Another interesting computational problem is finding a subgraph H in a given graph G. The simplest case of this finding the triangle, that is, the clique of size three. The fastest known quantum algorithm for this finds a triangle in $O(n^{5/4})$ quantum queries [319], improving upon [276, 175, 171, 70, 152, 21]. Stronger quantum query complexity upper bounds are known when the graphs are sufficiently sparse [319, 320]. Classically, triangle finding requires $\mathrm{\Omega }(n^2)$ queries [21]. More generally, a quantum computer can find an arbitrary subgraph of k vertices using $O(n^{2-2/k-t})$ queries where $t=(2k-d-3)/(k(d+1)(m+2))$ and d and m are such that H has a vertex of degree d and m+d edges [153]. This improves on the previous algorithm of [70]. In some cases, this query complexity is beaten by the quantum algorithm of [140], which finds H using $\tilde{O}\left(n^{\frac{3}{2}-\frac{1}{\mathrm{vc}(H)+1}}\right)$ queries, provided G is sparse, where vc(H) is the size of the minimal vertex cover of H. A quantum algorithm for finding constant-sized sub-hypergraphs over 3-uniform hypergraphs in $O(n^{1.883})$ queries is given in [241].

Algorithm: Graph Properties in the Adjacency List Model
Speedup: Polynomial
Description: Let G be a graph of N vertices, M edges, and degree d. We are given access to an oracle which, when queried with the label of a vertex and $j\in \{1,2,\dots ,d\}$ outputs the jth neighbor of the vertex or null if the vertex has degree less than d. Suppose we are given the promise that G is either bipartite or is far from bipartite in the sense that a constant fraction of the edges would need to be removed to achieve bipartiteness. Then, as shown in [144], the quantum complexity of deciding bipartiteness is $\tilde{O}(N^{1/3})$. Also in [144], it is shown that distinguishing expander graphs from graphs that are far from being expanders has quantum complexity $\tilde{O}(N^{1/3})$ and $\tilde{\mathrm{\Omega }}(N^{1/4})$, whereas the classical complexity is $\tilde{\mathrm{\Theta }}(\sqrt{N})$. The key quantum algorithmic tool is Ambainis' algorithm for element distinctness. In [34], it is shown that finding a minimal spanning tree has quantum query complexity $\mathrm{\Theta }(\sqrt{NM})$, deciding graph connectivity has quantum query complexity $\mathrm{\Theta }(N)$ in the undirected case, and $\tilde{\mathrm{\Theta }}(\sqrt{NM})$ in the directed case, and computing the lowest weight path from a given source to all other vertices on a weighted graph has quantum query complexity $\tilde{\mathrm{\Theta }}(\sqrt{NM})$. In [317] quantum algorithms are given for st-connectivity, deciding bipartiteness, and deciding whether a graph is a forest, which run in $\tilde{O}(N\sqrt{d})$ time and use only logarithmically many qubits.

Algorithm: Welded Tree
Speedup: Superpolynomial
Implementation: Classiq
Description: Some computational problems can be phrased in terms of the query complexity of finding one's way through a maze. That is, there is some graph G to which one is given oracle access. When queried with the label of a given node, the oracle returns a list of the labels of all adjacent nodes. The task is, starting from some source node (i.e. its label), to find the label of a certain marked destination node. As shown by Childs et al. [26], quantum computers can exponentially outperform classical computers at this task for at least some graphs. Specifically, consider the graph obtained by joining together two depth-n binary trees by a random "weld" such that all nodes but the two roots have degree three. Starting from one root, a quantum computer can find the other root using poly(n) queries, whereas this is provably impossible using classical queries.

Algorithm: Collision Finding and Element Distinctness
Speedup: Polynomial
Description: Suppose we are given oracle access to a two to one function f on a domain of size N. The collision problem is to find a pair $x,y\in \{1,2,\dots ,N\}$ such that f(x) = f(y). The classical randomized query complexity of this problem is $\mathrm{\Theta }(\sqrt{N})$, whereas, as shown by Brassard et al., a quantum computer can achieve this using $O(N^{1/3})$ queries [18]. (See also [315].) Removing the promise that f is two-to-one yields a problem called element distinctness, which has $\mathrm{\Theta }(N)$ classical query complexity. Improving upon [21], Ambainis gives a quantum algorithm with query complexity of $O(N^{2/3})$ for element distinctness, which is optimal [7, 374]. The problem of deciding whether any k-fold collisions exist is called k-distinctness. Improving upon [7,154], the best quantum query complexity for k-distinctness is $O(n^{3/4-1/(4(2^k-1))})$ [172, 173]. The series of works [7, 363], culminating in [464], show that this is also the quantum time complexity for all k, up to logarithmic factors. Given two functions f and g, on domains of size N and M, respectively a claw is a pair x,y such that f(x) = g(y). In the case that N=M, the algorithm of [7] solves claw-finding in $O(N^{2/3})$ queries, improving on the previous $O(N^{3/4}\log N)$ quantum algorithm of [21]. Further work gives improved query complexity for various parameter regimes in which $N\ne M$ [364, 365]. More generally, a related problem to element distinctness, is, given oracle access to a sequence, to estimate the $k^{\mathrm{th}}$ frequency moment $F_k=\sum _j{n}_j^k$, where $n_j$ is the number of times that j occurs in the sequence. An approximately quadratic speedup for this problem is obtained in [277]. See also graph collision.

Algorithm: Graph Collision
Speedup: Polynomial
Description: We are given an undirected graph of n vertices and oracle access to a labelling of the vertices by 1 and 0. The graph collision problem is, by querying this oracle, to decide whether there exist a pair of vertices, connected by an edge, both of which are labelled 1. One can embed Grover's unstructured search problem as an instance of graph collision by choosing the star graph, labelling the center 1, and labelling the remaining vertices by the database entries. Hence, this problem has quantum query complexity $\mathrm{\Omega }(\sqrt{n})$ and classical query complexity $\mathrm{\Theta }(n)$. In [70], Magniez, Nayak, and Szegedy gave a $O(N^{2/3})$-query quantum algorithm for graph collision on general graphs. This remains the best upper bound on quantum query complexity for this problem on general graphs. However, stronger upper bounds have been obtained for several special classes of graphs. Specifically, the quantum query complexity on a graph G is $\tilde{O}(\sqrt{n}+\sqrt{l})$ where l is the number of non-edges in G [161], $O(\sqrt{n}{\alpha }^{1/6})$ where $\alpha$ is the size of the largest independent set of G [172], $O(\sqrt{n}+\sqrt{{\alpha }^{\ast }})$ where ${\alpha }^{\ast }$ is the maximum total degree of any independent set of G [200], and $O(\sqrt{n}{t}^{1/6})$ where t is the treewidth of G [201]. Furthermore, the quantum query complexity is $\tilde{O}(\sqrt{n})$ with high probability for random graphs in which the presence or absence of an edge between each pair of vertices is chosen independently with fixed probability, (i.e. Erdős-Rényi graphs) [200]. See [201] for a summary of these results as well as new upper bounds for two additional classes of graph that are too complicated to describe here.

Algorithm: Matrix Commutativity
Speedup: Polynomial
Description: We are given oracle access to k matrices, each of which are $n\times n$. Given integers $i,j\in \{1,2,\dots ,n\}$, and $x\in \{1,2,\dots ,k\}$ the oracle returns the ij matrix element of the $x^{\mathrm{th}}$ matrix. The task is to decide whether all of these k matrices commute. As shown by Itakura [54], this can be achieved on a quantum computer using $O(k^{4/5}{n}^{9/5})$ queries, whereas classically this requires $\mathrm{\Omega }(k{n}^2)$ queries.

Algorithm: Group Commutativity
Speedup: Polynomial
Description: We are given a list of k generators for a group G and access to a blackbox implementing group multiplication. By querying this blackbox we wish to determine whether the group is commutative. The best known classical algorithm is due to Pak and requires O(k) queries. Magniez and Nayak have shown that the quantum query complexity of this task is $\tilde{\mathrm{\Theta }}(k^{2/3})$ [139].

Algorithm: Hidden Nonlinear Structures
Speedup: Superpolynomial
Description: Any Abelian group G can be visualized as a lattice. A subgroup H of G is a sublattice, and the cosets of H are all the shifts of that sublattice. The Abelian hidden subgroup problem is normally solved by obtaining superposition over a random coset of the Hidden subgroup, and then taking the Fourier transform so as to sample from the dual lattice. Rather than generalizing to non-Abelian groups (see non-Abelian hidden subgroup), one can instead generalize to the problem of identifying hidden subsets other than lattices. As shown by Childs et al. [23] this problem is efficiently solvable on quantum computers for certain subsets defined by polynomials, such as spheres. Decker et al. showed how to efficiently solve some related problems in [31, 212].

Algorithm: Center of Radial Function
Speedup: Polynomial
Description: We are given an oracle that evaluates a function f from ${\mathbb{R}}^d$ to some arbitrary set S, where f is spherically symmetric. We wish to locate the center of symmetry, up to some precision. (For simplicity, let the precision be fixed.) In [110], Liu gives a quantum algorithm, based on a curvelet transform, that solves this problem using a constant number of quantum queries independent of d. This constitutes a polynomial speedup over the classical lower bound, which is $\mathrm{\Omega }(d)$ queries. The algorithm works when the function f fluctuates on sufficiently small scales, e.g., when the level sets of f are sufficiently thin spherical shells. The quantum algorithm is shown to work in an idealized continuous model, and nonrigorous arguments suggest that discretization effects should be small.

Algorithm: Group Order and Membership
Speedup: Superpolynomial
Description: Suppose a finite group G is given oracularly in the following way. To every element in G, one assigns a corresponding label. Given an ordered pair of labels of group elements, the oracle returns the label of their product. There are several classically hard problems regarding such groups. One is to find the group's order, given the labels of a set of generators. Another task is, given a bitstring, to decide whether it corresponds to a group element. The constructive version of this membership problem requires, in the yes case, a decomposition of the given element as a product of group generators. Classically, these problems cannot be solved using polylog(|G|) queries even if G is Abelian. For Abelian groups, quantum computers can solve these problems using polylog(|G|) queries by reduction to the Abelian hidden subgroup problem, as shown by Mosca [74]. Furthermore, as shown by Watrous [91], quantum computers can solve these problems using polylog(|G|) queries for any solvable group. For groups given as matrices over a finite field rather than oracularly, the order finding and constructive membership problems can be solved in polynomial time by using the quantum algorithms for discrete log and factoring [124]. See also group isomorphism.

Algorithm: Group Isomorphism
Speedup: Superpolynomial
Description: Let G be a finite group. To every element of G is assigned an arbitrary label (bit string). Given an ordered pair of labels of group elements, the group oracle returns the label of their product. Given access to the group oracles for two groups G and G', and a list of generators for each group, we must decide whether G and G' are isomorphic. For Abelian groups, we can solve this problem using poly(log |G|, log |G'|) queries to the oracle by applying the quantum algorithm of [127], which decomposes any Abelian group into a canonical direct product of cyclic groups. The quantum algorithm of [128] solves the group isomorphism problem using poly(log |G|, log |G'|) queries for a certain class of non-Abelian groups. Specifically, a group G is in this class if G has a normal Abelian subgroup A and an element y of order coprime to |A| such that G = A, y. Zatloukal has recently given an exponential quantum speedup for some instances of a problem closely related to group isomorphism, namely testing equivalence of group extensions [202].

Algorithm: Statistical Difference
Speedup: Polynomial
Description: Suppose we are given two black boxes A and B whose domain is the integers 1 through T and whose range is the integers 1 through N. By choosing uniformly at random among allowed inputs we obtain a probability distribution over the possible outputs. We wish to approximate to constant precision the L1 distance between the probability distributions determined by A and B. Classically the number of necessary queries scales essentially linearly with N. As shown in [117], a quantum computer can achieve this using $O(\sqrt{N})$ queries. Approximate uniformity and orthogonality of probability distributions can also be decided on a quantum computer using $O(N^{1/3})$ queries. The main tool is the quantum counting algorithm of [16]. A further improved quantum algorithm for this task is obtained in [265].

Algorithm: Finite Rings and Ideals
Speedup: Superpolynomial
Description: Suppose we are given black boxes implementing the addition and multiplication operations on a finite ring R, not necessarily commutative, along with a set of generators for R. With respect to addition, R forms a finite Abelian group (R,+). As shown in [119], on a quantum computer one can find in poly(log |R|) time a set of additive generators $\{h_1,\dots ,h_m\}\subset R$ such that $(R,+)\simeq ⟨h_1⟩\times \dots \times ⟨h_M⟩$ and m is polylogarithmic in |R|. This allows efficient computation of a multiplication tensor for R. As shown in [118], one can similarly find an additive generating set for any ideal in R. This allows one to find the intersection of two ideals, find their quotient, prove whether a given ring element belongs to a given ideal, prove whether a given element is a unit and if so find its inverse, find the additive and multiplicative identities, compute the order of an ideal, solve linear equations over rings, decide whether an ideal is maximal, find annihilators, and test the injectivity and surjectivity of ring homomorphisms. As shown in [120], one can also use a quantum computer to efficiently decide whether a given polynomial is identically zero on a given finite black box ring. Known classical algorithms for these problems scale as poly(|R|).

Algorithm: Counterfeit Coins
Speedup: Polynomial
Description: Suppose we are given N coins, k of which are counterfeit. The real coins are all of equal weight, and the counterfeit coins are all of some other equal weight. We have a pan balance and can compare the weight of any pair of subsets of the coins. Classically, we need $\mathrm{\Omega }(k\log (N/k))$ weighings to identify all of the counterfeit coins. We can introduce an oracle such that given a pair of subsets of the coins of equal cardinality, it outputs one bit indicating balanced or unbalanced. Building on previous work by Terhal and Smolin [137], Iwama et al. have shown [136] that on a quantum computer, one can identify all of the counterfeit coins using $O(k^{1/4})$ queries. The core techniques behind the quantum speedup are amplitude amplification and the Bernstein-Vazirani algorithm.

Algorithm: Matrix Rank
Speedup: Polynomial
Description: Suppose we are given oracle access to the (integer) entries of an $n\times m$ matrix A. We wish to determine the rank of the matrix. Classically this requires order nm queries. Building on [149], Belovs [150] gives a quantum algorithm that can use fewer queries given a promise that the rank of the matrix is at least r. Specifically, Belovs' algorithm uses $O(\sqrt{r(n-r+1)}LT)$ queries, where L is the root-mean-square of the reciprocals of the r largest singular values of A and T is a factor that depends on the sparsity of the matrix. For general A, $T=O(\sqrt{nm})$. If A has at most k nonzero entries in any row or column then $T=O(k\log (n+m))$. (To achieve the corresponding query complexity in the k-sparse case, the oracle must take a column index as input, and provide a list of the nonzero matrix elements from that column as output.) As an important special case, one can use these quantum algorithms for the problem of determining whether a square matrix is singular, which is sometimes referred to as the determinant problem. For general A the quantum query complexity of the determinant problem is no lower than the classical query complexity [151]. However, [151] does not rule out a quantum speedup given a promise on A, such as sparseness or lack of small singular values.

Algorithm: Matrix Multiplication over Semirings
Speedup: Polynomial
Description: A semiring is a set endowed with addition and multiplication operations that obey all the axioms of a ring except the existence additive inverses. Matrix multiplication over various semirings has many applications to graph problems. Matrix multiplication over semirings can be sped up by straightforward Grover improvements upon schoolbook multiplication, yielding a quantum algorithm that multiplies a pair of $n\times n$ matrices in $\tilde{O}(n^{5/2})$ time. For some semirings this algorithm outperforms the fastest known classical algorithms and for some semirings it does not [206]. A case of particular interest is the Boolean semiring, in which OR serves as addition and AND serves as multiplication. No quantum algorithm is known for Boolean semiring matrix multiplication in the general case that beats the best classical algorithm, which has complexity $n^{2.373}$. However, for sparse input our output, quantum speedups are known. Specifically, let A,B be n by n Boolean matrices. Let C=AB, and let l be the number of entries of C that are equal to 1 (i.e. TRUE). Improving upon [19, 155, 157], the best known upper bound on quantum query complexity is $\tilde{O}(n\sqrt{l})$, as shown in [161]. If instead the input matrices are sparse, a quantum speedup over the fastest known classical algorithm also has been found in a certain regime [206]. For detailed comparison to classical algorithms, see [155, 206]. Quantum algorithms have been found to perform matrix multiplication over the (max,min) semiring in $\tilde{O}(n^{2.473})$ time and over the distance dominance semiring in $\tilde{O}(n^{2.458})$ time [206]. The fastest known classical algorithm for both of these problems has $\tilde{O}(n^{2.687})$ complexity.

Algorithm: Subset finding
Speedup: Polynomial
Description: We are oracle access to a function $f:D\to R$ where D and R are finite sets. Some property $P\subset (D\times R{)}^k$ is specified, for example as an explicit list. Our task is to find a size-k subset of D satisfying P, i.e. $((x_1,f(x_1)),\dots ,(x_k,f(x_k)))\in P$, or reject if none exists. As usual, we wish to do this with the minimum number of queries to f. Generalizing the result of [7], it was shown in [162] that this can be achieved using $O(|D{|}^{k/(k+1)})$ quantum queries. As an noteworthy special case, this algorithm solves the k-subset-sum problem of finding k numbers from a list with some desired sum. A matching lower bound for the quantum query complexity is proven in [163].

Algorithm: Search with Wildcards
Speedup: Polynomial
Description: The search with wildcards problem is to identify a hidden n-bit string x by making queries to an oracle f. Given $S\subseteq \{1,2,\dots ,n\}$ and $y\in \{0,1{\}}^{|S|}$, f returns one if the substring of x specified by S is equal to y, and returns zero otherwise. Classically, this problem has query complexity $\mathrm{\Theta }(n)$. As shown in [167], the quantum query complexity of this problem is $\mathrm{\Theta }(\sqrt{n})$. Interestingly, this quadratic speedup is achieved not through amplitude amplification or quantum walks, but rather through use of the so-called Pretty Good Measurement. The paper [167] also gives a quantum speedup for the related problem of combinatorial group testing. This result and subsequent faster quantum algorithms for group testing are discussed in the entry on Junta Testing and Group Testing.

Algorithm: Network flows
Speedup: Polynomial
Description: A network is a directed graph whose edges are labeled with numbers indicating their carrying capacities, and two of whose vertices are designated as the source and the sink. A flow on a network is an assignment of flows to the edges such that no flow exceeds that edge's capacity, and for each vertex other than the source and sink, the total inflow is equal to the total outflow. The network flow problem is, given a network, to find the flow that maximizes the total flow going from source to sink. For a network with n vertices, m edges, and integer capacities of maximum magnitude U, [168] gives a quantum algorithm to find the maximal flow in time $O(min\{n^{7/6}\sqrt{m}{U}^{1/3},\sqrt{nU}m\}\times \log n)$. The network flow problem is closely related to the problem of finding a maximal matching of a graph, that is, a maximal-size subset of edges that connects each vertex to at most one other vertex. The paper [168] gives algorithms for finding maximal matchings that run in time $O(n\sqrt{m+n}\log n)$ if the graph is bipartite, and $O(n^2(\sqrt{m/n}+\log n)\log n)$ in the general case. The core of these algorithms is Grover search. The known upper bounds on classical complexity of the network flow and matching problems are complicated to state because different classical algorithms are preferable in different parameter regimes. However, in certain regimes, the above quantum algorithms beat all known classical algorithms. (See [168] for details.)

Algorithm: Electrical Resistance
Speedup: Exponential
Description: We are given oracle access to a weighted graph of n vertices and maximum degree d whose edge weights are to be interpreted as electrical resistances. Our task is to compute the resistance between a chosen pair of vertices. Wang gave two quantum algorithms in [210] for this task that run in time $\mathrm{poly}(\log n,d,1/\varphi ,1/ϵ)$, where $\varphi$ is the expansion of the graph, and the answer is to be given to within a factor of $1+ϵ$. Known classical algorithms for this problem are polynomial in n rather than $\log n$. One of Wang's algorithms is based on a novel use of quantum walks. The other is based on the quantum algorithm of [104] for solving linear systems of equations. The first quantum query complexity upper bounds for the electrical resistance problem in the adjacency query model are given in [280] using approximate span programs.

Algorithm: Junta Testing and Group Testing
Speedup: Polynomial
Description: A function $f:\{0,1{\}}^n\to \{0,1\}$ is a k-junta if it depends on at most k of its input bits. The k-junta testing problem is to decide whether a given function is a k-junta or is $ϵ$-far from any k-junta. Althoug it is not obvious, this problem is closely related to group testing. A group testing problem is defined by a function $f:\{1,2,\dots ,n\}\to \{0,1\}$. One is given oracle access to F, which takes as input subsets of $\{1,2,\dots ,n\}$. F(S) = 1 if there exists $x\in S$ such that f(x) = 1 and F(S) = 0 otherwise. In [266] a quantum algorithm is given solving the k-junta problem using $\tilde{O}(\sqrt{k/ϵ})$ queries and $\tilde{O}(n\sqrt{k/ϵ})$ time. This is a quadratic speedup over the classical complexity, and improves upon a previous quantum algorithm for k-junta testing given in [267]. A polynomial speedup for a gapped (i.e. approximation) version of group testing is also given in [266], improving upon the earlier results of [167,268].

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Approximation and Simulation Algorithms

Algorithm: Quantum Simulation
Speedup: Superpolynomial
Implementation: Classiq (Hamiltonian), Classiq (Thermal), PennyLane, Qrisp
Description: The exponential complexity of classically simulating quantum systems led Feynman to first propose that quantum computers might outperform classical computers on certain tasks [40]. It is now believed that for any physically realistic Hamiltonian H on n degrees of freedom, the corresponding time evolution operator $e^{-iHt}$ can be implemented using poly(n,t) gates. Unless BPP=BQP, this problem is not solvable in general on a classical computer in polynomial time. Many techniques for quantum simulation have been developed in the abstract for general classes of Hamiltonians. Specifically, [95,92,372,278] consider n-body first-quantized Hamiltonians consisting of a kinetic term $\sum _{i=1}^n{\mathrm{\nabla }}_i^2$ plus a potential term $V({\mathbf{x}}_1,\dots ,{\mathbf{x}}_n)$. The works [5,25,12,205,211,245,294,295] consider general sparse Hamiltonians. The works [170,244,382] consider Hamiltonians expressible as a linear combination of efficiently implementable unitary operators. Some works, such as [293,470,496], consider the problem of simulating Hamiltonians $H=\sum _j{H}_j$, where each $e^{i{H}_{j}t}$ is efficiently implementable, while remaining indifferent to the nature of the implementation. The works [468,469,467,466] consider the problems that arise in simulating time-dependent Hamiltonians. The works [478,479] show that geometrically local Hamiltonians on lattices of fixed dimension can be simulated more efficiently than general Hamiltonians. Specifically, they show that the quantum runtime in this case is essentially linear in the spacetime volume of the process being simulated. A related result on the efficiency implications of approximate locality in systems of interacting Bosons is given in [480]. If the state being simulated is low energy then the operator norm of the Hamiltonian yields only a loose upper bound on Trotter error; tighter upper bounds in this case are given in [481,482,483]. Other works consider specific physical settings, such as chemical dynamics [63,68,227,310,375], condensed matter physics [1,99, 145], relativistic quantum mechanics (the Dirac and Klein-Gordon equations) [367,369,370,371], open quantum systems [376, 377,378,379,458,499,501,503,504,505,506,507], quantum field theory [107,166,228,229,230,368,484], and conformal field theory [495]. Although the problem of finding ground energies of local Hamiltonians is QMA-complete and therefore probably requires exponential time on a quantum computer in the worst case, quantum algorithms have been developed to approximate ground states [102,231,232,233,234,235,308,321,322,373,380,381], low energy states [463], and thermal states [132,121,281,282,307,456,491,500,502] for some classes of Hamiltonians. Quantum algorithms have also been developed to prepare equilibrium states for some classes of master equations [430]. Efficient quantum algorithms have been also obtained for preparing certain classes of tensor network states [323,324,325,326,327,328]. Interestingly, simulating Hamiltonian time evolution, as well as some problems preparing ground and thermal states can all be done as special cases of the quantum singular value transformation [433].

Algorithm: Knot Invariants
Speedup: Superpolynomial
Description: As shown by Freedman [42, 41], et al., finding a certain additive approximation to the Jones polynomial of the plat closure of a braid at $e^{i2\pi /5}$ is a BQP-complete problem. This result was reformulated and extended to $e^{i2\pi /k}$ for arbitrary k by Aharonov et al. [4,2]. Wocjan and Yard further generalized this, obtaining a quantum algorithm to estimate the HOMFLY polynomial [93], of which the Jones polynomial is a special case. Recent work has yielded quantum algorithms for estimating other topological invariants including Betti numbers [510] and Khovanov Homology [511]. (See also topological data analysis under machine learning.) Aharonov et al. showed that, for planar graphs, quantum computers can in polynomial time estimate a certain additive approximation to the Tutte polynomial [3], which is an object that includes Jones polynomials as a special case. It is not fully understood for what range of parameters the approximation obtained in [3] is BQP-hard. (See also partition functions.) Polynomial-time quantum algorithms have also been discovered for additively approximating link invariants arising from quantum doubles of finite groups [174]. (This problem is not known to be BQP-hard.) As shown in [83], the problem of finding a certain additive approximation to the Jones polynomial of the trace closure of a braid at $e^{i2\pi /5}$ is DQC1-complete.

Algorithm: Three-manifold Invariants
Speedup: Superpolynomial
Description: The Turaev-Viro invariant is a function that takes three-dimensional manifolds as input and produces a real number as output. Homeomorphic manifolds yield the same number. Given a three-manifold specified by a Heegaard splitting, a quantum computer can efficiently find a certain additive approximation to its Turaev-Viro invariant, and this approximation is BQP-complete [129]. Earlier, in [114], a polynomial-time quantum algorithm was given to additively approximate the Witten-Reshitikhin-Turaev (WRT) invariant of a manifold given by a surgery presentation. Squaring the WRT invariant yields the Turaev-Viro invariant. However, it is unknown whether the approximation achieved in [114] is BQP-complete. A suggestion of a possible link between quantum computation and three-manifold invariants was also given in [115].

Algorithm: Partition Functions
Speedup: Superpolynomial
Description: For a classical system with a finite set of states S the partition function is $Z=\sum _{s\in S}{e}^{-E(s)/kT}$, where T is the temperature and k is Boltzmann's constant. Essentially every thermodynamic quantity can be calculated by taking an appropriate partial derivative of the partition function. The partition function of the Potts model is a special case of the Tutte polynomial. A quantum algorithm for approximating the Tutte polynomial is given in [3]. Some connections between these approaches are discussed in [67]. Additional algorithms for estimating partition functions on quantum computers are given in [112,113,45,47]. A BQP-completeness result (where the "energies" are allowed to be complex) is also given in [113]. A method for approximating partition functions by simulating thermalization processes is given in [121]. Polynomial speedups for the approximation of certain general classes partition functions are given in [122,471]. A method based on quantum walks, achieving polynomial speedup for evaluating partition functions is given in [265].

Algorithm: Zeta Functions
Speedup: Superpolynomial
Description: Let f(x,y) be a degree-d polynomial over a finite field ${\mathbb{F}}_p$. Let $N_r$ be the number of projective solutions to f(x,y = 0 over the extension field ${\mathbb{F}}_{p^r}$. The zeta function for f is defined to be $Z_f(T)=\exp \left(\sum _{r=1}^{\mathrm{\infty }}\frac{N_r}{r}{T}^r\right)$. Remarkably, $Z_f(T)$ always has the form $Z_f(T)=\frac{Q_f(T)}{(1-pT)(1-T)}$ where $Q_f(T)$ is a polynomial of degree 2g and $g=\frac{1}{2}(d-1)(d-2)$ is called the genus of f. Given $Z_f(T)$ one can easily compute the number of zeros of f over any extension field ${\mathbb{F}}_{p^r}$. One can similarly define the zeta function when the original field over which f is defined does not have prime order. As shown by Kedlaya [64], quantum computers can determine the zeta function of a genus g curve over a finite field ${\mathbb{F}}_{p^r}$ in $\mathrm{poly}(\log p,r,g)$ time. The fastest known classical algorithms are all exponential in either log(p) or g. In a diffent, but somewhat related contex, van Dam has conjectured that due to a connection between the zeros of Riemann zeta functions and the eigenvalues of certain quantum operators, quantum computers might be able to efficiently approximate the number of solutions to equations over finite fields [87].

Algorithm: Weight Enumerators
Speedup: Superpolynomial
Description: Let C be a code on n bits, i.e. a subset of ${\mathbb{Z}}_2^n$. The weight enumerator of C is $S_C(x,y)=\sum _{c\in C}{x}^{|c|}{y}^{n-|c|}$ where |c| denotes the Hamming weight of c. Weight enumerators have many uses in the study of classical codes. If C is a linear code, it can be defined by $C=\{c:Ac=0\}$ where A is a matrix over ${\mathbb{Z}}_2$ In this case $S_C(x,y)=\sum _{c:Ac=0}{x}^{|c|}{y}^{n-|c|}$. Quadratically signed weight enumerators (QWGTs) are a generalization of this: $S(A,B,x,y)=\sum _{c:Ac=0}(-1{)}^{c^{T}Bc}{x}^{|c|}{y}^{n-|c|}$. Now consider the following special case. Let A be an $n\times n$ matrix over ${\mathbb{Z}}_2$ such that diag(A)=I. Let lwtr(A) be the lower triangular matrix resulting from setting all entries above the diagonal in A to zero. Let l,k be positive integers. Given the promise that $|S(A,\mathrm{lwtr}(A),k,l)|\ge \frac{1}{2}(k^2+l^2{)}^{n/2}$ the problem of determining the sign of $S(A,\mathrm{lwtr}(A),k,l)$ is BQP-complete, as shown by Knill and Laflamme in [65]. The evaluation of QWGTs is also closely related to the evaluation of Ising and Potts model partition functions [67,45,46].

Algorithm: Simulated Annealing
Speedup: Polynomial
Description: In simulated annealing, one has a series of Markov chains defined by stochastic matrices $M_1,M_2,\dots ,M_n$. These are slowly varying in the sense that their limiting distributions $p{i}_1,{\pi }_2,\dots ,{\pi }_n$ satisfy $|{\pi }_{t+1}-{\pi }_t|<ϵ$ for some small $ϵ$. These distributions can often be thought of as thermal distributions at successively lower temperatures. If ${\pi }_1$ can be easily prepared, then by applying this series of Markov chains one can sample from ${\pi }_n$. Typically, one wishes for ${\pi }_n$ to be a distribution over good solutions to some optimization problem. Let ${\delta }_i$ be the gap between the largest and second largest eigenvalues of $M_i$. Let $\delta =\underset{i}{min}{\delta }_i$. The run time of this classical algorithm is proportional to $1/\delta$. Building upon results of Szegedy [135,85], Somma et al. have shown [84, 177] that quantum computers can sample from ${\pi }_n$ with a runtime proportional to $1/\sqrt{\delta }$. Additional methods by which classical Markov chain Monte Carlo algorithms can be sped up using quantum walks are given in [265].

Algorithm: String Rewriting
Speedup: Superpolynomial
Description: String rewriting is a fairly general model of computation. String rewriting systems (sometimes called grammars) are specified by a list of rules by which certain substrings are allowed to be replaced by certain other substrings. For example, context free grammars, are equivalent to the pushdown automata. In [59], Janzing and Wocjan showed that a certain string rewriting problem is PromiseBQP-complete. Thus quantum computers can solve it in polynomial time, but classical computers probably cannot. Given three strings s,t,t', and a set of string rewriting rules satisfying certain promises, the problem is to find a certain approximation to the difference between the number of ways of obtaining t from s and the number of ways of obtaining t' from s. Similarly, certain problems of approximating the difference in number of paths between pairs of vertices in a graph, and difference in transition probabilities between pairs of states in a random walk are also BQP-complete [58].

Algorithm: Matrix Powers
Speedup: Superpolynomial
Description: Quantum computers have an exponential advantage in approximating matrix elements of powers of exponentially large sparse matrices. Suppose we are have an $N\times N$ symmetric matrix A such that there are at most polylog(N) nonzero entries in each row, and given a row index, the set of nonzero entries can be efficiently computed. The task is, for any 1 < i < N, and any m polylogarithmic in N, to approximate $(A^m{)}_{ii}$ the $i^{\mathrm{th}}$ diagonal matrix element of $A^m$. The approximation is additive to within $b^mϵ$ where b is a given upper bound on |A| and $ϵ$ is of order 1/polylog(N). As shown by Janzing and Wocjan, this problem is PromiseBQP-complete, as is the corresponding problem for off-diagonal matrix elements [60]. Thus, quantum computers can solve it in polynomial time, but classical computers probably cannot.

Algorithm: Probabilistic Sampling
Speedup: Superpolynomial
Description: Although most computational problems are formulated either as decision problems or search problems, one can also consider the complexity of sampling from a given distribution of bit strings. In [474,473], it was shown that quantum computers can efficiently sample from certain distributions that cannot be exactly sampled from by any efficient classical randomized algorithm unless the Polynomial Hierarchy collapses to the third level. Sampling problems of this type have subsequently been used in experiments demonstrating the ability of present-day quantum computers to perform tasks that are beyond the reach of classical supercomputers. This is sometimes referred to as "quantum supremacy". Some quantum algorithms achieving polynomial speedup for sampling problems with known practical applications have also been devised [475].

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Optimization, Numerics, and Machine Learning

Algorithm:Polynomial Quantum Speedups for Constraint Satisfaction Problems
Speedup: Polynomial
Implementation: Classiq, PennyLane, Qrisp (Quantum Backtracking)
Description: Constraint satisfaction problems, many of which are NP-hard, are ubiquitous in computer science, a canonical example being 3-SAT. If one wishes to satisfy as many constraints as possible rather than all of them, these become combinatorial optimization problems. (See also adiabatic algorithms.) The brute force solution to constraint satisfaction problems can be quadratically sped up using Grover's algorithm. However, most constraint satisfaction problems are solvable by classical algorithms that (although still exponential-time) run more than quadratically faster than brute force checking of all possible solutions. Nevertheless, a polynomial quantum speedup over the fastest known classical algorithm for 3-SAT is given in [133], and polynomial quantum speedups for some other constraint satisfaction problems are given in [134,298,493,492]. In [423] a quadratic quantum speedup for approximate solutions to homogeneous QUBO/Ising problems is obtained by building upon the quantum algorithm for semidefinite programming. A commonly used classical algorithm for constraint satisfaction is backtracking, and for some problems this algorithm is the fastest known. A general quantum speedup for backtracking algorithms is given in [264] and further improved in [422].

Algorithm: Adiabatic Algorithms
Speedup: Unknown
Implementation: Classiq (Linear Solver)
Description: In adiabatic quantum computation one starts with an initial Hamiltonian whose ground state is easy to prepare, and slowly varies the Hamiltonian to one whose ground state encodes the solution to some computational problem. By the adiabatic theorem, the system will track the instantaneous ground state provided the variation of the Hamiltonian is sufficiently slow. The runtime of an adiabatic algorithm scales at worst as $1/{\gamma }^3$ where $\gamma$ is the minimum eigenvalue gap between the ground state and the first excited state [185]. If the Hamiltonian is varied sufficiently smoothly, one can improve this to $\tilde{O}(1/{\gamma }^2)$ [247]. Optimization by adiabatic quantum computation traces back to the pioneering works of [96, 186,199, 198, 509]. Adiabatic quantum algorithms for optimization problems typically use "stoquastic" Hamiltonians, which do not suffer from the sign problem. Such algorithms are sometimes referred to as quantum annealing. Adiabatic quantum computation with non-stoquastic Hamiltonians is as powerful as the quantum circuit model [97]. Adiabatic algorithms using stoquastic Hamiltonians are probably less powerful [183], but are likely more powerful than classical computation [429]. The asymptotic runtime of adiabatic optimization algorithms is notoriously difficult to analyze, but some progress has been achieved [179,180,181,182,187,188,189,190,191,226,508]. Adiabatic quantum computers can perform a process somewhat analogous to Grover search in $O(\sqrt{N})$ time [98]. Adiabatic quantum algorithms achieving quadratic speedup for a more general class of problems are constructed in [184] by adapting techniques from [85]. Adiabatic quantum algorithms have been proposed for several specific problems, including PageRank [176], machine learning [192, 195], finding Hadamard matrices [406], solving linear systems [517, 518] and graph problems [193, 194]. Some quantum simulation algorithms also use adiabatic state preparation.

Algorithm: Quantum Approximate Optimization
Speedup: Superpolynomial
Implementation: Classiq, Cirq, PennyLane, Qrisp
Description: For many combinatorial optimization problems, finding the exact optimal solution is NP-complete. There are also hardness-of-approximation results proving that finding an approximation with sufficiently small error bound is NP-complete. For certain problems there is a gap between the best error bound achieved by a polynomial-time classical approximation algorithm and the error bound proven to be NP-hard. In this regime there is potential for exponential speedup by quantum computation. In [242] a new quantum algorithmic technique called the Quantum Approximate Optimization Algorithm (QAOA) was proposed for finding approximate solutions to combinatorial optimization problems. In [243] it was subsequently shown that QAOA solves a combinatorial optimization problem called Max E3LIN2 with a better approximation ratio than any polynomial-time classical algorithm known at the time. However, an efficient classical algorithm achieving an even better approximation ratio (in fact, the approximation ratio saturating the limit set by hardness-of-approximation) was subsequently discovered [260]. Presently, the power of QAOA relative to classical computing is an active area of research [300,301,302, 314,451,452,476].

Algorithm: Gradient Estimation and Learning Polynomials
Speedup: Polynomial
Description: Suppose we are given a oracle for computing some smooth function $f:{\mathbb{R}}^d\to \mathbb{R}$ to precision $\pm ϵ$. The task is to estimate $\mathrm{\nabla }f$ at some specified point ${\mathbf{x}}_0\in {\mathbb{R}}^d$. As shown in [61], a quantum computer can achieve this using one query, whereas a classical computer needs at least d+1 queries. In [436] the dependence on $ϵ$ was rigorously analyzed and quadratically improved relative to [61]. Analyses of the dependence on the smoothness of f are given in [437,438]. Extension to complex-valued f and an alternative method for improving $ϵ$-dependence are given in [439]. In [20], Bulger suggested potential applications for optimization problems. As shown in appendix D of [62], a quantum computer can use the gradient algorithm to find the minimum of a quadratic form in d dimensions using O(d) queries, whereas, as shown in [94], a classical computer needs at least $\mathrm{\Omega }(d^2)$ queries. The quantum algorithm of [62] can also extract all $d^2$ matrix elements of the quadratic form using O(d) queries, and more generally, all $d^n$ nth derivatives of a smooth function of d variables in $O(d^{n-1})$ queries. See also: convex optimization.

Algorithm: Semidefinite Programming
Speedup: Polynomial (with some exceptions)
Description: Given a list of m + 1 Hermitian $n\times n$ matrices $C,A_1,A_2,\dots ,A_m$ and m numbers $b_1,\dots ,b_m$, the problem of semidefinite programming is to find the positive semidefinite $n\times n$ matrix X that maximizes tr(CX) subject to the constraints $\mathrm{tr}(A_{j}X)\le b_j$ for $j=1,2,\dots ,m$. Semidefinite programming has many applications in operations research, combinatorial optimization, and quantum information, and it includes linear programming as a special case. Introduced in [313], and subsequently improved in [383, 425], quantum algorithms are now known that can approximately solve semidefinite programs to within $\pm ϵ$ in time $O(\sqrt{m}\log m\cdot \mathrm{poly}(\log n,r,{ϵ}^{-1}))$, where r is the rank of the semidefinite program. This constitutes a quadratic speedup over the fastest classical algorithms when r is small compared to n. The quantum algorithm is based on amplitude amplification and quantum Gibbs sampling [121, 307]. In a model in which input is provided in the form of quantum states the quantum algorithm for semidefinite programming can achieve superpolynomial speedup, as discussed in [383], although recent dequantization results [421] delineate limitations on the context in which superpolynomial quantum speedup for semidefinite programs is possible.

Algorithm: Convex Optimization
Speedup: Polynomial
Description: Remarkably general results in [418,419,420] give quantum speedups for convex optimization and volume estimation of convex bodies, as well as query complexity lower bounds. Roughly speaking these results show that for convex optimization and volume estimation in d dimensions one gets a quadratic speedup in d just as was found earlier for the special case of minimizing quadratic forms. As shown in [130,146], quadratic forms and multilinear polynomials in d variables over a finite field may be extracted with a factor of d fewer quantum queries than are required classically. In [461] a quantum algorithm is given that achieves a polynomial speedup for linear programming. There are also quantum speedups for combinatorial optimization problems that, relative to the natural topology of the search space, lack local optima other than the global optimum. In particular, single query quantum algorithms for finding the minima of basins based on Hamming distance were given in [147,148,223]. Some no-go results on quantum speedup for general convex optimization in the presence of an oracle for gradients are given in [477]. A quadratic speedup for convex optimization problems arising in the context of linear regression is given in [497]. See also: Gradient Estimation and Semidefinite Programming.

Algorithm: Optimization by Decoded Quantum Interferometry
Speedup: Superpolynomial
Implementation: Classiq
Description: In [453] a method called Decoded Quantum Interferometry (DQI) is introduced which reduces optimization problems to decoding problems using quantum Fourier transforms. This produces approximate optima, where the number of constraints satisfied is determined by the number of errors that can be decoded. If each constraint depends on a limited number of variables then the resulting decoding problem is for a classical LDPC code. These can be decoded from large numbers of errors using efficient classical algorithms such as belief propagation. Additional structure in the constraints also translates over to the decoding problem and can in some cases be exploited. In particular, certain problems of finding a degree n polynomial over a finite field ${\mathbb{F}}_p$ that approximates as well as possible a given data set are reduced via DQI to decoding of Reed-Solomon codes. Becuase efficient classical algorithms are known that decode Reed-Solomon codes with many errors (half their distance), DQI achieves a very good approximation to the original polynomial regression problems, which is not known to be achievable by any classical polynomial time algorithm, thus achieving apparent exponential speedup. The results of [454] giving a quartic speedup for a problem of planted inference and the results of [455] giving an exponential separation between quantum and classical query complexity for a random oracle, although not originally phrased as quantum optimization algorithms, bear close relationships to DQI.

Algorithm: Linear Systems
Speedup: Superpolynomial
Implementation: Classiq (HHL), Classiq (QSVT), Cirq (HHL), Qrisp/Pennylane (HHL)
Description: We are given oracle access to an $n\times n$ matrix A and some description of a vector b. We wish to find some property of f(A)b for some efficiently computable function f. Suppose A is a Hermitian matrix with O(polylog n) nonzero entries in each row and condition number k. As shown in [104], a quantum computer can in $O(k^2\log n)$ time compute to polynomial precision various expectation values of operators with respect to the vector f(A)b (provided that a quantum state proportional to b is efficiently constructable). For certain functions, such as f(x)=1/x, this procedure can be extended to non-Hermitian and even non-square A. The runtime of this algorithm was subsequently improved to $O(k{\log }^{3}k\log n)$ in [138]. Exponentially improved scaling of runtime with precision was obtained in [263]. Further improvement was achieved in [494]. Some methods to extend this algorithm to apply to non-sparse matrices have been proposed [309,402], although these require certain partial sums of the matrix elements to be pre-computed. Extensions of this quantum algorithm have been applied to problems of estimating electromagnetic scattering crossections [249] (see also [369] for a different approach), solving linear differential equations [156, 296], estimating electrical resistance of networks [210], least-squares curve-fitting [169], solving Toeplitz systems [297], and machine learning [214,222,250,251,309]. However, the linear-systems-based quantum algorithms for recommendation systems [309] and principal component analysis [250] were subsequently "dequantized" by Tang [400, 401]. That is, Tang obtained polynomial time classical randomized algorithms for these problems, thus proving that the proposed quantum algorithms for these tasks do not achieve exponential speedup. Some limitations of the quantum machine learning algorithms based on linear systems are nicely summarized in [246]. In [220] it was shown that quantum computers can invert well-conditioned $n\times n$ matrices using only $O(\log n)$ qubits, whereas the best classical algorithm uses order ${\log }^{2}n$ bits. Subsequent improvements to this quantum algorithm are given in [279]. Variants of the linear systems problem, including the computation of Moore-Penrose pseudoinverses, can be obtained as special cases of the quantum singular value transformation [433].

Algorithm: Machine Learning
Speedup: Varies
Implementation: Classiq (QSVM), Classiq (Autoencoder), PennyLane
Description: Maching learning encompasses a wide variety of computational problems and can be attacked by a wide variety of algorithmic techniques. This entry summarizes quantum algorithmic techniques for improved machine learning. Many of the quantum algorithms here are cross-listed under other headings. In [214,250,251,309,338,339,359,403], quantum algorithms for solving linear systems [104] are applied to speed up cluster-finding, principal component analysis, binary classification, training of neural networks, and various forms of regression, provided the data satisfies certain conditions. (See also [433] for subsequent improvements to quantum principal component analysis.) However, a number of quantum machine learning algorithms based on linear systems have subsequently been "dequantized". Specifically, Tang showed in [400, 401] that the problems of recommendation systems and principal component analysis solved by the quantum algorithms of [251,309] can in fact also be solved in polynomial time randomized classical algorithms. The works [222,487,488,489,490] give quantum algorithms for topological data analysis by persistent homology. A cluster-finding method not based on the linear systems algorithm of [104] is given in [336]. The papers [192,195,344,345,346,348] explore the use of adiabatic optimization techniques for the training of classifiers. In [456] quantum Ising machines are used for the training of classical neural networks. In [221], a method is proposed for training Boltzmann machines by manipulating coherent quantum states with amplitudes proportional to the Boltzmann weights. Polynomial speedups can be obtained by applying Grover search and related techniques such as amplitude amplification to amenable subroutines of state of the art classical machine learning algorithms. See, for example [358,340,341,342,343]. Other quantum machine learning algorithms not falling into one of the above categories include [337,349]. Some limitations of quantum machine learning algorithms are nicely summarized in [246]. Many other quantum query algorithms that extract hidden structure of the black-box function could be cast as machine learning algorithms. See for example [146,23,11,31,212]. Query algorithms for learning the majority and "battleship" functions are given in [224]. Large quantum advantages for learning from noisy oracles are given in [236,237]. In [428] quantum kernel estimation is used to implement a support-vector classifier solving a learning problem that is provably as hard as discrete logarithm. Several recent review articles [299,332,333] and a book [331] are available which summarize the state of the field. There is a related body of work, not strictly within the standard setting of quantum algorithms, regarding quantum learning in the case that the data itself is quantum coherent. See e.g. [350,334,335,351,352,353,354,355,356,357].

Algorithm: Tensor Principal Component Analysis
Speedup: Polynomial (quartic)
Description: In [424] a quantum algorithm is given for an idealized problem motivated by machine learning applications on high-dimensional data sets. Consider $T=\lambda v_{\mathrm{sig}}^{\otimes p}+G$ where G is a p-index tensor of Gaussian random variables, symmetrized over all permutations of indices, and $v_{\mathrm{sig}}$ is an N-dimensional vector of magnitude $\sqrt{N}$. The task is to recover $v_{\mathrm{sig}}$. Consider $\lambda =\alpha N^{-p/4}$. The best classical algorithms succeed when $\alpha \gg 1$ and have time and space complexity that scale exponentially in ${\alpha }^{-1}$. The quantum algorithm of [424] solves this problem in polynomial space and with runtime scaling quartically better in ${\alpha }^{-1}$ than the classical spectral algorithm. The quantum algorithm works by encoding the problem into the eigenspectrum of a many-body Hamiltonian and applying phase estimation together with amplitude amplification.

Algorithm: Solving Linear Differential Equations
Speedup: Superpolynomial
Implementation: Classiq
Description: Consider linear first order differential equation $\frac{d}{dt}\mathbf{x}=A(t)\mathbf{x}+\mathbf{b}(t)$, where $\mathbf{x}$ and $\mathbf{b}$ are N-dimensional vectors and A is an $N\times N$ matrix. Given an initial condition $\mathbf{x}(0)$ one wishes to compute the solution $\mathbf{x}(t)$ at some later time t to some precision $ϵ$ in the sense that the normalized vector $x(t)/\Vert x(t)\Vert$ produced has distance at most $ϵ$ from the exact solution. In [156], Berry gives a quantum algorithm for this problem that runs in time $O(t^2\mathrm{poly}(1/ϵ)\mathrm{poly}\log N)$, whereas the fastest classical algorithms run in time $O(t\mathrm{poly}N)$. The final result is produced in the form of a quantum superposition state on $O(\log N)$ qubits whose amplitudes contain the components of $\mathbf{x}(t)$. The algorithm works by reducing the problem to linear algebra via a high-order finite difference method and applying the quantum linear algebra primitive of [104]. In [410] an improved quantum algorithm for this problem was given which brings the epsilon dependence down to $\mathrm{poly}\log (1/ϵ)$. The quantum eigenvalue processing technique of [459] also yields efficiency improvements for solving linear differential equations. In [440] it is shown that quantum simulation of the linear ODEs governing coupled classical harmonic oscillators can be solved exponentially faster on quantum computers than classical computers when the connectivity is an expander graph. Related quantum algorithms have been developed to efficiently simulate certain dynamical systems with exponentially many fermionic degrees of freedom [513], bosonic degrees of freedom [514], and more general degrees of freedom [512]. Partial differential equations can be reduced to ordinary differential equations through discretization, and higher order differential equations can be reduced to first order through additiona of auxiliary variables. Consequently, these more general problems can be solved through the methods of [156, 104]. However, quantum algorithms designed to solve these problems directly may be more efficient (and for specific problems one may analyze the complexity of tasks that are unspecified in a more general formulation such as preparation of relevant initial states). In [442], the scaling with precision is improved for simulating second order elliptic linear partial differential equations. In [249] a quantum algorithm is given which solves the wave equation by applying finite-element methods to reduce it to linear algebra and then applying the quantum linear algebra algorithm of [104] with preconditioning. In [369] a quantum algorithm is given for solving the wave equation by discretizing it with finite differences and massaging it into the form of a Schrodinger equation which is then simulated using the method of [245]. The problem solved by [369] is not equivalent to that solved by [249] because in [249] the problem is reduced to a time-indepent one through assuming sinusoidal time dependence and applying separation of variables, whereas [369] solves the time-dependent problem. The quantum speedup achieved over classical methods for solving the wave equation in d-dimensions is polynomial for fixed d but expontial in d. Concrete resource estimates for quantum algorithms to solve differential equations are given in [412, 413, 414]. A quantum algorithm for solving linear partial differential equations using continuous-variable quantum computing is given in [415]. In [296] quantum finite element methods are analyzed in a general setting. A quantum spectral method for solving differential equations is given in [416]. Analysis of quantum algorithms applied to the heat equation in d dimensions is given in [446], finding that the speedup is at most quadratic.

Algorithm: Solving Nonlinear Differential Equations
Speedup: Superpolynomial
Description: A quantum algorithm for solving nonlinear differential equations, in the sense of obtaining a solution encoded in the amplitudes, is described in [411], which has exponential scaling in t. In [426] a method is introduced for solving nonlinear differential equations using Carleman linearization, whose runtime scales as $O(t^2)$ when the differential equation satisfies the conditions necessary for the quantum algorithm to apply. Further analysis of and improvement to Carleman-based quantum algorithms are given in [434,441]. In [427], by drawing upon intuition from the nonlinear Schrodinger equation, another quantum algorithm is given for solving nonlinear differential equations that also scales as $O(t^2)$. Note that, quantum algorithms for solving nonlinear equations generally encode the solution into the amplitudes of a quantum state, and thus unitarity implies that the algorithms will not be efficient if these solutions have norm that grows or shrinks too rapidly. A quantum algorithm for solving a nonlinear PDE called the Vlasov equation, which arises in plasma physics, is given in [417]. Several works give quantum algorithms to replace Monte Carlo simulations of nonlinear differential equations, reducing the runtime-dependence on precision from $O({ϵ}^{-2})$ to $O({ϵ}^{-1})$. Specifically, [443, 447, 448, 450] do so in the context of finance (including stochastic differential equations), and [444, 445] do so in the context of fluid dynamics. A quantum heuristic for solving the Black-Scholes equation by relating it to imaginary-time Schrodinger evolution is proposed in [449].

Algorithm: Quantum Dynamic Programming
Speedup: Polynomial
Description: In [409] the authors introduce a problem called path-in-the-hypercube. In this problem, one given a subgraph of the hypercube and asked whether there is a path along this subgraph that starts from the all zeros vertex, ends at the all ones vertex, and makes only Hamming weight increasing moves. (The vertices of the hypercube graph correspond to bit strings of length n and the hypercube graph joins vertices of Hamming distance one.) Many NP-complete problems for which the best classical algorithm is dynamic programming can be modeled as instances of path-in-the-hypercube. By combining Grover search with dynamic programming methods, a quantum algorithm can solve path-in-the-hypercube in time $O^{\ast }({1.817}^n)$, where the notation $O^{\ast }$ indicates that polynomial factors are being omitted. The fastest known classical algorithm for this problem runs in time $O^{\ast }(2^n)$. Using this primitive quantum algorithms can be constructed that solve vertex ordering problems in $O^{\ast }({1.817}^n)$ vs. $O^{\ast }(2^n)$ classically, graph bandwidth in $O^{\ast }({2.946}^n)$ vs. $O^{\ast }({4.383}^n)$ classically, travelling salesman and feedback arc set in $O^{\ast }({1.729}^n)$ vs. $O^{\ast }(2^n)$ classically, and minimum set cover in $O(\mathrm{poly}(m,n){1.728}^n)$ vs. $O(nm{2}^n)$ classically.

Algorithm: Computing the Principal Eigenvector
Speedup: Polynomial
Description: Given query access to the matrix elements of a $d\times d$ Hermitian matrix A, our problem is to obtain the principal eigenvector of A, that is, the eigenvector corresponding to the largest eigenvalue. If we wished to obtain this eigenvector as a quantum state then this would be equivalent to the ground state problem. However, we instead want a classical description of the eigenvector. Quantum algorithms solving this and some related problems are given in [462]. The quantum of [462] runs in time $\tilde{O}(d^{3/2})$ whereas the best classical algorithm runs in time $\tilde{O}(d^2)$.

Algorithm: Approximating Nash Equilibria
Speedup: Polynomial
Description: Given oracle access to an $m\times n$ payoff matrix of a zero-sum game, we wish to compute an $ϵ$-approximate Nash equilibrium. Classically, the best algorithm for this has runtime $\tilde{O}((m+n){ϵ}^{-2})$. The quantum algorithm of [485], improving on the prior result of [486] achieves this in runtime $\tilde{O}(\sqrt{m+n}{ϵ}^{-2.5}+{ϵ}^{-3})$ by making a connection between Nash equilibria and Gibbs sampling.

Algorithm: Lattice Problems by Filtering
Speedup: Exponential
Description: We are given a set of basis vectors in ${\mathbb{Z}}^n$ and we consider the lattice $\mathcal{L}$ defined by all integer linear combinations of these basis vectors. In general, finding the shortest nonzero vector in $\mathcal{L}$ is an NP-hard problem. Given a target point $\mathbf{x}\in {\mathbb{Z}}^n$, finding the nearest element of $\mathcal{L}$ is also NP-hard. There are various problems of finding approximately optimal solutions, or finding solutions given certain promises on the $\mathcal{L}$ that are not known to be NP-hard yet lack known polynomial time classical solutions. Such problems, versions of which underlie many post-quantum cryptosystems, provide very important targets for quantum algorithms research. In [498], Chen, Liu, and Zhandry obtained polynomial time quantum algorithms for some lattice problems that do not have known polynomial-time classical solutions. These quantum algorithms apply in a parameter regime that does not break the any of the main candidate post-quantum cryptosystems. A key ingredient in these algorithms is the observation from [78, 5]. that the quantum Fourier transform allows efficient reductions in both directinos between shortest vector problems on a lattice and nearest vector problems on the dual lattice. In [498], the authors add also a new ingredient, which is a filtering technique that uses quantum measurements in nontrivial bases to improve the parameters of these reductions in such a way as to achieve quantum advantage.

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Acknowledgments

I thank the following people for contributing their expertise (in chronological order).

• Daniel Lidar
• Wim van Dam
• Geordie Rose
• Yi-Kai Liu
• Robin Kothari
• Martin Schwarz
• Dorit Aharonov
• Alessandro Cosentino
• Andrew Childs
• Stacey Jeffery
• Lov Grover
• Eduin H. Serna
• Charles Greathouse
• Juani Bermejo-Vega
• Luis Kowada
• Keith Britt
• Aram Harrow
• Zafer Gedik
• David Cornwell
• Cedric Lin
• Shelby Kimmel
• Jeremy Singer
• Dan Boneh
• Rich Schroeppel
• Yuan Su
• Tim Stevens
• Martin Ekerå
• Igor Shparlinski
• Timothy Chase
• Alejandro Pozas-Kerstjens
• Nikhil Vyas
• Kevin Lui
• Vladimir Korepin
• Andriyan Suksmono
• Jack Hidary
• Donny Greenberg
• Nicola Vitucci
• Kunal Marwaha
• José Ignacio Espinoza Camacho
• Vincenzo Savona
• Barry Sanders
• Jeremy Wright
• Sarah Kaiser
• Benjamin Tokgöz
• Armando Bellante
• Seongbin Park
• Xujie Song
• Alexis Fimeyer
• Xiang Li
• Diego Garcia Martin
• Martin Larocca

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References

1

Daniel S. Abrams and Seth Lloyd
Simulation of many-body Fermi systems on a universal quantum computer.
Physical Review Letters, 79(13):2586-2589, 1997.
[arXiv:quant-ph/9703054]

2

Dorit Aharonov and Itai Arad
The BQP-hardness of approximating the Jones polynomial.
New Journal of Physics 13:035019, 2011.
[arXiv:quant-ph/0605181]

3

Dorit Aharonov, Itai Arad, Elad Eban, and Zeph Landau
Polynomial quantum algorithms for additive approximations of the Potts model and other points of the Tutte plane.
arXiv:quant-ph/0702008, 2007.

4

Dorit Aharonov, Vaughan Jones, and Zeph Landau
A polynomial quantum algorithm for approximating the Jones polynomial.
In Proceedings of the 38th ACM Symposium on Theory of Computing, 2006.
[arXiv:quant-ph/0511096]

5

Dorit Aharonov and Amnon Ta-Shma
Adiabatic quantum state generation and statistical zero knowledge.
In Proceedings of the 35th ACM Symposium on Theory of Computing, 2003.
[arXiv:quant-ph/0301023]

6

A. Ambainis, H. Buhrman, P. Høyer, M. Karpinizki, and P. Kurur
Quantum matrix verification.
Unpublished Manuscript, 2002.

7

Andris Ambainis
Quantum walk algorithm for element distinctness.
SIAM Journal on Computing, 37:210-239, 2007.
[arXiv:quant-ph/0311001]

8

Andris Ambainis, Andrew M. Childs, Ben W.Reichardt, Robert Špalek, and Shengyu Zheng
Every AND-OR formula of size N can be evaluated in time $n^{1/2+o(1)}$ on a quantum computer.
In Proceedings of the 48th IEEE Symposium on the Foundations of Computer Science, pages 363-372, 2007.
[ arXiv:quant-ph/0703015 and arXiv:0704.3628]

9

Dave Bacon, Andrew M. Childs, and Wim van Dam
From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups.
In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science, pages 469-478, 2005.
[arXiv:quant-ph/0504083]

10

Michael Ben-Or and Avinatan Hassidim
Quantum search in an ordered list via adaptive learning.
arXiv:quant-ph/0703231, 2007.

11

Ethan Bernstein and Umesh Vazirani
Quantum complexity theory.
In Proceedings of the 25th ACM Symposium on the Theory of Computing, pages 11-20, 1993.

12

D.W. Berry, G. Ahokas, R. Cleve, and B. C. Sanders
Efficient quantum algorithms for simulating sparse Hamiltonians.
Communications in Mathematical Physics, 270(2):359-371, 2007.
[arXiv:quant-ph/0508139]

13

A. Berzina, A. Dubrovsky, R. Frivalds, L. Lace, and O. Scegulnaja
Quantum query complexity for some graph problems.
In Proceedings of the 30th Conference on Current Trends in Theory and Practive of Computer Science, pages 140-150, 2004.

14

D. Boneh and R. J. Lipton
Quantum cryptanalysis of hidden linear functions.
In Don Coppersmith, editor, CRYPTO '95, Lecture Notes in Computer Science, pages 424-437. Springer-Verlag, 1995.

15

M. Boyer, G. Brassard, P. Høyer, and A. Tapp
Tight bounds on quantum searching.
Fortschritte der Physik, 46:493-505, 1998.

16

G. Brassard, P. Høyer, and A. Tapp
Quantum counting.
arXiv:quant-ph/9805082, 1998.

17

Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp
Quantum amplitude amplification and estimation.
In Samuel J. Lomonaco Jr. and Howard E. Brandt, editors, Quantum Computation and Quantum Information: A Millennium Volume, volume 305 of AMS Contemporary Mathematics Series. American Mathematical Society, 2002.
[arXiv:quant-ph/0005055]

18

Gilles Brassard, Peter Høyer, and Alain Tapp
Quantum algorithm for the collision problem.
ACM SIGACT News, 28:14-19, 1997.
[arXiv:quant-ph/9705002]

19

Harry Buhrman and Robert Špalek
Quantum verification of matrix products.
In Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms, pages 880-889, 2006.
[arXiv:quant-ph/0409035]

20

David Bulger
Quantum basin hopping with gradient-based local optimisation.
arXiv:quant-ph/0507193, 2005.

21

Harry Burhrman, Christoph Dürr, Mark Heiligman, Peter Høyer, Frédéric Magniez, Miklos Santha, and Ronald de Wolf
Quantum algorithms for element distinctness.
In Proceedings of the 16th IEEE Annual Conference on Computational Complexity, pages 131-137, 2001.
[arXiv:quant-ph/0007016]

22

Dong Pyo Chi, Jeong San Kim, and Soojoon Lee
Notes on the hidden subgroup problem on some semi-direct product groups.
Phys. Lett. A 359(2):114-116, 2006.
[arXiv:quant-ph/0604172]

23

A. M. Childs, L. J. Schulman, and U. V. Vazirani
Quantum algorithms for hidden nonlinear structures.
In Proceedings of the 48th IEEE Symposium on Foundations of Computer Science, pages 395-404, 2007.
[arXiv:0705.2784]

24

Andrew Childs and Troy Lee
Optimal quantum adversary lower bounds for ordered search.
Proceedings of ICALP 2008
[arXiv:0708.3396]

25

Andrew M. Childs
Quantum information processing in continuous time.
PhD thesis, MIT, 2004.

26

Andrew M. Childs, Richard Cleve, Enrico Deotto, Edward Farhi, Sam Gutmann, and Daniel A. Spielman
Exponential algorithmic speedup by quantum walk.
In Proceedings of the 35th ACM Symposium on Theory of Computing, pages 59-68, 2003.
[arXiv:quant-ph/0209131]

27

Andrew M. Childs, Richard Cleve, Stephen P. Jordan, and David Yonge-Mallo
Discrete-query quantum algorithm for NAND trees.
Theory of Computing, 5:119-123, 2009.
[arXiv:quant-ph/0702160]

28

Andrew M. Childs and Wim van Dam
Quantum algorithm for a generalized hidden shift problem.
In Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms, pages 1225-1232, 2007.
[arXiv:quant-ph/0507190]

29

Richard Cleve, Dmitry Gavinsky, and David L. Yonge-Mallo
Quantum algorithms for evaluating MIN-MAX trees.
In Theory of Quantum Computation, Communication, and Cryptography, pages 11-15,
Springer, 2008. (LNCS Vol. 5106)
[arXiv:0710.5794]

30

J. Niel de Beaudrap, Richard Cleve, and John Watrous
Sharp quantum versus classical query complexity separations.
Algorithmica, 34(4):449-461, 2002.
[arXiv:quant-ph/0011065v2]

31

Thomas Decker, Jan Draisma, and Pawel Wocjan
Quantum algorithm for identifying hidden polynomials.
Quantum Information and Computation, 9(3):215-230, 2009.
[arXiv:0706.1219]

32

David Deutsch
Quantum theory, the Church-Turing principle, and the universal quantum computer.
Proceedings of the Royal Society of London Series A, 400:97-117, 1985.

33

David Deutsch and Richard Jozsa
Rapid solution of problems by quantum computation.
Proceedings of the Royal Society of London Series A, 493:553-558, 1992.

34

Christoph Dürr, Mark Heiligman, Peter Høyer, and Mehdi Mhalla
Quantum query complexity of some graph problems.
SIAM Journal on Computing, 35(6):1310-1328, 2006.
[arXiv:quant-ph/0401091]

35

Christoph Dürr and Peter Høyer
A quantum algorithm for finding the minimum.
arXiv:quant-ph/9607014, 1996.

36

Christoph Dürr, Mehdi Mhalla, and Yaohui Lei
Quantum query complexity of graph connectivity.
arXiv:quant-ph/0303169, 2003.

37

Mark Ettinger, Peter Høyer, and Emanuel Knill
The quantum query complexity of the hidden subgroup problem is polynomial.
Information Processing Letters, 91(1):43-48, 2004.
[arXiv:quant-ph/0401083]

38

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann
A quantum algorithm for the Hamiltonian NAND tree.
Theory of Computing 4:169-190, 2008.
[arXiv:quant-ph/0702144]

39

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser
Invariant quantum algorithms for insertion into an ordered list.
arXiv:quant-ph/9901059, 1999.

40

Richard P. Feynman
Simulating physics with computers.
International Journal of Theoretical Physics, 21(6/7):467-488, 1982.

41

Michael Freedman, Alexei Kitaev, and Zhenghan Wang
Simulation of topological field theories by quantum computers.
Communications in Mathematical Physics, 227:587-603, 2002.

42

Michael Freedman, Michael Larsen, and Zhenghan Wang
A modular functor which is universal for quantum computation.
Comm. Math. Phys. 227(3):605-622, 2002.
[arXiv:quant-ph/0001108]

43

K. Friedl, G. Ivanyos, F. Magniez, M. Santha, and P. Sen
Hidden translation and translating coset in quantum computing.
SIAM Journal on Computing Vol. 43, pp. 1-24, 2014.
Appeared earlier in Proceedings of the 35th ACM Symposium on Theory of Computing, pages 1-9, 2003.
[arXiv:quant-ph/0211091]

44

D. Gavinsky
Quantum solution to the hidden subgroup problem for poly-near-Hamiltonian-groups.
Quantum Information and Computation, 4:229-235, 2004.

45

Joseph Geraci
A new connection between quantum circuits, graphs and the Ising partition function
Quantum Information Processing, 7(5):227-242, 2008.
[arXiv:0801.4833]

46

Joseph Geraci and Frank Van Bussel
A theorem on the quantum evaluation of weight enumerators for a certain class of cyclic Codes with a note on cyclotomic cosets.
arXiv:cs/0703129, 2007.

47

Joseph Geraci and Daniel A. Lidar
On the exact evaluation of certain instances of the Potts partition function by quantum computers.
Comm. Math. Phys. Vol. 279, pg. 735, 2008.
[arXiv:quant-ph/0703023]

48

Lov K. Grover
Quantum mechanics helps in searching for a needle in a haystack.
Physical Review Letters, 79(2):325-328, 1997.
[arXiv:quant-ph/9605043]

49

Sean Hallgren
Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem.
In Proceedings of the 34th ACM Symposium on Theory of Computing, 2002.

50

Sean Hallgren
Fast quantum algorithms for computing the unit group and class group of a number field.
In Proceedings of the 37th ACM Symposium on Theory of Computing, 2005.

51

Sean Hallgren, Alexander Russell, and Amnon Ta-Shma
Normal subgroup reconstruction and quantum computation using group representations.
SIAM Journal on Computing, 32(4):916-934, 2003.

52

Mark Heiligman
Quantum algorithms for lowest weight paths and spanning trees in complete graphs.
arXiv:quant-ph/0303131, 2003.

53

Yoshifumi Inui and François Le Gall
Efficient quantum algorithms for the hidden subgroup problem over a class of semi-direct product groups.
Quantum Information and Computation, 7(5/6):559-570, 2007.
[arXiv:quant-ph/0412033]

54

Yuki Kelly Itakura
Quantum algorithm for commutativity testing of a matrix set.
Master's thesis, University of Waterloo, 2005.
[arXiv:quant-ph/0509206]

55

Gábor Ivanyos, Frédéric Magniez, and Miklos Santha
Efficient quantum algorithms for some instances of the non-abelian hidden subgroup problem.
In Proceedings of the 13th ACM Symposium on Parallel Algorithms and Architectures, pages 263-270, 2001.
[arXiv:quant-ph/0102014]

56

Gábor Ivanyos, Luc Sanselme, and Miklos Santha
An efficient quantum algorithm for the hidden subgroup problem in extraspecial groups.
In Proceedings of the 24th Symposium on Theoretical Aspects of Computer Science, 2007.
[arXiv:quant-ph/0701235]

57

Gábor Ivanyos, Luc Sanselme, and Miklos Santha
An efficient quantum algorithm for the hidden subgroup problem in nil-2 groups.
In LATIN 2008: Theoretical Informatics, pg. 759-771, Springer (LNCS 4957).
[arXiv:0707.1260]

58

Dominik Janzing and Pawel Wocjan
BQP-complete problems concerning mixing properties of classical random walks on sparse graphs.
arXiv:quant-ph/0610235, 2006.

59

Dominik Janzing and Pawel Wocjan
A promiseBQP-complete string rewriting problem.
Quantum Information and Computation, 10(3/4):234-257, 2010.
[arXiv:0705.1180]

60

Dominik Janzing and Pawel Wocjan
A simple promiseBQP-complete matrix problem.
Theory of Computing, 3:61-79, 2007.
[arXiv:quant-ph/0606229]

61

Stephen P. Jordan
Fast quantum algorithm for numerical gradient estimation.
Physical Review Letters, 95:050501, 2005.
[arXiv:quant-ph/0405146]

62

Stephen P. Jordan
Quantum Computation Beyond the Circuit Model.
PhD thesis, Massachusetts Institute of Technology, 2008.
[arXiv:0809.2307]

63

Ivan Kassal, Stephen P. Jordan, Peter J. Love, Masoud Mohseni, and Alán Aspuru-Guzik
Quantum algorithms for the simulation of chemical dynamics.
Proc. Natl. Acad. Sci. Vol. 105, pg. 18681, 2008.
[arXiv:0801.2986]

64

Kiran S. Kedlaya
Quantum computation of zeta functions of curves.
Computational Complexity, 15:1-19, 2006.
[arXiv:math/0411623]

65

E. Knill and R. Laflamme
Quantum computation and quadratically signed weight enumerators.
Information Processing Letters, 79(4):173-179, 2001.
[arXiv:quant-ph/9909094]

66

Greg Kuperberg
A subexponential-time quantum algorithm for the dihedral hidden subgroup problem.
SIAM Journal on Computing, 35(1):170-188, 2005.
[arXiv:quant-ph/0302112]

67

Daniel A. Lidar
On the quantum computational complexity of the Ising spin glass partition function and of knot invariants.
New Journal of Physics Vol. 6, pg. 167, 2004.
[arXiv:quant-ph/0309064]

68

Daniel A. Lidar and Haobin Wang
Calculating the thermal rate constant with exponential speedup on a quantum computer.
Physical Review E, 59(2):2429-2438, 1999.
[arXiv:quant-ph/9807009]

69

Chris Lomont
The hidden subgroup problem - review and open problems.
arXiv:quant-ph/0411037, 2004.

70

Frédéric Magniez, Miklos Santha, and Mario Szegedy
Quantum algorithms for the triangle problem.
SIAM Journal on Computing, 37(2):413-424, 2007.
[arXiv:quant-ph/0310134]

71

Carlos Magno, M. Cosme, and Renato Portugal
Quantum algorithm for the hidden subgroup problem on a class of semidirect product groups.
arXiv:quant-ph/0703223, 2007.

72

Cristopher Moore, Daniel Rockmore, Alexander Russell, and Leonard Schulman
The power of basis selection in Fourier sampling: the hidden subgroup problem in affine groups.
In Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms, pages 1113-1122, 2004.
[arXiv:quant-ph/0211124]

73

M. Mosca
Quantum searching, counting, and amplitude amplification by eigenvector analysis.
In R. Freivalds, editor, Proceedings of International Workshop on Randomized Algorithms, pages 90-100, 1998.

74

Michele Mosca
Quantum Computer Algorithms.
PhD thesis, University of Oxford, 1999.

75

Ashwin Nayak and Felix Wu
The quantum query complexity of approximating the median and related statistics.
In Proceedings of 31st ACM Symposium on the Theory of Computing, 1999.
[arXiv:quant-ph/9804066]

76

Michael A. Nielsen and Isaac L. Chuang.
Quantum Computation and Quantum Information.
Cambridge University Press, Cambridge, UK, 2000.

77

Erich Novak
Quantum complexity of integration.
Journal of Complexity, 17:2-16, 2001.
[arXiv:quant-ph/0008124]

78

Oded Regev
Quantum computation and lattice problems.
In Proceedings of the 43rd Symposium on Foundations of Computer Science, 2002.
[arXiv:cs/0304005]

79

Oded Regev
A subexponential time algorithm for the dihedral hidden subgroup problem with polynomial space.
arXiv:quant-ph/0406151, 2004.

80

Ben Reichardt and Robert Špalek
Span-program-based quantum algorithm for evaluating formulas.
Proceedings of STOC 2008
[arXiv:0710.2630]

81

Martin Roetteler and Thomas Beth
Polynomial-time solution to the hidden subgroup problem for a class of non-abelian groups.
arXiv:quant-ph/9812070, 1998.

82

Peter W. Shor
Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer.
SIAM Journal on Computing, 26(5):1484-1509, 1997.
[arXiv:quant-ph/9508027]

83

Peter W. Shor and Stephen P. Jordan
Estimating Jones polynomials is a complete problem for one clean qubit.
Quantum Information and Computation, 8(8/9):681-714, 2008.
[arXiv:0707.2831]

84

R. D. Somma, S. Boixo, and H. Barnum
Quantum simulated annealing.
arXiv:0712.1008, 2007.

85

M. Szegedy
Quantum speed-up of Markov chain based algorithms.
In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, pg. 32, 2004.

86

Wim van Dam
Quantum algorithms for weighing matrices and quadratic residues.
Algorithmica, 34(4):413-428, 2002.
[arXiv:quant-ph/0008059]

87

Wim van Dam
Quantum computing and zeros of zeta functions.
arXiv:quant-ph/0405081, 2004.

88

Wim van Dam and Sean Hallgren
Efficient quantum algorithms for shifted quadratic character problems.
arXiv:quant-ph/0011067, 2000.

89

Wim van Dam, Sean Hallgren, and Lawrence Ip
Quantum algorithms for some hidden shift problems.
SIAM Journal on Computing, 36(3):763-778, 2006.
[arXiv:quant-h/0211140]

90

Wim van Dam and Gadiel Seroussi
Efficient quantum algorithms for estimating Gauss sums.
arXiv:quant-ph/0207131, 2002.

91

John Watrous
Quantum algorithms for solvable groups.
In Proceedings of the 33rd ACM Symposium on Theory of Computing, pages 60-67, 2001.
[arXiv:quant-ph/0011023]

92

Stephen Wiesner
Simulations of many-body quantum systems by a quantum computer.
arXiv:quant-ph/9603028, 1996.

93

Pawel Wocjan and Jon Yard
The Jones polynomial: quantum algorithms and applications in quantum complexity theory.
Quantum Information and Computation 8(1/2):147-180, 2008.
[arXiv:quant-ph/0603069]

94

Andrew Yao
On computing the minima of quadratic forms.
In Proceedings of the 7th ACM Symposium on Theory of Computing, pages 23-26, 1975.

95

Christof Zalka
Efficient simulation of quantum systems by quantum computers.
Proceedings of the Royal Society of London Series A, 454:313, 1996.
[arXiv:quant-ph/9603026]

96

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Michael Sipser
Quantum computation by adiabatic evolution.
arXiv:quant-ph/0001106, 2000.

97

Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev
Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation.
SIAM Journal on Computing, 37(1):166-194, 2007.
[arXiv:quant-ph/0405098]

98

Jérémie Roland and Nicolas J. Cerf
Quantum search by local adiabatic evolution.
Physical Review A, 65(4):042308, 2002.
[arXiv:quant-ph/0107015]

99

L.-A. Wu, M.S. Byrd, and D. A. Lidar
Polynomial-Time Simulation of Pairing Models on a Quantum Computer.
Physical Review Letters, 89(6):057904, 2002.
[arXiv:quant-ph/0108110]

100

Eli Biham, Ofer Biham, David Biron, Markus Grassl, and Daniel Lidar
Grover's quantum search algorithm for an arbitrary initial amplitude distribution.
Physical Review A, 60(4):2742, 1999.
[arXiv:quant-ph/9807027 and arXiv:quant-ph/0010077]

101

Andrew Childs, Shelby Kimmel, and Robin Kothari
The quantum query complexity of read-many formulas
In Proceedings of ESA 2012, pg. 337-348, Springer. (LNCS 7501)
[arXiv:1112.0548]

102

Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, and Martin Head-Gordon
Simulated quantum computation of molecular energies.
Science, 309(5741):1704-1707, 2005.
[arXiv:quant-ph/0604193]

103

A. M. Childs, A. J. Landahl, and P. A. Parrilo
Quantum algorithms for the ordered search problem via semidefinite programming.
Physical Review A, 75 032335, 2007.
[arXiv:quant-ph/0608161]

104

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd
Quantum algorithm for solving linear systems of equations.
Physical Review Letters 15(103):150502, 2009.
[arXiv:0811.3171]

105

Martin Roetteler
Quantum algorithms for highly non-linear Boolean functions.
Proceedings of SODA 2010
[arXiv:0811.3208]

106

Stephen P. Jordan
Fast quantum algorithms for approximating the irreducible representations of groups.
arXiv:0811.0562, 2008.

107

Tim Byrnes and Yoshihisa Yamamoto
Simulating lattice gauge theories on a quantum computer.
Physical Review A, 73, 022328, 2006.
[arXiv:quant-ph/0510027]

108

D. Simon
On the Power of Quantum Computation.
In Proceedings of the 35th Symposium on Foundations of Computer Science, pg. 116-123, 1994.

109

John Proos and Christof Zalka
Shor's discrete logarithm quantum algorithm for elliptic curves.
Quantum Information and Computation, Vol. 3, No. 4, pg.317-344, 2003.
[arXiv:quant-ph/0301141]

110

Yi-Kai Liu
Quantum algorithms using the curvelet transform.
Proceedings of STOC 2009, pg. 391-400.
[arXiv:0810.4968]

111

Wim van Dam and Igor Shparlinski
Classical and quantum algorithms for exponential congruences.
Proceedings of TQC 2008, pg. 1-10.
[arXiv:0804.1109]

112

Itai Arad and Zeph Landau
Quantum computation and the evaluation of tensor networks.
SIAM Journal on Computing, 39(7):3089-3121, 2010.
[arXiv:0805.0040]

113

M. Van den Nest, W. Dür, R. Raussendorf, and H. J. Briegel
Quantum algorithms for spin models and simulable gate sets for quantum computation.
Physical Review A, 80:052334, 2009.
[arXiv:0805.1214]

114

Silvano Garnerone, Annalisa Marzuoli, and Mario Rasetti
Efficient quantum processing of 3-manifold topological invariants.
Advances in Theoretical and Mathematical Physics, 13(6):1601-1652, 2009.
[arXiv:quant-ph/0703037]

115

Louis H. Kauffman and Samuel J. Lomonaco Jr.
q-deformed spin networks, knot polynomials and anyonic topological quantum computation.
Journal of Knot Theory, Vol. 16, No. 3, pg. 267-332, 2007.
[arXiv:quant-ph/0606114]

116

Arthur Schmidt and Ulrich Vollmer
Polynomial time quantum algorithm for the computation of the unit group of a number field.
In Proceedings of the 37th Symposium on the Theory of Computing, pg. 475-480, 2005.

117

Sergey Bravyi, Aram Harrow, and Avinatan Hassidim
Quantum algorithms for testing properties of distributions.
IEEE Transactions on Information Theory 57(6):3971-3981, 2011.
[arXiv:0907.3920]

118

Pawel M. Wocjan, Stephen P. Jordan, Hamed Ahmadi, and Joseph P. Brennan
Efficient quantum processing of ideals in finite rings.
arXiv:0908.0022, 2009.

119

V. Arvind, Bireswar Das, and Partha Mukhopadhyay
The complexity of black-box ring problems.
In Proceedings of COCCOON 2006, pg 126-145.

120

V. Arvind and Partha Mukhopadhyay
Quantum query complexity of multilinear identity testing.
In Proceedings of STACS 2009, pg. 87-98.

121

David Poulin and Pawel Wocjan
Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer.
Physical Review Letters 103:220502, 2009.
[arXiv:0905.2199]

122

Pawel Wocjan, Chen-Fu Chiang, Anura Abeyesinghe, and Daniel Nagaj
Quantum speed-up for approximating partition functions.
Physical Review A 80:022340, 2009.
[arXiv:0811.0596]

123

Ashley Montanaro
Quantum search with advice.
In Proceedings of the 5th conference on Theory of quantum computation, communication, and cryptography (TQC 2010)
[arXiv:0908.3066]

124

Laszlo Babai, Robert Beals, and Akos Seress
Polynomial-time theory of matrix groups.
In Proceedings of STOC 2009, pg. 55-64.

125

Peter Shor
Algorithms for Quantum Computation: Discrete Logarithms and Factoring.
In Proceedings of FOCS 1994, pg. 124-134.

126

Aaron Denney, Cristopher Moore, and Alex Russell
Finding conjugate stabilizer subgroups in PSL(2;q) and related groups.
Quantum Information and Computation 10(3):282-291, 2010.
[arXiv:0809.2445]

127

Kevin K. H. Cheung and Michele Mosca
Decomposing finite Abelian groups.
Quantum Information and Computation 1(2):26-32, 2001.
[arXiv:cs/0101004]

128

François Le Gall
An efficient quantum algorithm for some instances of the group isomorphism problem.
In Proceedings of STACS 2010.
[arXiv:1001.0608]

129

Gorjan Alagic, Stephen Jordan, Robert Koenig, and Ben Reichardt
Approximating Turaev-Viro 3-manifold invariants is universal for quantum computation.
Physical Review A 82, 040302(R), 2010.
[arXiv:1003.0923]

130

Martin Rötteler
Quantum algorithms to solve the hidden shift problem for quadratics and for functions of large Gowers norm.
In Proceedings of MFCS 2009, pg 663-674.
[arXiv:0911.4724]

131

Arthur Schmidt
Quantum Algorithms for many-to-one Functions to Solve the Regulator and the Principal Ideal Problem.
arXiv:0912.4807, 2009.

132

K. Temme, T.J. Osborne, K.G. Vollbrecht, D. Poulin, and F. Verstraete
Quantum Metropolis Sampling.
Nature, Vol. 471, pg. 87-90, 2011.
[arXiv:0911.3635]

133

Andris Ambainis
Quantum Search Algorithms.
SIGACT News, 35 (2):22-35, 2004.
[arXiv:quant-ph/0504012]

134

Nicolas J. Cerf, Lov K. Grover, and Colin P. Williams
Nested quantum search and NP-hard problems.
Applicable Algebra in Engineering, Communication and Computing, 10 (4-5):311-338, 2000.

135

Mario Szegedy
Spectra of Quantized Walks and a $\sqrt{\delta ϵ}$ rule.
arXiv:quant-ph/0401053, 2004.

136

Kazuo Iwama, Harumichi Nishimura, Rudy Raymond, and Junichi Teruyama
Quantum Counterfeit Coin Problems.
In Proceedings of 21st International Symposium on Algorithms and Computation (ISAAC2010), LNCS 6506, pp.73-84, 2010.
[arXiv:1009.0416]

137

Barbara Terhal and John Smolin
Single quantum querying of a database.
Physical Review A 58:1822, 1998.
[arXiv:quant-ph/9705041]

138

Andris Ambainis
Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations.
arXiv:1010.4458, 2010.

139

Frédéric Magniez and Ashwin Nayak
Quantum complexity of testing group commutativity.
In Proceedings of 32nd International Colloquium on Automata, Languages and Programming. LNCS 3580, pg. 1312-1324, 2005.
[arXiv:quant-ph/0506265]

140

Andrew Childs and Robin Kothari
Quantum query complexity of minor-closed graph properties.
In Proceedings of the 28th Symposium on Theoretical Aspects of Computer Science (STACS 2011), pg. 661-672
[arXiv:1011.1443]

141

Frédéric Magniez, Ashwin Nayak, Jérémie Roland, and Miklos Santha
Search via quantum walk.
In Proceedings STOC 2007, pg. 575-584.
[arXiv:quant-ph/0608026]

142

Dmitry Gavinsky, Martin Roetteler, and Jérémy Roland
Quantum algorithm for the Boolean hidden shift problem.
In Proceedings of the 17th annual international conference on Computing and combinatorics (COCOON '11), 2011.
[arXiv:1103.3017]

143

Mark Ettinger and Peter Høyer
On quantum algorithms for noncommutative hidden subgroups.
Advances in Applied Mathematics, Vol. 25, No. 3, pg. 239-251, 2000.
[arXiv:quant-ph/9807029]

144

Andris Ambainis, Andrew Childs, and Yi-Kai Liu
Quantum property testing for bounded-degree graphs.
In Proceedings of RANDOM '11: Lecture Notes in Computer Science 6845, pp. 365-376, 2011.
[arXiv:1012.3174]

145

G. Ortiz, J.E. Gubernatis, E. Knill, and R. Laflamme
Quantum algorithms for Fermionic simulations.
Physical Review A 64: 022319, 2001.
[arXiv:cond-mat/0012334]

146

Ashley Montanaro
The quantum query complexity of learning multilinear polynomials.
Information Processing Letters, 112(11):438-442, 2012.
[arXiv:1105.3310]

147

Tad Hogg
Highly structured searches with quantum computers.
Physical Review Letters 80: 2473, 1998.

148

Markus Hunziker and David A. Meyer
Quantum algorithms for highly structured search problems.
Quantum Information Processing, Vol. 1, No. 3, pg. 321-341, 2002.

149

Ben Reichardt
Span programs and quantum query complexity: The general adversary bound is nearly tight for every Boolean function.
In Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS '09), pg. 544-551, 2009.
[arXiv:0904.2759]

150

Aleksandrs Belovs
Span-program-based quantum algorithm for the rank problem.
arXiv:1103.0842, 2011.

151

Sebastian Dörn and Thomas Thierauf
The quantum query complexity of the determinant.
Information Processing Letters Vol. 109, No. 6, pg. 305-328, 2009.

152

Aleksandrs Belovs
Span programs for functions with constant-sized 1-certificates.
In Proceedings of STOC 2012, pg. 77-84.
[arXiv:1105.4024]

153

Troy Lee, Frédéric Magniez, and Mikos Santha
A learning graph based quantum query algorithm for finding constant-size subgraphs.
Chicago Journal of Theoretical Computer Science, Vol. 2012, Article 10, 2012.
[arXiv:1109.5135]

154

Aleksandrs Belovs and Troy Lee
Quantum algorithm for k-distinctness with prior knowledge on the input.
arXiv:1108.3022, 2011.

155

François Le Gall
Improved output-sensitive quantum algorithms for Boolean matrix multiplication.
In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '12), 2012.

156

Dominic Berry
Quantum algorithms for solving linear differential equations.
J. Phys. A: Math. Theor.47, 105301, 2014.
[arXiv:1010.2745].

157

Virginia Vassilevska Williams and Ryan Williams
Subcubic equivalences between path, matrix, and triangle problems.
In 51st IEEE Symposium on Foundations of Computer Science (FOCS '10) pg. 645 - 654, 2010.

158

Ben W. Reichardt
Reflections for quantum query algorithms.
In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pg. 560-569, 2011.
[arXiv:1005.1601]

159

Ben W. Reichardt
Span-program-based quantum algorithm for evaluating unbalanced formulas.
arXiv:0907.1622, 2009.

160

Ben W. Reichardt
Faster quantum algorithm for evaluating game trees.
In Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pg. 546-559, 2011.
[arXiv:0907.1623]

161

Stacey Jeffery, Robin Kothari, and Frédéric Magniez
Improving quantum query complexity of Boolean matrix multiplication using graph collision.
In Proceedings of ICALP 2012, pg. 522-532.
[arXiv:1112.5855]

162

Andrew M. Childs and Jason M. Eisenberg
Quantum algorithms for subset finding.
Quantum Information and Computation 5(7):593-604, 2005.
[arXiv:quant-ph/0311038]

163

Aleksandrs Belovs and Robert Špalek
Adversary lower bound for the k-sum problem.
In Proceedings of ITCS 2013, pg. 323-328.
[arXiv:1206.6528]

164

Bohua Zhan, Shelby Kimmel, and Avinatan Hassidim
Super-polynomial quantum speed-ups for Boolean evaluation trees with hidden structure.
ITCS 2012: Proceedings of the 3rd Innovations in Theoretical Computer Science, ACM, pg. 249-265.
[arXiv:1101.0796]

165

Shelby Kimmel
Quantum adversary (upper) bound.
39th International Colloquium on Automata, Languages and Programming - ICALP 2012 Volume 7391, p. 557-568.
[arXiv:1101.0797]

166

Stephen Jordan, Keith Lee, and John Preskill
Quantum algorithms for quantum field theories.
Science, Vol. 336, pg. 1130-1133, 2012.
[arXiv:1111.3633]

167

Andris Ambainis and Ashley Montanaro
Quantum algorithms for search with wildcards and combinatorial group testing.
arXiv:1210.1148, 2012.

168

Andris Ambainis and Robert Špalek
Quantum algorithms for matching and network flows.
Proceedings of STACS 2007, pg. 172-183.
[arXiv:quant-ph/0508205]

169

Nathan Wiebe, Daniel Braun, and Seth Lloyd
Quantum data-fitting.
Physical Review Letters 109, 050505, 2012.
[arXiv:1204.5242]

170

Andrew Childs and Nathan Wiebe
Hamiltonian simulation using linear combinations of unitary operations.
Quantum Information and Computation 12, 901-924, 2012.
[arXiv:1202.5822]

171

Stacey Jeffery, Robin Kothari, and Frédéric Magniez
Nested quantum walks with quantum data structures.
In Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA'13), pg. 1474-1485, 2013.
[arXiv:1210.1199]

172

Aleksandrs Belovs
Learning-graph-based quantum algorithm for k-distinctness.
Proceedings of STOC 2012, pg. 77-84.
[arXiv:1205.1534]

173

Andrew Childs, Stacey Jeffery, Robin Kothari, and Frédéric Magniez
A time-efficient quantum walk for 3-distinctness using nested updates.
arXiv:1302.7316, 2013.

174

Hari Krovi and Alexander Russell
Quantum Fourier transforms and the complexity of link invariants for quantum doubles of finite groups.
Commun. Math. Phys. 334, 743-777, 2015
[arXiv:1210.1550]

175

Troy Lee, Frédéric Magniez, and Miklos Santha
Improved quantum query algorithms for triangle finding and associativity testing.
arXiv:1210.1014, 2012.

176

Silvano Garnerone, Paolo Zanardi, and Daniel A. Lidar
Adiabatic quantum algorithm for search engine ranking.
Physical Review Letters 108:230506, 2012.

177

R. D. Somma, S. Boixo, H. Barnum, and E. Knill
Quantum simulations of classical annealing.
Physical Review Letters 101:130504, 2008.
[arXiv:0804.1571]

178

Daniel J. Bernstein, Stacey Jeffery, Tanja Lange, and Alexander Meurer
Quantum algorithms for the subset-sum problem.
from cr.yp.to.

179

Boris Altshuler, Hari Krovi, and Jérémie Roland
Anderson localization casts clouds over adiabatic quantum optimization.
Proceedings of the National Academy of Sciences 107(28):12446-12450, 2010.
[arXiv:0912.0746]

180

Ben Reichardt
The quantum adiabatic optimization algorithm and local minima.
In Proceedings of STOC 2004, pg. 502-510. [Erratum].

181

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann
Quantum adiabatic evolution algorithms versus simulated annealing.
arXiv:quant-ph/0201031, 2002.

182

E. Farhi, J. Goldstone, D. Gosset, S. Gutmann, H. B. Meyer, and P. Shor
Quantum adiabatic algorithms, small gaps, and different paths.
Quantum Information and Computation, 11(3/4):181-214, 2011.
[arXiv:0909.4766]

183

Sergey Bravyi, David P. DiVincenzo, Roberto I. Oliveira, and Barbara M. Terhal
The Complexity of Stoquastic Local Hamiltonian Problems.
Quantum Information and Computation, 8(5):361-385, 2008.
[arXiv:quant-ph/0606140]

184

Rolando D. Somma and Sergio Boixo
Spectral gap amplification.
SIAM Journal on Computing, 42:593-610, 2013.
[arXiv:1110.2494]

185

Sabine Jansen, Mary-Beth Ruskai, Ruedi Seiler
Bounds for the adiabatic approximation with applications to quantum computation.
Journal of Mathematical Physics, 48:102111, 2007.
[arXiv:quant-ph/0603175]

186

E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem.
Science, 292(5516):472-475, 2001.
[arXiv:quant-ph/0104129]

187

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Daniel Nagaj
How to make the quantum adiabatic algorithm fail.
International Journal of Quantum Information, 6(3):503-516, 2008.
[arXiv:quant-ph/0512159]

188

Edward Farhi, Jeffrey Goldstone, Sam Gutmann, and Daniel Nagaj
Unstructured randomness, small gaps, and localization.
Quantum Information and Computation, 11(9/10):840-854, 2011.
[arXiv:1010.0009]

189

Edward Farhi, Jeffrey Goldstone, Sam Gutmann
Quantum adiabatic evolution algorithms with different paths.
arXiv:quant-ph/0208135, 2002.

190

Wim van Dam, Michele Mosca, and Umesh Vazirani
How powerful is adiabatic quantum computation?
In Proceedings of FOCS 2001, pg. 279-287.
arXiv:quant-ph/0206003 [See also this.]

191

E. Farhi, D. Gosset, I. Hen, A. W. Sandvik, P. Shor, A. P. Young, and F. Zamponi
The performance of the quantum adiabatic algorithm on random instances of two optimization problems on regular hypergraphs.
Physical Review A, 86:052334, 2012.
[arXiv:1208.3757]

192

Kristen L. Pudenz and Daniel A. Lidar
Quantum adiabatic machine learning.
Quantum Information Processing, 12:2027, 2013.
[arXiv:1109.0325]

193

Frank Gaitan and Lane Clark
Ramsey numbers and adiabatic quantum computing.
Physical Review Letters, 108:010501, 2012.
[arXiv:1103.1345]

194

Frank Gaitan and Lane Clark
Graph isomorphism and adiabatic quantum computing.
Physical Review A, 89(2):022342, 2014.
[arXiv:1304.5773]

195

Hartmut Neven, Vasil S. Denchev, Geordie Rose, and William G. Macready
Training a binary classifier with the quantum adiabatic algorithm.
arXiv:0811.0416, 2008.

196

Robert Beals
Quantum computation of Fourier transforms over symmetric groups.
In Proceedings of STOC 1997, pg. 48-53.

197

Dave Bacon, Isaac L. Chuang, and Aram W. Harrow
The quantum Schur transform: I. efficient qudit circuits.
In Proceedings of SODA 2007, pg. 1235-1244.
[arXiv:quant-ph/0601001]

198

S. Morita, H. Nishimori
Mathematical foundation of quantum annealing.
Journal of Methematical Physics, 49(12):125210, 2008.

199

A. B. Finnila, M. A. Gomez, C. Sebenik, C. Stenson, J. D. Doll
Quantum annealing: a new method for minimizing multidimensional functions.
Chemical Physics Letters, 219:343-348, 1994.

200

D. Gavinsky and T. Ito
A quantum query algorithm for the graph collision problem.
arXiv:1204.1527, 2012.

201

Andris Ambainis, Kaspars Balodis, Jānis Iraids, Raitis Ozols, and Juris Smotrovs
Parameterized quantum query complexity of graph collision.
arXiv:1305.1021, 2013.

202

Kevin C. Zatloukal
Classical and quantum algorithms for testing equivalence of group extensions.
arXiv:1305.1327, 2013.

203

Andrew Childs and Gábor Ivanyos
Quantum computation of discrete logarithms in semigroups.
arXiv:1310.6238, 2013.

204

Matan Banin and Boaz Tsaban
A reduction of semigroup DLP to classic DLP.
arXiv:1310.7903, 2013.

205

D. W. Berry, R. Cleve, and R. D. Somma
Exponential improvement in precision for Hamiltonian-evolution simulation.
arXiv:1308.5424, 2013.

206

François Le Gall and Harumichi Nishimura
Quantum algorithms for matrix products over semirings.
arXiv:1310.3898, 2013.

207

Nolan Wallach
A quantum polylog algorithm for non-normal maximal cyclic hidden subgroups in the affine group of a finite field.
arXiv:1308.1415, 2013.

208

Lov Grover
Fixed-point quantum search.
Phys. Rev. Lett. 95(15):150501, 2005.
[arXiv:quant-ph/0503205]

209

Tathagat Tulsi, Lov Grover, and Apoorva Patel
A new algorithm for fixed point quantum search.
Quantum Information and Computation 6(6):483-494, 2005.
[arXiv:quant-ph/0505007]

210

Guoming Wang
Quantum algorithms for approximating the effective resistances of electrical networks.
arXiv:1311.1851

211

Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma
Exponential improvement in precision for simulating sparse Hamiltonians
arXiv:1312.1414

212

Thomas Decker, Peter Høyer, Gabor Ivanyos, and Miklos Santha
Polynomial time quantum algorithms for certain bivariate hidden polynomial problems
arXiv:1305.1543

213

Kirsten Eisenträger, Sean Hallgren, Alexei Kitaev, and Fang Song
A quantum algorithm for computing the unit group of an arbitrary degree number field
In Proceedings of STOC 2014 pg. 293-302.

214

Seth Lloyd, Masoud Mohseni, and Patrick Robentrost
Quantum algorithms for supervised and unsupervised machine learning
arXiv:1307.0411

215

Ashley Montanaro
Quantum pattern matching fast on average
arXiv:1408.1816

216

Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani
Strengths and weaknesses of quantum computing
SIAM J. Comput. 26(5):1524-1540, 1997
[arXiv:quant-ph/9701001]

217

H. Ramesh and V. Vinay
String matching in $\tilde{O}(\sqrt{n}+\sqrt{m})$ quantum time
Journal of Discrete Algorithms 1:103-110, 2003
[arXiv:quant-ph/0011049]

218

Greg Kuperberg
Another subexponential-time quantum algorithm for the dihedral hidden subgroup problem
In Proceedings of TQC pg. 20-34, 2013
[arXiv:1112.3333]

219

Peter Høyer, Jan Neerbek, and Yaoyun Shi
Quantum complexities of ordered searching, sorting, and element distinctness
In Proceedings of ICALP pg. 346-357, 2001
[arXiv:quant-ph/0102078]

220

Amnon Ta-Shma
Inverting well conditioned matrices in quantum logspace
In Proceedings of STOC 2013 pg. 881-890.

221

Nathan Wiebe, Ashish Kapoor, and Krysta Svore
Quantum deep learning
arXiv:1412.3489

222

Seth Lloyd, Silvano Garnerone, and Paolo Zanardi
Quantum algorithms for topological and geometric analysis of big data
arXiv:1408.3106

223

David A. Meyer and James Pommersheim
Single-query learning from abelian and non-abelian Hamming distance oracles
arXiv:0912.0583

224

Markus Hunziker, David A. Meyer, Jihun Park, James Pommersheim, and Mitch Rothstein
The geometry of quantum learning
Quantum Information Processing 9:321-341, 2010.
[arXiv:quant-ph/0309059]

225

Lawrence M. Ioannou and Michele Mosca
Limitations on some simple adiabatic quantum algorithms
International Journal of Quantum Information, 6(3):419-426, 2008.
[arXiv:quant-ph/0702241]

226

Michael Jarret and Stephen P. Jordan
Adiabatic optimization without local minima
Quantum Information and Computation, 15(3/4):0181-0199, 2015.
[arXiv:1405.7552]

227

Matthew B. Hastings, Dave Wecker, Bela Bauer, and Matthias Troyer
Improving quantum algorithms for quantum chemistry
Quantum Information and Computation, 15(1/2):0001-0021, 2015.
[arXiv:1403.1539]

228

Stephen P. Jordan, Keith S. M. Lee, and John Preskill
Quantum simulation of scattering in scalar quantum field theories
Quantum Information and Computation, 14(11/12):1014-1080, 2014.
[arXiv:1112.4833]

229

Stephen P. Jordan, Keith S. M. Lee, and John Preskill
Quantum algorithms for fermionic quantum field theories
arXiv:1404.7115

230

Gavin K. Brennen, Peter Rohde, Barry C. Sanders, and Sukhi Singh
Multi-scale quantum simulation of quantum field theory using wavelets
arXiv:1412.0750

231

Hefeng Wang, Sabre Kais, Alán Aspuru-Guzik, and Mark R. Hoffmann.
Quantum algorithm for obtaining the energy spectrum of molecular systems
Physical Chemistry Chemical Physics, 10(35):5388-5393, 2008.
[arXiv:0907.0854]

232

Ivan Kassal and Alán Aspuru-Guzik
Quantum algorithm for molecular properties and geometry optimization
Journal of Chemical Physics, 131(22), 2009.
[arXiv:0908.1921]

233

James D. Whitfield, Jacob Biamonte, and Alán Aspuru-Guzik
Simulation of electronic structure Hamiltonians using quantum computers
Molecular Physics, 109(5):735-750, 2011.
[arXiv:1001.3855]

234

Borzu Toloui and Peter J. Love
Quantum algorithms for quantum chemistry based on the sparsity of the CI-matrix
arXiv:1312.2529

235

James D. Whitfield
Spin-free quantum computational simulations and symmetry adapted states
Journal of Chemical Physics, 139(2):021105, 2013.
[arXiv:1306.1147]

236

Andrew W. Cross, Graeme Smith, and John A. Smolin
Quantum learning robust to noise
arXiv:1407.5088

237

Aram W. Harrow and David J. Rosenbaum
Uselessness for an oracle model with internal randomness
Quantum Information and Computation 14(7/8):608-624, 2014
[arXiv:1111.1462]

238

Jon R. Grice and David A. Meyer
A quantum algorithm for Viterbi decoding of classical convolutional codes
arXiv:1405.7479

239

Alexander Barg and Shiyu Zhou
A quantum decoding algorithm of the simplex code
Proceedings of the 36th Annual Allerton Conference, 1998
Available at author's homepage.

240

Guoming Wang
Span-program-based quantum algorithm for tree detection
arXiv:1309.7713, 2013.

241

François Le Gall, Harumichi Nishimura, and Seiichiro Tani
Quantum algorithm for finding constant-sized sub-hypergraphs over 3-uniform hypergraphs
In Proceedings of COCOON, 2014. pg. 429-440
[arXiv:1310.4127]

242

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann
A quantum approximate optimization algorithm
arXiv:1411.4028, 2014.

243

Edward Farhi, Jeffrey Goldstone, and Sam Gutmann
A quantum approximate optimization algorithm applied to a bounded occurrence constraint problem
arXiv:1412.6062, 2014.

244

Dominic W. Berry, Andrew M. Childs, Richard Cleve, Robin Kothari, and Rolando D. Somma
Simulating Hamiltonian dynamics with a truncated Taylor series
arXiv:1412.4687, 2014.

245

Dominic W. Berry, Andrew M. Childs, and Robin Kothari
Hamiltonian simulation with nearly optimal dependence on all parameters
arXiv:1501.01715, 2015.

246

Scott Aaronson
Read the fine print
Nature Physics 11:291-293, 2015.
[fulltext]

247

Alexander Elgart and George A. Hagedorn
A note on the switching adiabatic theorem
Journal of Mathematical Physics 53(10):102202, 2012.
[arXiv:1204.2318]

248

Daniel J. Bernstein, Johannes Buchmann, and Erik Dahmen, Eds.
Post-Quantum Cryptography
Springer, 2009.

249

B. D. Clader, B. C. Jacobs, and C. R. Sprouse
Preconditioned quantum linear system algorithm
Phys. Rev. Lett. 110:250504, 2013.
[arXiv:1301.2340]

250

S. Lloyd, M. Mohseni, and P. Rebentrost
Quantum principal component analysis
Nature Physics. 10(9):631, 2014.
[arXiv:1307.0401]

251

Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd
Quantum support vector machine for big data classification
Phys. Rev. Lett. 113, 130503, 2014.
[arXiv:1307.0471]

252

J. M. Pollard
Theorems on factorization and primality testing
Proceedings of the Cambridge Philosophical Society. 76:521-228, 1974.

253

L. Babai, R. Beals, and A. Seress
Polynomial-time theory of matrix groups
In Proceedings of STOC 2009, pg. 55-64.

254

Neil J. Ross and Peter Selinger
Optimal ancilla-free Clifford+T approximations of z-rotations
arXiv:1403.2975, 2014.

255

L. A. B. Kowada, C. Lavor, R. Portugal, and C. M. H. de Figueiredo
A new quantum algorithm for solving the minimum searching problem
International Journal of Quantum Information, Vol. 6, No. 3, pg. 427-436, 2008.

256

Sean Hallgren and Aram Harrow
Superpolynomial speedups based on almost any quantum circuit
Proceedings of ICALP 2008, pg. 782-795.
[arXiv:0805.0007]

257

Fernando G.S.L. Brandao and Michal Horodecki
Exponential quantum speed-ups are generic
Quantum Information and Computation, Vol. 13, Pg. 0901, 2013
[arXiv:1010.3654]

258

Scott Aaronson and Andris Ambainis
Forrelation: A problem that optimally separates quantum from classical computing.
arXiv:1411.5729, 2014.

259

Z. Gedik
Computational speedup with a single qutrit
arXiv:1403.5861, 2014.

260

Boaz Barak, Ankur Moitra, Ryan O'Donnell, Prasad Raghavendra, Oded Regev, David Steurer, Luca Trevisan, Aravindan Vijayaraghavan, David Witmer, and John Wright
Beating the random assignment on constraint satisfaction problems of bounded degree
arXiv:1505.03424, 2015.

261

David Cornwell
Amplified Quantum Transforms
arXiv:1406.0190, 2015.

262

T. Laarhoven, M. Mosca, and J. van de Pol
Solving the shortest vector problem in lattices faster using quantum search
Proceedings of PQCrypto13, pp. 83-101, 2013.
[arXiv:1301.6176]

263

Andrew M. Childs, Robin Kothari, and Rolando D. Somma
Quantum linear systems algorithm with exponentially improved dependence on precision
arXiv:1511.02306, 2015.

264

Ashley Montanaro
Quantum walk speedup of backtracking algorithms
arXiv:1509.02374, 2015.

265

Ashley Montanaro
Quantum speedup of Monte Carlo methods
arXiv:1504.06987, 2015.

266

Andris Ambainis, Aleksandrs Belovs, Oded Regev, and Ronald de Wolf
Efficient quantum algorithms for (gapped) group testing and junta testing
arXiv:1507.03126, 2015.

267

A. Atici and R. A. Servedio
Quantum algorithms for learning and testing juntas
Quantum Information Processing, 6(5):323-348, 2007.
[arXiv:0707.3479]

268

Aleksandrs Belovs
Quantum algorithms for learning symmetric juntas via the adversary bound
Computational Complexity, 24(2):255-293, 2015.
(Also appears in proceedings of CCC'14).
[arXiv:1311.6777]

269

Stacey Jeffery and Shelby Kimmel
NAND-trees, average choice complexity, and effective resistance
arXiv:1511.02235, 2015.

270

Scott Aaronson, Shalev Ben-David, and Robin Kothari
Separations in query complexity using cheat sheets
arXiv:1511.01937, 2015.

271

Frédéric Grosshans, Thomas Lawson, François Morain, and Benjamin Smith
Factoring safe semiprimes with a single quantum query
arXiv:1511.04385, 2015.

272

Agnis Āriņš
Span-program-based quantum algorithms for graph bipartiteness and connectivity
arXiv:1510.07825, 2015.

273

Juan Bermejo-Vega and Kevin C. Zatloukal
Abelian hypergroups and quantum computation
arXiv:1509.05806, 2015.

274

Andrew Childs and Jeffrey Goldstone
Spatial search by quantum walk
Physical Review A, 70:022314, 2004.
[arXiv:quant-ph/0306054]

275

Shantanav Chakraborty, Leonardo Novo, Andris Ambainis, and Yasser Omar
Spatial search by quantum walk is optimal for almost all graphs
arXiv:1508.01327, 2015.

276

François Le Gall
Improved quantum algorithm for triangle finding via combinatorial arguments
In Proceedings of the 55th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pg. 216-225, 2014.
[arXiv:1407.0085]

277

Ashley Montanaro
The quantum complexity of approximating the frequency moments
arXiv:1505.00113, 2015.

278

Rolando D. Somma
Quantum simulations of one dimensional quantum systems
arXiv:1503.06319, 2015.

279

Bill Fefferman and Cedric Yen-Yu Lin
A complete characterization of unitary quantum space
arXiv:1604.01384, 2016.

280

Tsuyoshi Ito and Stacey Jeffery
Approximate span programs
arXiv:1507.00432, 2015.

281

Arnau Riera, Christian Gogolin, and Jens Eisert
Thermalization in nature and on a quantum computer
Physical Review Letters, 108:080402 (2012)
[arXiv:1102.2389]

282

Michael J. Kastoryano and Fernando G. S. L. Brandao
Quantum Gibbs Samplers: the commuting case
Communications in Mathematical Physics, 344(3):915-957 (2016)
[arXiv:1409.3435]

283

Andrew M. Childs, David Jao, and Vladimir Soukharev
Constructing elliptic curve isogenies in quantum subexponential time
Journal of Mathematical Cryptology, 8(1):1-29 (2014)
[arXiv:1012.4019]

284

Markus Grassl, Brandon Langenberg, Martin Roetteler, and Rainer Steinwandt
Applying Grover's algorithm to AES: quantum resource estimates
arXiv:1512.04965, 2015.

285

M. Ami, O. Di Matteo, V. Gheorghiu, M. Mosca, A. Parent, and J. Schanck
Estimating the cost of generic quantum pre-image attacks on SHA-2 and SHA-3
arXiv:1603.09383, 2016.

286

Marc Kaplan, Gaetan Leurent, Anthony Leverrier, and Maria Naya-Plasencia
Quantum differential and linear cryptanalysis
arXiv:1510.05836, 2015.

287

Scott Fluhrer
Quantum Cryptanalysis of NTRU
Cryptology ePrint Archive: Report 2015/676, 2015.

288

Marc Kaplan
Quantum attacks against iterated block ciphers
arXiv:1410.1434, 2014.

289

H. Kuwakado and M. Morii
Quantum distinguisher between the 3-round Feistel cipher and the random permutation
In Proceedings of IEEE International Symposium on Information Theory (ISIT), pg. 2682-2685, 2010.

290

H. Kuwakado and M. Morii
Security on the quantum-type Even-Mansour cipher
In Proceedings of International Symposium on Information Theory and its Applications (ISITA), pg. 312-316, 2012.

291

Martin Roetteler and Rainer Steinwandt
A note on quantum related-key attacks
arXiv:1306.2301, 2013.

292

Thomas Santoli and Christian Schaffner
Using Simon's algorithm to attack symmetric-key cryptographic primitives
arXiv:1603.07856, 2016.

293

Rolando D. Somma
A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation
arXiv:1512.03416, 2015.

294

Guang Hao Low and Isaac Chuang
Optimal Hamiltonian simulation by quantum signal processing
arXiv:1606.02685, 2016.

295

Dominic W. Berry and Leonardo Novo
Corrected quantum walk for optimal Hamiltonian simulation
arXiv:1606.03443, 2016.

296

Ashley Montanaro and Sam Pallister
Quantum algorithms and the finite element method
arXiv:1512.05903, 2015.

297

Lin-Chun Wan, Chao-Hua Yu, Shi-Jie Pan, Fei Gao, and Qiao-Yan Wen
Quantum algorithm for the Toeplitz systems
arXiv:1608.02184, 2016.

298

Salvatore Mandra, Gian Giacomo Guerreschi, and Alan Aspuru-Guzik
Faster than classical quantum algorithm for dense formulas of exact satisfiability and occupation problems
arXiv:1512.00859, 2015.

299

J. Adcock, E. Allen, M. Day, S. Frick, J. Hinchliff, M. Johnson, S. Morley-Short, S. Pallister, A. Price, and S. Stanisic
Advances in quantum machine learning
arXiv:1512.02900, 2015.

300

Cedric Yen-Yu Lin and Yechao Zhu
Performance of QAOA on typical instances of constraint satisfaction problems with bounded degree
arXiv:1601.01744, 2016.

301

Dave Wecker, Matthew B. Hastings, and Matthias Troyer
Training a quantum optimizer
arXiv:1605.05370, 2016.

302

Edward Farhi and Aram W. Harrow
Quantum supremacy through the quantum approximate optimization algorithm
arXiv:1602.07674, 2016.

303

Thomas G. Wong
Quantum walk search on Johnson graphs
arXiv:1601.04212, 2016.

304

Jonatan Janmark, David A. Meyer, and Thomas G. Wong
Global symmetry is unnecessary for fast quantum search
Physical Review Letters 112:210502, 2014.
[arXiv:1403.2228]

305

David A. Meyer and Thomas G. Wong
Connectivity is a poor indicator of fast quantum search
Physical Review Letters 114:110503, 2014.
[arXiv:1409.5876]

306

Thomas G. Wong
Spatial search by continuous-time quantum walk with multiple marked vertices
Quantum Information Processing 15(4):1411-1443, 2016.
[arXiv:1501.07071]

307

Anirban Naryan Chowdhury and Rolando D. Somma
Quantum algorithms for Gibbs sampling and hitting-time estimation
arXiv:1603.02940, 2016.

308

Edward Farhi, Shelby Kimmel, and Kristan Temme
A quantum version of Schoning's algorithm applied to quantum 2-SAT
arXiv:1603.06985, 2016.

309

Iordanis Kerenidis and Anupam Prakash
Quantum recommendation systems
Innovations in Theoretical Computer Science (ITCS 2017), LIPIcs, vol. 67, pg. 1868-8969.
[arXiv:1603.08675]

310

Markus Reiher, Nathan Wiebe, Krysta M. Svore, Dave Wecker, and Matthias Troyer
Elucidating reaction mechanisms on quantum computers
arXiv:1605.03590, 2016.

311

Aram W. Harrow and Ashley Montanaro
Sequential measurements, disturbance, and property testing
arXiv:1607.03236, 2016.

312

Martin Roetteler
Quantum algorithms for abelian difference sets and applications to dihedral hidden subgroups
arXiv:1608.02005, 2016.

313

Fernando G.S.L. Brandao and Krysta Svore
Quantum speed-ups for semidefinite programming
arXiv:1609.05537, 2016.

314

Z-C Yang, A. Rahmani, A. Shabani, H. Neven, and C. Chamon
Optimizing variational quantum algorithms using Pontryagins's minimum principle
arXiv:1607.06473, 2016.

315

Gilles Brassard, Peter Høyer, and Alain Tapp
Quantum cryptanalysis of hash and claw-free functions
In Proceedings of the 3rd Latin American symposium on Theoretical Informatics (LATIN'98), pg. 163-169, 1998.

316

Daniel J. Bernstein
Cost analysis of hash collisions: Will quantum computers make SHARCS obsolete?
In Proceedings of the 4th Workshop on Special-purpose Hardware for Attacking Cryptographic Systems (SHARCS'09), pg. 105-116, 2009.
[available here]

317

Chris Cade, Ashley Montanaro, and Aleksandrs Belovs
Time and space efficient quantum algorithms for detecting cycles and testing bipartiteness
arXiv:1610.00581, 2016.

318

A. Belovs and B. Reichardt
Span programs and quantum algorithms for st-connectivity and claw detection
In European Symposium on Algorithms (ESA'12), pg. 193-204, 2012.
[arXiv:1203.2603]

319

Titouan Carette, Mathieu Laurière, and Frédéric Magniez
Extended learning graphs for triangle finding
arXiv:1609.07786, 2016.

320

F. Le Gall and N. Shogo
Quantum algorithm for triangle finding in sparse graphs
In Proceedings of the 26th International Symposium on Algorithms and Computation (ISAAC'15), pg. 590-600, 2015.

321

Or Sattath and Itai Arad
A constructive quantum Lovász local lemma for commuting projectors
Quantum Information and Computation, 15(11/12)987-996pg, 2015.
[arXiv:1310.7766]

322

Martin Schwarz, Toby S. Cubitt, and Frank Verstraete
An information-theoretic proof of the constructive commutative quantum Lovász local lemma
arXiv:1311.6474

323

C. Shoen, E. Solano, F. Verstraete, J. I. Cirac, and M. M. Wolf
Sequential generation of entangled multi-qubit states
Physical Review Letters, 95:110503, 2005.
[arXiv:quant-ph/0501096]

324

C. Shoen, K. Hammerer, M. M. Wolf, J. I. Cirac, and E. Solano
Sequential generation of matrix-product states in cavity QED
Physical Review A, 75:032311, 2007.
[arXiv:quant-ph/0612101]

325

Yimin Ge, András Molnár, and J. Ignacio Cirac
Rapid adiabatic preparation of injective PEPS and Gibbs states
Physical Review Letters, 116:080503, 2016.
[arXiv:1508.00570]

326

Martin Schwarz, Kristan Temme, and Frank Verstraete
Preparing projected entangled pair states on a quantum computer
Physical Review Letters, 108:110502, 2012.
[arXiv:1104.1410]

327

Martin Schwarz, Toby S. Cubitt, Kristan Temme, Frank Verstraete, and David Perez-Garcia
Preparing topological PEPS on a quantum computer
Physical Review A, 88:032321, 2013.
[arXiv:1211.4050]

328

M. Schwarz, O. Buerschaper, and J. Eisert
Approximating local observables on projected entangled pair states
arXiv:1606.06301, 2016.

329

Jean-François Biasse and Fang Song
Efficient quantum algorithms for computing class groups and solving the principal ideal problem in arbitrary degree number fields
Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '16), pg. 893-902, 2016.

330

Peter Høyer and Mojtaba Komeili
Efficient quantum walk on the grid with multiple marked elements
Proceedings of the 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), 42, 2016.
[arXiv:1612.08958]

331

Peter Wittek
Quantum Machine Learning: what quantum computing means to data mining
Academic Press, 2014.

332

Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione
An introduction to quantum machine learning
Contemporary Physics, 56(2):172, 2014.
[arXiv:1409.3097]

333

J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, N. Wiebe, and S. Lloyd
Quantum machine learning
arXiv:1611.09347

334

Esma Aïmeur, Gilles Brassard, and Sébastien Gambs
Machine learning in a quantum world
In Advances in Artificial Intelligence: 19th Conference of the Canadian Society for Computational Studies of Intelligence pg. 431-442, Springer, 2006.

335

Vedran Dunjko, Jacob Taylor, and Hans Briegel
Quantum-enhanced machine learning
Phys. Rev. Lett 117:130501, 2016.

336

Nathan Wiebe, Ashish Kapoor, and Krysta Svore
Quantum algorithms for nearest-neighbor methods for supervised and unsupervised learning
Quantum Information and Computation 15(3/4): 0318-0358, 2015.
[arXiv:1401.2142]

337

Seokwon Yoo, Jeongho Bang, Changhyoup Lee, and Junhyoug Lee
A quantum speedup in machine learning: finding a N-bit Boolean function for a classification
New Journal of Physics 6(10):103014, 2014.
[arXiv:1303.6055]

338

Maria Schuld, Ilya Sinayskiy, and Francesco Petruccione
Prediction by linear regression on a quantum computer
Physical Review A 94:022342, 2016.
[arXiv:1601.07823]

339

Zhikuan Zhao, Jack K. Fitzsimons, and Joseph F. Fitzsimons
Quantum assisted Gaussian process regression
arXiv:1512.03929

340

Esma Aïmeur, Gilles Brassard, and Sébastien Gambs
Quantum speed-up for unsupervised learning
Machine Learning, 90(2):261-287, 2013.

341

Nathan Wiebe, Ashish Kapoor, and Krysta Svore
Quantum perceptron models
Advances in Neural Information Processing Systems 29 (NIPS 2016), pg. 3999–4007, 2016.
[arXiv:1602.04799]

342

G. Paparo, V. Dunjko, A. Makmal, M. Martin-Delgado, and H. Briegel
Quantum speedup for active learning agents
Physical Review X4(3):031002, 2014.
[arXiv:1401.4997]

343

Daoyi Dong, Chunlin Chen, Hanxiong Li, and Tzyh-Jong Tarn
Quantum reinforcement learning
IEEE Transactions on Systems, Man, and Cybernetics- Part B (Cybernetics)38(5):1207, 2008.

344

Daniel Crawford, Anna Levit, Navid Ghadermarzy, Jaspreet S. Oberoi, and Pooya Ronagh
Reinforcement learning using quantum Boltzmann machines
arXiv:1612.05695, 2016.

345

Steven H. Adachi and Maxwell P. Henderson
Application of Quantum Annealing to Training of Deep Neural Networks
arXiv:1510.06356, 2015.

346

M. Benedetti, J. Realpe-Gómez, R. Biswas, and A. Perdomo-Ortiz
Quantum-assisted learning of graphical models with arbitrary pairwise connectivity
arXiv:1609.02542, 2016.

348

M. H. Amin, E. Andriyash, J. Rolfe, B. Kulchytskyy, and R. Melko
Quantum Boltzmann machine
arXiv:1601.02036, 2016.

349

Peter Wittek and Christian Gogolin
Quantum enhanced inference in Markov logic networks
Scientific Reports7:45672, 2017.
[arXiv:1611.08104]

350

N. H. Bshouty and J. C. Jackson
Learning DNF over the uniform distribution using a quantum example oracle
SIAM Journal on Computing28(3):1136-1153, 1999.

351

Srinivasan Arunachalam and Ronald de Wolf
A survey of quantum learning theory
arXiv:1701.06806, 2017.

352

Rocco A. Servedio and Steven J. Gortler
Equivalences and separations between quantum and classical learnability
SIAM Journal on Computing, 33(5):1067-1092, 2017.

353

Srinivasan Arunachalam and Ronald de Wolf
Optimal quantum sample complexity of learning algorithms
arXiv:1607.00932, 2016.

354

Alex Monràs, Gael Sentís, and Peter Wittek
Inductive quantum learning: why you are doing it almost right
arXiv:1605.07541, 2016.

355

A. Bisio, G. Chiribella, G. M. D'Ariano, S. Facchini, and P. Perinotti
Optimal quantum learning of a unitary transformation
Physical Review A 81:032324, 2010.
[arXiv:0903.0543]

356

M. Sasaki, A. Carlini, and R. Jozsa
Quantum template matching
Physical Review A 64:022317, 2001.
[arXiv:quant-ph/0102020]

357

Masahide Sasaki and Alberto Carlini
Quantum learning and universal quantum matching machine
Physical Review A 66:022303, 2002.
[arXiv:quant-ph/0202173]

358

Esma Aïmeur, Gilles Brassard, and Sébastien Gambs
Quantum clustering algorithms
In Proceedings of the 24th International Conference on Machine Learning (ICML), pg. 1-8, 2007.

359

Iordanis Kerenidis and Anupam Prakash
Quantum gradient descent for linear systems and least squares
arXiv:1704.04992, 2017.

360

Dan Boneh and Mark Zhandry
Quantum-secure message authentication codes
In Proceedings of Eurocrypt, pg. 592-608, 2013.

361

A. M. Childs, W. van Dam, S-H Hung, and I. E. Shparlinski
Optimal quantum algorithm for polynomial interpolation
In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP), pg. 16:1-16:13, 2016.
[arXiv:1509.09271]

362

Volker Strassen
Einige Resultate über Berechnungskomplexität
In Jahresbericht der Deutschen Mathematiker-Vereinigung, 78(1):1-8, 1976/1977.

363

Stacey Jeffery
Frameworks for Quantum Algorithms
PhD thesis, U. Waterloo, 2014.

364

Seiichiro Tani
An improved claw finding algorithm using quantum walk
In Mathematical Foundations of Computer Science (MFCS), pg. 536-547, 2007.
[arXiv:0708.2584]

365

K. Iwama and A. Kawachi
A new quantum claw-finding algorithm for three functions
New Generation Computing, 21(4):319-327, 2003.

366

D. J. Bernstein, N. Heninger, P. Lou, and L. Valenta
Post-quantum RSA
IACR e-print 2017/351, 2017.

367

Francois Fillion-Gourdeau, Steve MacLean, and Raymond Laflamme
Quantum algorithm for the dsolution of the Dirac equation
arXiv:1611.05484, 2016.

368

Ali Hamed Moosavian and Stephen Jordan
Faster quantum algorithm to simulate Fermionic quantum field theory
arXiv:1711.04006, 2017.

369

Pedro C.S. Costa, Stephen Jordan, and Aaron Ostrander
Quantum algorithm for simulating the wave equation
arXiv:1711.05394, 2017.

370

Jeffrey Yepez
Highly covariant quantum lattice gas model of the Dirac equation
arXiv:1106.0739, 2011.

371

Jeffrey Yepez
Quantum lattice gas model of Dirac particles in 1+1 dimensions
arXiv:1307.3595, 2013.

372

Bruce M. Boghosian and Washington Taylor
Simulating quantum mechanics on a quantum computer
Physica D 120:30-42, 1998.
[arXiv:quant-ph/9701019]

373

Yimin Ge, Jordi Tura, and J. Ignacio Cirac
Faster ground state preparation and high-precision ground energy estimation on a quantum computer
arXiv:1712.03193, 2017.

374

Renato Portugal
Element distinctness revisited
arXiv:1711.11336, 2017.

375

Kanav Setia and James D. Whitfield
Bravyi-Kitaev superfast simulation of fermions on a quantum computer
arXiv:1712.00446, 2017.

376

Richard Cleve and Chunhao Wang
Efficient quantum algorithms for simulating Lindblad evolution
arXiv:1612.09512, 2016.

377

M. Kliesch, T. Barthel, C. Gogolin, M. Kastoryano, and J. Eisert
Dissipative quantum Church-Turing theorem
Physical Review Letters 107(12):120501, 2011.
[arXiv:1105.3986]

378

A. M. Childs and T. Li
Efficient simulation of sparse Markovian quantum dynamics
arXiv:1611.05543, 2016.

379

R. Di Candia, J. S. Pedernales, A. del Campo, E. Solano, and J. Casanova
Quantum simulation of dissipative processes without reservoir engineering
Scientific Reports 5:9981, 2015.

380

R. Babbush, D. Berry, M. Kieferová, G. H. Low, Y. Sanders, A. Sherer, and N. Wiebe
Improved techniques for preparing eigenstates of Fermionic Hamiltonians
arXiv:1711.10460, 2017.

381

D. Poulin, A. Kitaev, D. S. Steiger, M. B. Hasting, and M. Troyer
Fast quantum algorithm for spectral properties
arXiv:1711.11025, 2017.

382

Guang Hao Low and Isaac Chuang
Hamiltonian simulation by qubitization
arXiv:1610.06546, 2016.

383

F.G.S.L. Brandão, A. Kalev, T. Li, C. Y.-Y. Lin, K. M. Svore, and X. Wu
Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning
Proceedings of ICALP 2019
[arXiv:1710.02581]

384

M. Ekerå and J. Håstad
Quantum Algorithms for Computing Short Discrete Logarithms and Factoring RSA Integers
Proceedings of PQCrypto 2017, pg. 347-363. (LNCS Volume 10346), 2017.

385

M. Ekerå
On post-processing in the quantum algorithm for computing short discrete logarithms
IACR ePrint Archive Report 2017/1122, 2017.

386

D. J. Bernstein, J.-F. Biasse, and M. Mosca
A low-resource quantum factoring algorithm
Proceedings of PQCrypto 2017, pg. 330-346 (LNCS Volume 10346), 2017.

387

Jianxin Chen, Andrew M. Childs, and Shih-Han Hung
Quantum algorithm for multivariate polynomial interpolation
Proceedings of the Royal Society A, 474:20170480, 2017.
arXiv:1701.03990

388

Lisa Hales and Sean Hallgren
An improved quantum Fourier transform algorithm and applications.
In Proceedings of FOCS 2000, pg. 515-525.

389

Igor Shparlinski and Arne Winterhof
Quantum period reconstruction of approximate sequences
Information Processing Letters, 103:211-215, 2007.

390

Alexander Russell and Igor E. Shparlinski
Classical and quantum function reconstruction via character evaluation
Journal of Complexity, 20:404-422, 2004.

391

Sean Hallgren, Alexander Russell, and Igor Shparlinski
Quantum noisy rational function reconstruction
Proceedings of COCOON 2005, pg. 420-429.

392

G. Ivanyos, M. Karpinski, M. Santha, N. Saxena, and I. Shparlinski
Polynomial interpolation and identity testing from high powers over finite fields
Algorithmica, 80:560-575, 2017.

393

Qi Cheng
Primality Proving via One Round in ECPP and One Iteration in AKS
Journal of Cryptology, Volume 20, Issue 3, pg. 375-387, July 2007.

394

Daniel J. Bernstein
Proving primality in essentially quartic random time
Mathematics of Computation, Vol. 76, pg. 389-403, 2007.

395

F. Morain
Implementing the asymptotically fast version of the elliptic curve primality proving algorithm
Mathematics of Computation, Vol. 76, pg. 493-505, 2007.

396

Alvaro Donis-Vela and Juan Carlos Garcia-Escartin
A quantum primality test with order finding
arXiv:1711.02616, 2017.

397

H. F. Chau and H.-K. Lo
Primality test via quantum factorization
International Journal of Modern Physics C, Vol. 8, No. 2, pg. 131-138, 1997.
[arXiv:quant-ph/9508005]

398

David Harvey and Joris Van Der Hoeven
Integer multiplication in time $O(n\log n)$
hal-02070778, 2019.

399

Charles Greathouse
personal communication, 2019.

400

Ewin Tang
A quantum-inspired classical algorithm for recommendation systems
In Proceedings of STOC 2019, pg. 217-228.
[arXiv:1807.04271]

401

Ewin Tang
Quantum-inspired classical algorithms for principal component analysis and supervised clustering
arXiv:1811.00414, 2018.

402

L. Wossnig, Z. Zhao, and A. Prakash
A quantum linear system algorithm for dense matrices
Physical Review Letters vol. 120, no. 5, pg. 050502, 2018.
arXiv:1704.06174, 2017.

403

Zhikuan Zhao, Alejandro Pozas-Kerstjens, Patrick Rebentrost, and Peter Wittek
Bayesian Deep Learning on a Quantum Computer
Quantum Machine Intelligence vol. 1, pg. 41-51, 2019.
[arXiv:1806.11463]

404

Anja Becker, Jean-Sebastien Coron, and Antoine Joux
Improved generic algorithms for hard knapsacks
Proceedings of Eurocrypt 2011 pg. 364-385
[IACR eprint 2011/474]

405

Kun Zhang and Vladimir E. Korepin
Low depth quantum search algorithm
arXiv:1908.04171, 2019.

406

Andriyan Bayo Suksmono and Yuichiro Minato
Finding Hadamard matrices by a quantum annealing machine
Scientific Reports 9:14380, 2019.
[arXiv:1902.07890]

407

Gábor Ivanyos, Anupam Prakash, and Miklos Santha
On learning linear functions from subset and its applications in quantum computing
26th Annual European Symposium on Algorithms (ESA 2018), LIPIcs volume 112, 2018.
[arXiv:1806.09660]

408

Gábor Ivanyos
On solving systems of random linear disequations
Quantum Information and Computation, 8(6):579-594, 2008.
[arXiv:0704.2988]

409

A. Ambainis, K. Balodis, J. Iraids, M. Kokainis, K. Prusis, and J. Vihrovs
Quantum speedups for exponential-time dynamic programming algorithms
Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 19), pg. 1783-1793, 2019.
[arXiv:1807.05209]

410

Dominic W. Berry, Andrew M. Childs, Aaron Ostrander, and Guoming Wang
Quantum algorithm for linear differential equations with exponentially improved dependence on precision
Communications in Mathematical Physics, 356(3):1057-1081, 2017.
[arXiv:1701.03684]

411

Sarah K. Leyton and Tobias J. Osborne
Quantum algorithm to solve nonlinear differential equations
arXiv:0812.4423

412

Y. Cao, A. Papageorgiou, I. Petras, J. Traub, and S. Kais
Quantum algorithm and circuit design solving the Poisson equation
New Journal of Physics 15(1):013021, 2013.
[arXiv:1207.2485]

413

S. Wang, Z. Wang, W. Li, L. Fan, Z. Wei, and Y. Gu
Quantum fast Poisson solver: the algorithm and modular circuit design
arXiv:1910.09756, 2019.

414

A. Scherer, B. Valiron, S.-C. Mau, S. Alexander, E. van den Berg, and T. Chapuran
Concrete resource analysis of the quantum linear system algorithm used to compute the electromagnetic scattering crossection of a 2D target
Quantum Information Processing 16:60, 2017.
[arXiv:1505.06552]

415

Juan Miguel Arrazola, Timjan Kalajdziavski, Christian Weedbrook, and Seth Lloyd
Quantum algorithm for nonhomogeneous linear partial differential equations
Physical Review A 100:032306, 2019.
[arXiv:1809.02622]

416

Andrew Childs and Jin-Peng Liu
Quantum spectral methods for differential equations
arXiv:1901.00961

417

Alexander Engle, Graeme Smith, and Scott E. Parker
A quantum algorithm for the Vlasov equation
arXiv:1907.09418

418

Shouvanik Chakrabarti, Andrew M. Childs, Tongyang Li, and Xiaodi Wu
Quantum algorithms and lower bounds for convex optimization
arXiv:1809.01731

419

S. Chakrabarti, A. M. Childs, S.-H. Hung, T. Li, C. Wang, and X. Wu
Quantum algorithm for estimating volumes of convex bodies
arXiv:1908.03903

420

Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf
Convex optimization using quantum oracles
arXiv:1809.00643

421

Nai-Hui Chia, Andráas Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, and Chunhao Wang
Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning
Proceedings of STOC 2020, pg. 387-400
[arXiv:1910.06151]

422

Andris Ambainis and Martins Kokainis
Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games
Proceedings of STOC 2017, pg. 989-1002
[arXiv:1704.06774]

423

Fernando G.S L. Brandão, Richard Kueng, Daniel Stilck França
Faster quantum and classical SDP approximations for quadratic binary optimization
arXiv:1909.04613

424

Matthew B. Hastings
Classical and Quantum Algorithms for Tensor Principal Component Analysis
Quantum 4:237, 2020.
[arXiv:1907.12724]

425

Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf
Quantum SDP-Solvers: Better upper and lower bounds
Quantum 4:230, 2020.
[arXiv:1705.01843]

426

J-P Liu, H. Kolden, H. Krovi, N. Loureiro, K. Trivisa, and A. M. Childs
Efficient quantum algorithm for dissipative nonlinear differential equations
arXiv:2011.03185

427

S. Lloyd, G. De Palma, C. Gokler, B. Kiani, Z-W Liu, M. Marvian, F. Tennie, and T. Palmer
Quantum algorithm for nonlinear differential equations
arXiv:2011.06571

428

Yunchao Liu, Srinivasan Arunachalam, and Kristan Temme
A rigorous and robust quantum speed-up in supervised machine learning
arXiv:2010.02174

429

Matthew B. Hastings
The power of adiabatic quantum computation with no sign problem
arXiv:2005.03791

430

Nathan Ramusat and Vincenzo Savona
A quantum algorithm for the direct estimation of the steady state of open quantum systems
arXiv:2008.07133

431

Craig Gidney and Martin Ekera
How to factor 2048 bit RSA integers in 8 hours using 20 million noisy qubits
Quantum 5:433, 2021.
[arXiv:1905.09749]

432

Martin Roetteler, Michael Naehrig, Krysta M. Svore, and Kristin Lauter
Quantum resource estimates for computing elliptic curve discrete logarithms
Proceedings of ASIACRYPT 2017
[arXiv:1706.06752]

433

András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
Proceedings of STOC 2019, pg. 193-204
[arXiv:1806.01838]

434

Dong An, Di Fang, Stephen Jordan, Jin-Peng Liu, Guang Hao Low, and Jiasu Wang
Efficient quantum algorithm for nonlinear reaction-diffusion equations and energy estimation
arXiv:2205.01141, 2022.

435

Pradeep Niroula and Yunseong Nam
A quantum algorithm for string matching
NPJ Quantum Information, 7:37, 2021.

436

András Gilyén, Srininvasan Arunachalam, and Nathan Wiebe
Optimizing quantum optimization algorithms via faster quantum gradient computation
Proceedings SODA 2019, pp. 1425-1444
[arXiv:1711.00465]

437

Arjan Cornelissen
Quantum gradient estimation of Gevrey functions
arXiv:1909.13528, 2019.

438

Pan Gao, Keren Li, Shijie Wei, Jiancun Gao, and Guilu Long
Quantum gradient algorithm for general polynomials
Physical Review A 103:042403, 2021.
[arXiv:2004.11086]

439

Yuxin Zhang and Changpeng Shao
Quantum spectral method for gradient and Hessian estimation
arXiv:2407.03833, 2024.

440

Ryan Babbush, Dominic W. Berry, Robin Kothari, Rolando D. Somma, and Nathan Wiebe
Exponential quantum speedup in simulating coupled classical oscillators
Physical Review X 13:041041, 2024.
[arXiv:2303.13012]

441

Hari Krovi
Improved quantum algorithms for linear and nonlinear differential equations
Quantum 7:913, 2023.
[arXiv:2202.01054]

442

Andrew M. Childs, Jin-Peng Liu, and Aaron Ostrander
High-precision quantum algorithms for partial differential equations
Quantum 5:574, 2021.
[arXiv:2002.07868]

443

Dong An, Noah Linden, Jin-Peng Liu, Ashley Montanaro, Changpeng Shao, and Jiasu Wang
Quantum-accelerated multilevel Monte Carlo methods for stochastic differential equations in mathematical finance
Quantum 5:481, 2021.
[arXiv:2012.06283]

444

G. Xu, A. J. Daley, P. Givi, and R. D. Somma
Turbulent mixing simulation via a quantum algorithm
AIAA Journal 56(2):687-699, 2018.

445

G. Xu, A. J. Daley, P. Givi, and R. D. Somma
Quantum algorithm for the computation of the reactant conversion rate in homogeneous turbulence
Combustion Theory and Modelling 23(6):1090-1104, 2018.

446

Noah Linden, Ashley Montanaro, Changpeng Shao
Quantum vs. classical algorithms for solving the heat equation
Communications in Mathematical Physics 395:601, 2022.
[arXiv:2004.06516]

447

K. Kaneko, K. Miyamoto, N. Takeda, and K. Yoshino
Quantum speedup of Monte Carlo integration in the directino of dimension and its application to finance
Quantum Information Processing 20:185, 2021.
[arXiv:2011.02165]

448

P. Rebentrost, B. Gupt, and T. R. Bromley
Quantum computational finance: Monte Carlo pricing of financial derivatives
Physical Review A 98(2):022321, 2018.
[arXiv:1805.00109]

449

Javier Gonzalez-Conde, Ángel Rodríguez-Rozas, Enrique Solano, and Mikel Sanz
Efficient Hamiltonian simulation for solving option price dynamics
Physical Review Research 5:043220, 2024.
[arXiv:2101.04023]

450

Adam Bouland, Wim van Dam, Hamed Joorati, Iordanis Kerenidis, Anupam Prakash
Prospects and challenges of quantum finance
arXiv:2011.06492, 2020.

451

R. Shaydulin, C. Li, S. Chakrabarti, M. DeCross, D. Herman, N. Kumar, J. Larson, D. Lykov, P. Minssen, Y. Sun, Y. Alexeev, J. M. Dreiling, J. P. Gaebler, T. M. Gatterman, J. A. Gerber, K. Gilmore, D. Gresh, N. Hewitt, C. V. Horst, S. Hu, J. Johansen, M. Matheny, T. Mengle, M. Mills, S. A. Moses, B. Neyenhuis, P. Siegfried, R. Yalovetzky, and M. Pistoia
Evidence of scaling advantage for the quantum approximate optimization algorithm on a classically intractable problem
Science Advances 10(22):eadm6761, 2024.
[arXiv:2308.02342]

452

Joao Basso, Edward Farhi, Kunal Marwaha, Benjamin Villalonga, and Leo Zhou
The Quantum Approximate Optimization Algorithm at high depth for MaxCut on large-girth regular graphs and the Sherrington-Kirkpatrick model
Proceedings of TQC22 7:1-7:21, 2022.
[arXiv:2110.14206]

453

Stephen P. Jordan, Noah Shutty, Mary Wootters, Adam Zalcman, Alexander Schmidhuber, Robbie King, Sergei V. Isakov, and Ryan Babbush
Optimization by Decoded Quantum Interferometry
arXiv:2408.08292, 2024.

454

Alexander Schmidhuber, Ryan O'Donnell, Robin Kothari, Ryan Babbush
Quartic quantum speedups for planted inference
arXiv:2406.19378, 2024.

455

Takashi Yamakawa and Mark Zhandry
Verifiable Quantum Advantage without Structure
Journal of the ACM 71(3):1-50.
[arXiv:2204.02063]

456

Xujie Song, Tong Liu, Shengbo Eben Li, Jingliang Duan, Wenxuan Wang, and Keqiang Li
Training multi-layer neural networks on Ising machine
arXiv:2311.03408.

457

Chi-Fang Chen, Michael J. Kastoryano, Fernando G.S.L. Brandão, András Gilyén
Quantum Thermal State Preparation
arXiv:2303.18224.

458

Dong An, Jin-Peng Liu and Lin Lin
Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost
Physical Review Letters 131(15):150603, 2023.
[arXiv:2303.01029]

459

Guang Hao Low and Yuan Su
Quantum eigenvalue processing
arXiv:2401.06240, 2024.

460

Sergey Bravyi, Anirban Chowdhury, David Gosset, Vojtěch Havlíček, and Guanyu Zhu
Quantum complexity of the Kronecker coefficients
PRX Quantum 5(1):010329, 2023.
[arXiv:2302.11454]

461

Simon Apers and Sander Gribling
Quantum speedups for linear programming via interior point methods
arXiv:2311.03215, 2023.

462

Yanlin Chen, András Gilyén, and Ronald de Wolf
A quantum speed-up for approximating the top eigenvectors of a matrix
arXiv:2405.14765, 2024.

463

Chi-Fang Chen, Alexander M. Dalzell, Mario Berta, Fernando G. S. L. Brandão, and Joel A. Tropp
Sparse random Hamiltonians are quantumly easy
Physical Review X 14(1):011014, 2024.
[arXiv:2302.03394]

464

Stacey Jeffery and Sebastian Zur
Multidimensional quantum walks
Proceedings of STOC23, 1125-1130, 2023.
[arXiv:2208.13492]

465

Robin Kothari and Ryan O'Donnell
Mean estimation when you have the source code; or, quantum Monte Carlo methods
Proceedings of SODA23, 1186-1215, 2023.
[arXiv:2208.07544]

466

Kaoru Mizuta and Keisuke Fujii
Optimal Hamiltonian simulation for time-periodic systems
Quantum, 7:962, 2023.
[arXiv:2209.05048]

467

Dominic W. Berry, Andrew M. Childs, Yuan Su, Xin Wang, and Nathan Wiebe
Time-dependent Hamiltonian simulation with L1-norm scaling
Quantum, 4:254, 2020.
[arXiv:1906.07115]

468

David Poulin, Angie Qarry, Rolando Somma, and Frank Verstraete
Quantum simulation of time-dependent Hamiltonians and the convenient illusion of Hilbert space
Physical Review Letters, 106(17):170501, 2011.
[arXiv:1102.1360]

469

Mária Kieferová, Artur Scherer, and Dominic W. Berry
Simulating the dynamics of time-dependent Hamiltonians with a truncated Dyson series
Physical Review A, 99(4):042314, 2019.
[arXiv:1805.00582]

470

Guang Hao Low and Nathan Wiebe
Hamiltonian simulation in the interaction picture
arXiv:1805.00675, 2018.

471

Arjan Cornelissen and Yassine Hamoudi
A sublinear-time quantum algorithm for approximating partition functions
Proceedings of SODA23, 1245-1264, 2023.
[arXiv:2207.08643]

472

Arjan Cornelissen, Yassine Hamoudi, Sofiene Jerbi
Near-optimal quantum algorithms for multivariate mean estimation
Proceedings of STOC22, 33-43, 2022.
[arXiv:2111.09787]

473

Scott Aaronson and Alex Arkhipov
The computational complexity of linear optics
Proceedings of STOC11, 333-342, 2011.
[arXiv:1011.3245]

474

Dan Shepherd and Michael J. Bremner
Temporally unstructured quantum computation
Proceedings of the Royal Society A, 465(2105):1413-1439, 2009.
[arXiv:0809.0847]

475

Andrew M. Childs, Tongyang Li, Jin-Peng Liu, Chunhao Wang, Ruizhe Zhang
Quantum algorithms for sampling log-concave distributions and estimating normalizing constants
Advances in Neural Information Processing Systems (NeurIPS), 35:23205-23217, 2022.
[arXiv:2210.06539]

476

Sami Boulebnane and Ashley Montanaro
Solving boolean satisfiability problems with the quantum approximate optimization algorithm
arXiv:2208.06909, 2022.

477

Ankit Garg, Robin Kothari, Praneeth Netrapalli, and Suhail Sherif
No quantum speedup over gradient descent for non-smooth convex optimization
arXiv:2010.01801, 2020.

478

Jeongwan Haah, Matthew B. Hastings, Robin Kothari, and Guang Hao Low
Quantum algorithm for simulating real time evolution of lattice Hamiltonians
SIAM Journal on Computing, 52(6):10.1137, 2018.
[arXiv:1801.03922]

479

Andrew M. Childs and Yuan Su
Nearly optimal lattice simulation by product formulas
Physical Review Letters, 123(5):050503, 2019.
[arXiv:1901.00564]

480

Tomotaka Kuwahara, Tan Van Vu, and Keiji Saito
Effective light cone and digital quantum simulation of interacting bosons
Nature Communications, 15:2520, 2024.
[arXiv:2206.14736]

481

Burak Şahinoğlu and Rolando D. Somma
Hamiltonian simulation in the low-energy subspace
npj Quantum Information, 7:119, 2021.
[arXiv:2006.02660]

482

Weiyuan Gong, Shuo Zhou3, and Tongyang Li
Complexity of digital quantum simulation in the low-energy subspace: applications and a lower bound
Quantum, 8:1409, 2024.
[arXiv:2312.08867]

483

Kasra Hejazi, Modjtaba Shokrian Zini, and Juan Miguel Arrazola
Better bounds for low-energy product formulas
arXiv:2402.10362, 2024.

484

Yu Tong, Victor V. Albert, Jarrod R. McClean, John Preskill, and Yuan Su
Provably accurate simulation of gauge theories and bosonic systems
Quantum, 6:816, 2022.
[arXiv:2110.06942]

485

Adam Bouland, Yosheb Getachew, Yujia Jin, Aaron Sidford, and Kevin Tian
Quantum Speedups for zero-sum games via improved dynamic Gibbs sampling
Proceedings of ICML23, 2023.
[arXiv:2301.03763]

486

Joran van Apeldoorn and András Gilyén
Quantum algorithms for zero-sum games
arXiv:1904.03180, 2019.

487

Sam McArdle, András Gilyén, and Mario Berta
A streamlined quantum algorithm for topological data analysis with exponentially fewer qubits
arXiv:2209.12887, 2022.

488

Bernardo Ameneyro, Vasileios Maroulas, and George Siopsis
Quantum persistent homology
Journal of Applied and Computational Topology, 1-20, 2024.
[arXiv:2202.12965]

489

Ryu Hayakawa
Quantum algorithm for persistent Betti numbers and topological data analysis
Quantum, 6:873, 2022.
[arXiv:2111.00433]

490

Dominic W. Berry, Yuan Su, Casper Gyurik, Robbie King, Joao Basso, Alexander Del Toro Barba, Abhishek Rajput, Nathan Wiebe, Vedran Dunjko, and Ryan Babbush
Analyzing prospects for quantum advantage in topological data analysis
PRX Quantum, 5:010319, 2022.
[arXiv:2209.13581]

491

Zoe Holmes, Gopikrishnan Muraleedharan, Rolando D. Somma, Yigit Subasi, and Burak Şahinoğlu
Quantum algorithms from fluctuation theorems: Thermal-state preparation
Quantum, 6:825, 2022.
[arXiv:2203.08882]

492

Alexander M. Dalzell, Nicola Pancotti, Earl T. Campbell, and Fernando G.S.L. Brandão
Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end
Proceedings of STOC23, 1131 - 1144, 2023.
[arXiv:2212.01513]

493

M. B. Hastings
A short path quantum algorithm for exact optimization
Quantum, 2:78, 2018.
[arXiv:1802.10124]

494

Pedro C.S. Costa, Dong An, Yuval R. Sanders, Yuan Su, Ryan Babbush, and Dominic W. Berry
Optimal Scaling Quantum Linear-Systems Solver via Discrete Adiabatic Theorem
PRX Quantum, 3:040303, 2022.
[arXiv:2111.08152]

495

Tobias J. Osborne and Alexander Stottmeister
Quantum simulation of conformal field theory
arXiv:2109.14214, 2021.

496

Changhao Yi and Elizabeth Crosson
Spectral analysis of product formulas for quantum simulation
npj Quantum Information, 8:38, 2022.
[arXiv:2102.12655]

497

Yanlin Chen and Ronald de Wolf
Quantum algorithms and lower bounds for linear regression with norm constraints
arXiv:2110.13086, 2021.

498

Yilei Chen, Qipeng Liu, and Mark Zhandry
Quantum algorithms for variants of average-case lattice problems via filtering
Proceedings of EUROCRYPT22, 372 - 401, 2022.
[arXiv:2108.11015]

499

Xiang Li, Su-Xiang Lyu, Yao Wang, Rui-Xue Xu, Xiao Zheng, and YiJing Yan
Towards Quantum Simulation of Non-Markovian Open Quantum Dynamics: A Universal and Compact Theory
Physical Review A, 110:03620, 2024.
[arXiv:2401.17255]

500

Chi-Fang Chen, Michael J. Kastoryano, and András Gilyén
An efficient and exact noncommutative quantum Gibbs sampler
arXiv:2311.09207, 2023.

501

Peter L. Walters and Fei Wang
Path integral quantum algorithm for simulating non-Markovian quantum dynamics in open quantum systems
Physical Review Research, 6:013135, 2024.

502

Jiaqing Jiang and Sandy Irani
Quantum Metropolis Sampling via Weak Measurement
arXiv:2406.16023, 2024.

503

Matthew Pocrnic, Dvira Segal, and Nathan Wiebe
Quantum Simulation of Lindbladian Dynamics via Repeated Interactions
arXiv:2312.05371, 2023.

504

Mekena Metcalf, Emma Stone, Katherine Klymko, Alexander F Kemper, Mohan Sarovar, and Wibe A de Jong
Quantum Markov chain Monte Carlo with digital dissipative dynamics on quantum computers
Quantum Science and Technology, 7(2):025017, 2022.

505

Dhrumil Patel and Mark M. Wilde
Wave Matrix Lindbladization I: Quantum Programs for Simulating Markovian Dynamics
Open Systems & Information Dynamics, 30(2):2350010, 2023.

506

Dhrumil Patel and Mark M. Wilde
Wave Matrix Lindbladization II: General Lindbladians, Linear Combinations, and Polynomials
Open Systems & Information Dynamics, 30(2):2350014, 2023.

507

Xiantao Li and Chunhao Wang
Succinct Description and Efficient Simulation of Non-Markovian Open Quantum Systems
Communications in Mathematical Physics, 401:147-183, 2023.

508

Bin Yan and Nikolai A. Sinitsyn
Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian
Nature Communications, 13:2212, 2022.

509

Tadashi Kadowaki and Hidetoshi Nishimori
Quantum Annealing in the Transverse Ising Model
Physical Review E, 58:5355, 1998.
[arXiv:cond-mat/9804280]

510

Chris Cade and P. Marcos Crichigno
Complexity of Supersymmetric Systems and the Cohomology Problem
Quantum, 8:1325, 2024.
[arXiv:2107.00011]

511

Alexander Schmidhuber, Michele Reilly, Paolo Zanardi, Seth Lloyd, and Aaron Lauda
A quantum algorithm for Khovanov homology
arXiv:2501.12378, 2025.

512

Rolando D. Somma, Robbie King, Robin Kothari, Thomas O'Brien, and Ryan Babbush
Shadow Hamiltonian Simulation
arXiv:2407.21775, 2024.

513

Maarten Stroeks, Daan Lenterman, Barbara Terhal, Yaroslav Herasymenko
Solving Free Fermion Problems on a Quantum Computer
arXiv:2409.04550, 2024.

514

Alice Barthe, M. Cerezo, Andrew T. Sornborger, Martín Larocca, and Diego García-Martín
Gate-Based Quantum Simulation of Gaussian Bosonic Circuits on Exponentially Many Modes
Physical Review Letters, 134:070604, 2025.
[arXiv:2407.06290]

515

Greta Panova
Polynomial time classical versus quantum algorithms for representation theoretic multiplicities
arXiv:2502.20253, 2025.

516

Martin Larocca, Vojtech Havlicek
Quantum Algorithms for Representation-Theoretic Multiplicities
arXiv:2407.17649, 2024.

517

Dong An and Lin Lin
Quantum Linear System Solver Based on Time-optimal Adiabatic Quantum Computing and Quantum Approximate Optimization Algorithm
ACM Transactions on Quantum Computing, 3(2):1–28, 2022.
[arXiv:1909.05500]

518

Paolo C. S. Costa, Dong An, Yuval R. Sanders, Yuan Su, Ryan Babbush, and Dominic W. Berry
Optimal Scaling Quantum Linear-Systems Solver via Discrete Adiabatic Theorem
PRX Quantum, 3:040303, 2022.
[arXiv:2111.08152]