State preparation by shallow circuits using feed forward

Quantum 8, 1552 (2024).

https://doi.org/10.22331/q-2024-12-09-1552

Fault tolerant quantum computers repetitively apply a four-step procedure: First, perform a few one and two-qubit quantum gates. Second, perform a syndrome measurement on a subset of the qubits. Third, perform fast classical computations to establish if and where errors occurred. And, fourth, correct the errors with a correction step. The next iteration applies the same procedure with new one and two-qubit gates. Even though current error-rates prohibit this procedure to work and fault tolerant quantum computing remains a distant goal, the same procedure can already prove useful today. In this work we make use of this four-step scheme not to carry out fault-tolerant computations, but to enhance short, $constant$-depth, quantum circuits that perform 1 qubit gates and $nearest-neighbor$ 2 qubit gates.

We introduce a new computational model called $textit{Local Alternating Quantum Classical Computations}$ $textsf{(LAQCC)}$. In this model, qubits are placed in a grid and they can only interact with their direct neighbors; the quantum circuits are of constant depth with intermediate measurements; a classical controller can perform log-depth computations on these intermediate measurement outcomes and control future quantum operations based on the outcome. This model fits naturally between quantum algorithms in the NISQ era and full-fledged fault-tolerant quantum computation. We first prove that any Clifford circuit has an equivalent $textsf{LAQCC}$ circuit, and that any $textsf{LAQCC}$ circuit can be simulated by a $mathsf{QNC^1}$circuit. Next, we conjecture the non-simulatability of $textsf{LAQCC}$ by showing that $textsf{LAQCC}$ contains the class of Instantaneous Quantum Polynomial-time circuits. We also show that any $textsf{LAQCC}$ circuit with polynomial-sized quantum circuits and unbounded classical computations is contained in the class of quantum circuits equipped with post-selection gates with respect to the task of state preparation. We continue by presenting $textsf{LAQCC}$ implementations of different subroutines, including OR-gates, quantum Fourier transforms and Threshold gates. These subroutines prove vital in constructing three state preparation routines in the main part of this work. Preparing a uniform superposition uses constant-depth arithmetic gates, combined with an exact Grover implementation by Long. For the $W$-state, we employ a compress-uncompress method to switch between a binary and one-hot encoding. This method extends to the more generalized Dicke-states, the superposition of $n$-bit strings of Hamming weight $k$, for $k=mathcal{O}(sqrt{n})$, but fails for higher $k$ due to the birthday paradox. We extend this protocol to a protocol that prepares many-body scar states, highly excited states with low entanglement and longer coherence times than states with the same energy density. We present a circuit for preparing Dicke-states for larger $k$ requiring log-depth circuits that maps between the factoradic number system and the combinatorial number system.