Dictionary-based Block Encoding of Sparse Matrices with Low Subnormalization and Circuit Depth

Quantum 9, 1805 (2025).

https://doi.org/10.22331/q-2025-07-22-1805

Block encoding severs as an important data input model in quantum algorithms, enabling quantum computers to simulate non-unitary operators effectively. In this paper, we propose an efficient block-encoding protocol for sparse matrices based on a novel data structure, called the dictionary data structure, which classifies all non-zero elements according to their values and indices. Non-zero elements with the same values, lacking common column and row indices, belong to the same classification in our block-encoding protocol’s dictionary. When compiled into the $textit{{U(2), CNOT}}$ gate set, the protocol queries a $2^n times 2^n$ sparse matrix with $s$ non-zero elements at a circuit depth of $mathcal{O}(log(ns))$, utilizing $mathcal{O}(n^2s)$ ancillary qubits. This offers an exponential improvement in circuit depth relative to the number of system qubits, compared to existing methods [1,2] with a circuit depth of $mathcal{O}(n)$. Moreover, in our protocol, the subnormalization, a scaled factor that influences the measurement probability of ancillary qubits, is minimized to $sum_{l=0}^{s_0}vert A_lvert$, where $s_0$ denotes the number of classifications in the dictionary and $A_l$ represents the value of the $l$-th classification. Furthermore, we show that our protocol connects to linear combinations of unitaries $(LCU)$ and the sparse access input model $(SAIM)$. To demonstrate the practical utility of our approach, we provide several applications, including Laplacian matrices in graph problems and discrete differential operators.