Block encoding of sparse matrices with a periodic diagonal structure

TL;DR

This paper introduces an efficient quantum circuit for block encoding sparse matrices with a periodic diagonal structure, enabling improved quantum algorithms for differential equations with polynomial or linear gate complexity.

Contribution

It provides a novel quantum circuit for block encoding such matrices using the LCU framework, with significant computational advantages over general methods.

Findings

Polynomial gate complexity for banded matrices

Linear gate complexity for simple diagonal matrices

Validated through numerical simulations

Abstract

Block encoding is a successful technique used in several powerful quantum algorithms. In this work we provide an explicit quantum circuit for block encoding a sparse matrix with a periodic diagonal structure. The proposed methodology is based on the linear combination of unitaries (LCU) framework and on an efficient unitary operator used to project the complex exponential at a frequency $\omega$ multiplied by the computational basis into its real and imaginary components. We demonstrate a distinct computational advantage with a $\mathcal{O}(\text{poly}(n))$ gate complexity, where $n$ is the number of qubits, in the worst-case scenario used for banded matrices, and $\mathcal{O}(n)$ when dealing with a simple diagonal matrix, compared to the exponential scaling of general-purpose methods for dense matrices. Various applications for the presented methodology are discussed in the context of solving differential problems such as the advection-diffusion-reaction (ADR) dynamics, using quantum algorithms with optimal scaling, e.g., quantum singular value transformation (QSVT). Numerical results are used to validate the analytical formulation.