Quantum algorithm for linear matrix equations

TL;DR

This paper presents a quantum algorithm that efficiently solves the Sylvester matrix equation, enabling faster computation of matrix entries than classical methods, with potential applications in control and physics.

Contribution

The authors develop a quantum algorithm for the Sylvester equation that constructs the solution as a block-encoded matrix, achieving exponential speedups in certain properties of the solution.

Findings

Quantum algorithm constructs solution matrix efficiently

Complexity is nearly linear in condition number and logarithmic in dimension

Applicable to solving BQP-complete problems

Abstract

We describe an efficient quantum algorithm for solving the linear matrix equation AX+XB=C, where A, B, and C are given complex matrices and X is unknown. This is known as the Sylvester equation, a fundamental equation with applications in control theory and physics. Our approach constructs the solution matrix X/x in a block-encoding, where x is a rescaling factor needed for normalization. This allows us to obtain certain properties of the entries of X exponentially faster than would be possible from preparing X as a quantum state. The query and gate complexities of the quantum circuit that implements this block-encoding are almost linear in a condition number that depends on A and B, and depend logarithmically in the dimension and inverse error. We show how our quantum circuits can solve BQP-complete problems efficiently, discuss potential applications and extensions of our approach, its connection to Riccati equation, and comment on open problems.