Quantum algorithm for solving linear systems of equations

TL;DR

The paper tackles the problem of estimating statistics of the solution to a linear system $A\vec{x}=\vec{b}$ when $A$ is Hermitian and sparse, focusing on the statistic $\vec{x}^T M\vec{x}$ rather than the full solution. It introduces a quantum algorithm that combines Hamiltonian simulation, phase estimation, and a filter-based non-unitary inversion to prepare a state $|x\rangle$ proportional to $A^{-1}|b\rangle$ and then estimate $\langle x|M|x\rangle$ efficiently. The key contributions are a polylogarithmic-in-$N$ runtime (up to polynomial factors in the condition number $\kappa$ and precision $\epsilon$) and a demonstration of near-optimality: classical algorithms cannot match this scaling in general, and improvements would imply major complexity-theoretic collapses. The results establish quantum linear-algebra techniques that yield exponential speedups in suitable regimes and lay groundwork for extracting solution features via quantum measurements rather than full state reconstruction.

Abstract

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b, find a vector x such that Ax=b. We consider the case where one doesn't need to know the solution x itself, but rather an approximation of the expectation value of some operator associated with x, e.g., x'Mx for some matrix M. In this case, when A is sparse, N by N and has condition number kappa, classical algorithms can find x and estimate x'Mx in O(N sqrt(kappa)) time. Here, we exhibit a quantum algorithm for this task that runs in poly(log N, kappa) time, an exponential improvement over the best classical algorithm.

Theorems & Definitions (10)

• Theorem 1: Error bounds
• Lemma 2
• proof
• Lemma 3
• proof
• Theorem 4
• Theorem 5
• proof
• Theorem 6
• proof