Efficient quantum algorithms for solving quantum linear system problems

TL;DR

This paper introduces two efficient quantum algorithms for solving linear systems by transforming the problem into finding a specific singular vector, achieving optimal query complexity and simplicity over previous methods.

Contribution

The paper presents two novel quantum algorithms for linear system solving that are simpler and achieve optimal query complexity in condition number, improving upon prior approaches.

Findings

Both algorithms achieve optimal query complexity in the condition number.

The first algorithm uses quantum eigenstate filtering with $O(s\,\kappa\,\log(1/\epsilon))$ complexity.

The second algorithm employs quantum resonant transition with $O(s\kappa + \log(1/\epsilon)/\log\log(1/\epsilon))$ complexity.

Abstract

We transform the problem of solving linear system of equations $A\mathbf{x}=\mathbf{b}$ to a problem of finding the right singular vector with singular value zero of an augmented matrix $C$, and present two quantum algorithms for solving this problem. The first algorithm solves the problem directly by applying the quantum eigenstate filtering algorithm with query complexity of $O\left( s\kappa \log \left( 1/\epsilon \right) \right) $ for a $s$-sparse matrix $C$, where $\kappa $ is the condition number of the matrix $A$, and $\epsilon $ is the desired precision. The second algorithm uses the quantum resonant transition approach, the query complexity scales as $O\left[s\kappa + \log\left( 1/\epsilon \right)/\log \log \left( 1/\epsilon \right) \right] $. Both algorithms meet the optimal query complexity in $\kappa $, and are simpler than previous algorithms.