Tight Quantum Depth Lower Bound for Solving Systems of Linear Equations

TL;DR

This paper establishes a fundamental lower bound on the depth of quantum algorithms for solving linear systems, showing that parallelization cannot significantly reduce the depth beyond a certain limit.

Contribution

It proves a tight lower bound on the quantum query depth for solving linear systems, highlighting inherent limitations of parallel quantum algorithms.

Findings

Quantum algorithms have a depth lower bound of Ω(κ) for linear systems.

Parallelism does not substantially reduce the quantum query depth.

The result is tight up to a constant factor.

Abstract

Since Harrow, Hassidim, and Lloyd (2009) showed that a system of linear equations with $N$ variables and condition number $\kappa$ can be solved on a quantum computer in $\operatorname{poly}(\log(N), \kappa)$ time, exponentially faster than any classical algorithms, its improvements and applications have been extensively investigated. The state-of-the-art quantum algorithm for this problem is due to Costa, An, Sanders, Su, Babbush, and Berry (2022), with optimal query complexity $\Theta(\kappa)$. An important question left is whether parallelism can bring further optimization. In this paper, we study the limitation of parallel quantum computing on this problem. We show that any quantum algorithm for solving systems of linear equations with time complexity $\operatorname{poly}(\log(N), \kappa)$ has a lower bound of $\Omega(\kappa)$ on the depth of queries, which is tight up to a constant factor.