Constant Factor Analysis of Optimal Quantum Linear Solvers in Practice

TL;DR

This paper compares two quantum linear solver methods, analyzing their constant factors and performance in different scenarios, revealing when each method is more efficient.

Contribution

It provides a comprehensive numerical comparison of the adiabatic and shortcut quantum linear solvers, highlighting their relative efficiencies under various conditions.

Findings

Adiabatic solver performs slightly better when the solution norm is unknown.

Shortcut method outperforms when the solution norm is known for non-Hermitian matrices.

Constant factors are crucial for practical efficiency of quantum linear solvers.

Abstract

Optimal quantum linear equation solvers provide complexity $O(\kappa\log(1/\epsilon))$, where $\kappa$ is the condition number and $\epsilon$ is the allowable error. The optimal solver using a discrete adiabatic approach [PRX Quantum 3, 040303 (2022)] has large analytically proven constant factors for the upper bound on the complexity. The constant factors were later found to be about 1,200 times smaller in numerical testing [Quantum 9, 1887 (2025)]. This meant it is about an order of magnitude more efficient than using a randomised approach from [PRX Quantum 6, 040373 (2025)], which has far smaller analytically proven constant factors. Recently, a ``Shortcut'' method has been found to provide an optimal solver which also has small proven constant factors. In the present work, we conduct a comprehensive numerical analysis comparing this method with the adiabatic solver for two families of random linear systems. We find that, in the case where the solution norm is unknown, the adiabatic solver provides slightly better performance. If the solution norm is known, then the shortcut method provides significantly better performance for non-Hermitian matrices.