Constrained Optimal Polynomials for Quantum Linear System Solvers

TL;DR

This paper introduces constrained optimal polynomials for quantum linear system solvers, improving approximation accuracy and noise robustness by leveraging classical Krylov methods and spectral moment reconstruction.

Contribution

It develops two classes of polynomial-based quantum solvers, CUP and CAP, that outperform standard methods under noise and spectral structure considerations.

Findings

CUP achieves robust performance across generic spectra.

CAP further improves accuracy by exploiting spectral structure.

Numerical experiments show lower error than existing QSVT and Chebyshev methods.

Abstract

Quantum linear system solvers typically realize the inverse map as a polynomial transformation of the spectrum, so their practical cost hinges on implementing this transformation at a low polynomial degree. We introduce constrained optimal polynomials as a framework for this task, drawing on classical Krylov subspace theory. Within this framework, we develop two classes of solvers. Constrained Uniform Polynomial (CUP) solvers optimize the tradeoff between approximation accuracy and block encoding normalization under a uniform spectral model consistent with the available bounds. Constrained Adaptive Polynomial (CAP) solvers retain this structure but replace the uniform model with a probability measure reconstructed from spectral moments via a maximum entropy ansatz, where the moments are extracted from QSVT measurements. Numerical experiments under hardware and stochastic noise show that these methods achieve lower error than standard QSVT-based and Chebyshev-iteration-type solvers, particularly in noise-limited regimes. CUP offers robust performance under generic spectra, while CAP provides further improvement when the spectral structure can be exploited.