Randomized semi-quantum matrix processing

TL;DR

This paper introduces a hybrid quantum-classical method for matrix function simulation that reduces circuit depth and noise sensitivity, leveraging randomization and a modified Hadamard test, with applications in quantum algorithms.

Contribution

It proposes a novel randomized semi-quantum framework that improves early fault-tolerant quantum hardware efficiency over traditional methods.

Findings

Reduces average circuit depth, lowering noise sensitivity.

Achieves quadratic speed-ups in key parameters.

Removes approximation-error dependence in applications.

Abstract

We present a hybrid quantum-classical framework for simulating generic matrix functions more amenable to early fault-tolerant quantum hardware than standard quantum singular-value transformations. The method is based on randomization over the Chebyshev approximation of the target function while keeping the matrix oracle quantum, and is assisted by a variant of the Hadamard test that removes the need for post-selection. The resulting statistical overhead is similar to the fully quantum case and does not incur any circuit depth degradation. On the contrary, the average circuit depth is shown to get smaller, yielding equivalent reductions in noise sensitivity, as explicitly shown for depolarizing noise and coherent errors. We apply our technique to partition-function estimation, linear system solvers, and ground-state energy estimation. For these cases, we prove advantages on average depths, including quadratic speed-ups on costly parameters and even the removal of the approximation-error dependence.