Halving the Cost of Quantum Algorithms with Randomization

TL;DR

This paper introduces Stochastic Quantum Signal Processing, integrating randomized compiling with QSP to quadratically reduce error and nearly halve query complexity across various quantum algorithms.

Contribution

It presents a novel method combining randomized compiling with QSP, achieving quadratic error suppression and broad applicability to quantum algorithms.

Findings

Error suppression reduces query complexity by nearly 50%.

Applicable to algorithms for time evolution, phase estimation, and matrix inversion.

Quadratic error reduction enhances quantum algorithm efficiency.

Abstract

Quantum signal processing (QSP) provides a systematic framework for implementing a polynomial transformation of a linear operator, and unifies nearly all known quantum algorithms. In parallel, recent works have developed randomized compiling, a technique that promotes a unitary gate to a quantum channel and enables a quadratic suppression of error (i.e., $\epsilon \rightarrow O(\epsilon^2)$) at little to no overhead. Here we integrate randomized compiling into QSP through Stochastic Quantum Signal Processing. Our algorithm implements a probabilistic mixture of polynomials, strategically chosen so that the average evolution converges to that of a target function, with an error quadratically smaller than that of an equivalent individual polynomial. Because nearly all QSP-based algorithms exhibit query complexities scaling as $O(\log(1/\epsilon))$ -- stemming from a result in functional analysis -- this error suppression reduces their query complexity by a factor that asymptotically approaches $1/2$. By the unifying capabilities of QSP, this reduction extends broadly to quantum algorithms, which we demonstrate on algorithms for real and imaginary time evolution, phase estimation, ground state preparation, and matrix inversion.