Prospects of Quantum Error Mitigation for Quantum Signal Processing

TL;DR

This paper investigates the effectiveness of quantum error mitigation, specifically zero-noise-extrapolation, on quantum signal processing algorithms for Hamiltonian simulation under depolarizing noise, highlighting regimes where QEM is practical.

Contribution

It provides an analysis of QEM performance on QSP-based Hamiltonian simulation, identifying noise and depth regimes where error mitigation yields useful results.

Findings

QEM can recover noiseless expectation values in certain noise regimes.

Performance depends on noise level and circuit depth.

ZNE may be ineffective beyond specific noise thresholds.

Abstract

Quantum error mitigation (QEM) protocols have provably exponential bounds on the cost scaling; however, exploring which regimes QEM can recover usable results is still of sizable interest. The expected absence of complete error correction for near-term and intermediate-term quantum devices means that QEM protocols will remain relevant for devices with low enough error rates to attempt small examples of fault-tolerant algorithms. Herein, we are interested in the performance of QEM with a template for quantum algorithms, quantum signal processing (QSP). QSP-based algorithms are designed with an especially simple relation between the circuit depths and the algorithm's parameters, including the required precision. Hence, they may be a useful playground for exploring QEM's practical performance and costs for a range of controlled parameters. As a preliminary step in this direction, this work explores the performance of zero-noise-extrapolation (ZNE) on a Hamiltonian simulation algorithm designed within QSP under local depolarizing noise. We design a QSP-based Hamiltonian simulation of a modified Ising model under depolarizing noise for low precision and varying simulation times. We quantify for which noise and depth regimes our ZNE protocol can recover an approximation of the noiseless expectation value. We discuss existing bounds on the sample budget, eventually using a fixed number of shots. While this does not guarantee the success of QEM, it gives us usable results in relevant cases. Finally, we briefly discuss and present a numerical study on the region where ZNE is unusable, even given an unlimited sample budget.