Classical algorithms for estimating expectation values in linear-optical circuits

TL;DR

This paper introduces classical algorithms for efficiently approximating expectation values and output probabilities in linear-optical circuits, challenging claims of quantum advantage in photonic systems.

Contribution

It presents novel classical algorithms for estimating expectation values, transition amplitudes, and related matrix functions in linear-optical circuits, including near-Clifford circuits.

Findings

Photonic variational algorithms may not achieve quantum supremacy due to classical simulability.

Boson sampling with polynomially sparse output distributions can be efficiently simulated classically.

The algorithms enable efficient approximation of transition amplitudes and matrix functions like the hafnian.

Abstract

We present a classical algorithm for approximating the expectation values of observables in linear-optical circuits with arbitrary product input states, achieving additive-error accuracy. This result indicates that current applications of photonic systems aimed at demonstrating practical quantum supremacy through expectation value estimation, such as photonic variational algorithms, may face challenges in attaining the computational advantage. It also implies the output probabilities of boson sampling with arbitrary product input states can be efficiently approximated by our method, resulting that boson sampling becomes efficiently simulable when its output probability distribution is polynomially sparse. We also develop an efficient classical algorithm for estimating transition amplitudes of arbitrary product states in linear-optical circuits. This provides additive-error approximation algorithms for matrix functions associated with linear-optical circuits, such as the (loop-)hafnian, which are of independent interest. As an application, it solves the generalized molecular vibronic spectra problem (Oh et al., 2024), previously suggested as a candidate for practical quantum advantage. Finally, we extend our framework to near-Clifford circuits, enabling classical approximation of their expectation values.