Classical algorithms for estimating expectation values in linear-optical circuits

TL;DR

This work presents a classical algorithm for approximating the expectation values of observables in linear-optical circuits with arbitrary product input states, achieving additive-error accuracy, and extends the framework to near-Clifford circuits, enabling classical approximation of their expectation values.

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.

Figures (1)

• Figure 1: Summary of classical algorithms introduced in the main text. (a) A schematic of the general expectation value of an $M$-mode linear-optical circuit (Theorem \ref{['th:exp']}). If $\hat{O}_i$ are photon-number projectors $|m_i\rangle \langle m_i|_i$, the expectation values represent marginal output probabilities of a boson sampling circuit, resulting in an efficient simulation of a sparse boson sampling (Corollary \ref{['co:sparse']}). (b) A transition amplitude of a linear-optical circuit (Theorem \ref{['th:ggurvits']}). When $|\phi_i \rangle$'s are single-photon Fock states and $|\psi_i\rangle$'s are squeezed vacuum states, the amplitude is proportional to the hafnian of a general complex symmetric matrix (Corollary \ref{['co:haf']}). A transition amplitude is reduced to a phase shifter expectation value, Eq. \ref{['eq:fourier']}, with $|\phi_i\rangle=|\psi_i\rangle$ and $\hat{U} \rightarrow \hat{U}^\dagger e^{i\hat{\bm{n}}\cdot \bm{\phi}}\hat{U}$ (Corollary \ref{['co:LCBS']}).

Theorems & Definitions (9)

• Theorem 1: Expectation-value approximation in linear-optical circuit
• proof : Proof sketch
• Corollary 1: Sparse boson sampling
• Theorem 2: Transition amplitude approximation in linear-optical circuit
• proof : Proof Sketch
• Corollary 2: Hafnian approximation
• Corollary 3: Phase-shifter expectation value
• Theorem 3: Expectation value approximation in near-Clifford circuit
• proof : Proof Sketch