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arXiv:2502.12882 · Physical Review Letters

Classical Algorithms for Estimating Expectation Values in Linear Optical Circuits

Anonymous, Changhun Oh

Published Aug 13, 2026• 0 citations
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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 advantage 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, so 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 [C. Oh et al., Nat. Phys. 20, 225 (2024)10.1038/s41567-023-02308-9], 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.

DOI
10.1103/bl9r-2gyb

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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 advantage 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, so 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 [C. Oh et al., Nat. Phys. 20, 225 (2024)10.1038/s41567-023-02308-9], 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.
Boson Sampling is a prominent candidate for the demonstration of quantum computational advantage but it remains unclear whether a large-scale boson sampler can find useful computational applications. The challenge is that to use a boson sampler to estimate, for example, a physical observable, it is necessary to coarse grain the outcome distribution, owing to the exponential size of the outcome space and the anti-concentration properties of the Boson Sampling distribution. In this work, we analyse the complexity of a specific type of coarse-graining of Boson Sampling distributions based on linear functions of the output photon occupation numbers, which we refer to as linear statistics. We present an efficient classical algorithm for approximating linear statistics of boson samplers within additive error for different kinds of input states, such as Fock states and squeezed states. This allows us to unify in the same framework recent results on efficient quantum-inspired classical algorithms for simulating molecular vibronic spectra, or efficient classical approximations of coarse-grained distributions based on detector binning. Additionally, we show how our algorithm can be used to classically evaluate a proposed one-way function based on Boson Sampling, while other proposals for cryptographic applications escape our classical simulation techniques. We leave open the question of classical simulability of non-linear statistics and connect it to the problem of computing transition amplitudes of linear-optical circuits with one layer of interactions.
Abstract Quasiprobability representations are important tools for analyzing a quantum system, such as a quantum state or a quantum circuit. In this work, we propose classical algorithms specialized for approximating outcome probabilities of a linear optical circuit using quasiprobability distributions. Notably, we can reduce the negativity bound of a circuit from exponential to at most polynomial for specific cases by modulating the shapes of quasiprobability distributions thanks to the symmetry of the linear optical transformation in the phase space. Consequently, our scheme provides an efficient estimation of outcome probabilities within an additive-error whose precision depends on the classicality of the input state. When the classicality is high enough, we reach a polynomial-time estimation algorithm of a probability within a multiplicative-error by an efficient sampling from a log-concave function. By choosing appropriate input states and measurements, our results provide plenty of quantum-inspired classical algorithms for approximating various matrix functions beating best-known results. Moreover, we give sufficient conditions for the classical simulability of Gaussian Boson sampling using our approximating algorithm for any (marginal) outcome probability under the poly-sparse condition.
Classical simulability of quantum circuits plays a central role in characterizing and quantifying quantum computational advantage. Typically, the presence of non-Clifford gates introduces an exponential runtime overhead for classically estimating expectation values of observables. In this work, we introduce an efficient simulation framework that integrates the sum-over-Clifford method with quasi-probability-based stabilizer simulation to directly estimate expectation values. To optimize efficiency, we incorporate a circuit-adaptive reduction via variance estimation protocol, which terminates simulations once an empirical confidence bound is sufficient to guarantee the prescribed error tolerance. Our method circumvents overly conservative Hoeffding's inequality, thereby reducing the dependence on the gate-wise stabilizer extent from quadratic to linear in the low-variance regime, while maintaining the same space complexity as previous approaches. We explicitly prove the sample-complexity reduction for random quantum circuits and T-doped Clifford circuits and also empirically validate these advantages in the quantum approximate optimization algorithm and quantum kernel method.
Abstract Quasiprobability representation is an important tool for analyzing a quantum system, such as a quantum state or a quantum circuit. In this work, we propose classical algorithms specialized for approximating outcome probabilities of a linear optical circuit using sss-parameterized quasiprobability distributions. Notably, we can reduce the negativity bound of a circuit from exponential to at most polynomial for specific cases by modulating the shapes of quasiprobability distributions thanks to the symmetry of the linear optical transformation in the phase space. Consequently, our scheme renders an efficient estimation of outcome probabilities within an additive error whose precision depends on the classicality of the circuit. When the classicality is high enough, we reach a polynomial-time estimation algorithm of a probability within a multiplicative error by an efficient sampling from the log-concave function. By choosing appropriate input states and measurements, our results provide plenty of quantum-inspired classical algorithms for approximating various matrix functions beating best-known results. Moreover, we give sufficient conditions for the classical simulability of Gaussian boson sampling using the approximating algorithm for any (marginal) outcome probability under the poly-sparse condition.
Gaussian building blocks are essential for photonic quantum information processing, and universality can be practically achieved by equipping Gaussian circuits with adaptive measurement and feedforward. The number of adaptive steps then provides a natural parameter for computational power. Rather than assessing power only through sampling problems—the usual benchmark—we follow the ongoing shift toward tasks of practical relevance and study the quantum mean-value problem, i.e., estimating observable expectation values that underpin simulation and variational algorithms. More specifically, we analyze bosonic circuits with adaptivity and prove that when the number of adaptive measurements is small, the mean-value problem admits efficient classical algorithms even if a large amount of non-Gaussian resources are present in the input state, whereas less constrained regimes are computationally hard. This yields a task-level contrast with sampling, where non-Gaussian ingredients alone often induce hardness, and provides a clean complexity boundary parameterized by the number of adaptive measurement-and-feedforward steps between classical simulability and quantum advantage. Beyond the main result, we introduce classical techniques—including a generalization of Gurvits’ second algorithm to arbitrary product inputs and Gaussian circuits—for computing the marginal quantities needed by our estimators, which may be of independent interest.
Gaussian boson sampling (GBS) seeks quantum advantage by sampling photon-number patterns generated with squeezed inputs and passive linear optics. Many proposed GBS applications instead target function computation by applying functions to mode-resolved photon-number outcomes---a natural form of experimental postprocessing that produces classical outputs. Sampling hardness alone, however, does not determine the complexity of these tasks. By analyzing the irreducible decomposition of fixed-photon-number operator spaces, we prove that the expectation value of every such function in the average case over passive linear-optical networks can be classically evaluated for inputs with finite squeezing strength. We also provide a classical algorithm that estimates this value to inverse-polynomial additive error. The result provides new theoretical tools for analyzing linear-optical quantum systems, helps clarify the origin of current GBS hardness evidence, and inspires new applications of GBS with genuine quantum advantages.