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arXiv:2510.26408 · Journal of the Optical Society of America B

Feynman path sum approach for simulation of linear optics

W. F. Balthazar, Quinn Palmer, Alex E. Jones, Jacob F. F. Bulmer, Ernesto F. Galvão

Published Jul 15, 2026• 0 citations
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Abstract

The Feynman path integral formalism has inspired the development of memory-efficient and parallelizable classical algorithms for simulating quantum computers. We adapt this approach for the calculation of probability amplitudes of linear-optical boson sampling experiments, which involve Fock-state inputs, linear optical circuits, and photo-detection at the output. We describe this simulation method and compare it with alternative approaches. Additionally, we implement a linear-optical Feynman path simulator in open-source C code, enhancing its performance using tensor contraction techniques. Our method is benchmarked for low-depth linear optical circuits, where it offers advantages in runtime and memory efficiency.

DOI
10.1364/josab.599855

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The Feynman path integral formalism has inspired the development of memory-efficient and parallelizable classical algorithms for simulating quantum computers. We adapt this approach for the calculation of probability amplitudes of linear-optical boson sampling experiments, which involve Fock-state inputs, linear optical circuits, and photo-detection at the output. We describe this simulation method and compare it with alternative approaches. Additionally, we implement a linear-optical Feynman path simulator in open-source C code, enhancing its performance using tensor contraction techniques. Our method is benchmarked for low-depth linear optical circuits, where it offers advantages in runtime and memory efficiency.
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We propose an approximate classical algorithm for estimating quantum circuits' transition amplitudes, ⟨Ψf∣U^∣Ψ0⟩\langle Ψ_f | \hat{U} | Ψ_0 \rangle⟨Ψf​∣U^∣Ψ0​⟩, based on Monte Carlo sums over bidirectionally sampled Feynman paths. The algorithm generates Feynman paths, i.e., sequences of intermediate computational basis states, by combining information from random walks initiated at both the input and output states, Ψ0Ψ_0Ψ0​ and ΨfΨ_fΨf​, respectively. It further combines this bidirectional approach with Multiple Importance Sampling, a powerful variance reduction technique that avoids variance peaks caused by divisions by the very low sampling probabilities associated with hard to reach paths. Empirical data demonstrate the superior convergence rate exhibited by the bidirectional algorithm over a unidirectional, forward-only sampling approach. This superiority is due to the effectiveness of the proposed algorithm in finding and weighting paths with non-zero amplitude in an extremely sparse path space. Sum of paths algorithms exhibit very favourable space complexity. Monte Carlo replaces exhaustive enumeration of the exponentially large path space by stochastic sampling, the computational cost being determined by the variance of the estimator. This work demonstrates these two features by presenting results on very large quantum circuits, with up to 4096 qubits.
We propose an approximate classical algorithm for estimating quantum circuits' transition amplitudes, ⟨Ψf∣U^∣Ψ0⟩\langle Ψ_f | \hat{U} | Ψ_0 \rangle⟨Ψf​∣U^∣Ψ0​⟩, based on Monte Carlo sums over bidirectionally sampled Feynman paths. The algorithm generates Feynman paths, i.e., sequences of intermediate computational basis states, by combining information from random walks initiated at both the input and output states, Ψ0Ψ_0Ψ0​ and ΨfΨ_fΨf​, respectively. It further combines this bidirectional approach with Multiple Importance Sampling, a powerful variance reduction technique that avoids variance peaks caused by divisions by the very low sampling probabilities associated with hard to reach paths. Empirical data demonstrate the superior convergence rate exhibited by the bidirectional algorithm over a unidirectional, forward-only sampling approach. This superiority is due to the effectiveness of the proposed algorithm in finding and weighting paths with non-zero amplitude in an extremely sparse path space. Sum of paths algorithms exhibit very favourable space complexity. Monte Carlo replaces exhaustive enumeration of the exponentially large path space by stochastic sampling, the computational cost being determined by the variance of the estimator. This work demonstrates these two features by presenting results on very large quantum circuits, with up to 4096 qubits.
Sampling from probability distributions of quantum circuits is a fundamentally and practically important task which can be used to demonstrate quantum supremacy using noisy intermediate-scale quantum devices. In the present work, we examine classical simulability of sampling from the output photon-number distribution of linear-optical circuits composed of random beam splitters with equally distributed squeezed vacuum states and single-photon states input. We provide efficient classical algorithms to simulate linear-optical random circuits and show that the algorithms' error is exponentially small up to a depth less than quadratic in the distance between sources using a classical random walk behavior of random linear-optical circuits. Notably, the average-case depth allowing an efficient classical simulation is larger than the worst-case depth limit, which is linear in the distance. Besides, our results together with the hardness of boson sampling give a lower-bound on the depth for constituting global Haar-random unitary circuits.
Classical simulation of quantum computation has often been viewed as the method to determine where the horizon of quantum supremacy is located---that is, where quantum computation can no longer be simulated by classical methods. As of now, the 50 qubit threshold for quantum supremacy has been determined largely by the state vector simulation method's exponential space demands placing an upper bound on simulation memory capabilities. To investigate this claim, we present and test an implementation of a known path integral simulation algorithm running in linear space; the method is based on recursively traversing the underlying computation tree for quantum algorithms and summing over possible amplitudes. We find that the implementation is able to simulate the hidden subgroup method (HSP) standard method---a notable class of circuits including Shor's algorithm amongst others---in a reasonable amount of time using extremely low memory, as well as other circuits with similar parameters. The performance results of this algorithm suggest that it can serve as a feasible alternative to state vector simulation and that with respect to the HSP standard method, quantum supremacy may be more accurately measured using the recursive path-summing method on large numbers of qubits, compared to the state vector method.