Bidirectional Path Integral Monte Carlo Simulation of Quantum Circuits
Abstract
We propose an approximate classical algorithm for estimating quantum circuits' transition amplitudes, ⟨ Ψf | Û | 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, Ψ0 and Ψf, 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.
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BibTeX
@misc{santos2026bidirectional,
title = {{Bidirectional Path Integral Monte Carlo Simulation of Quantum Circuits}},
author = {Luis Paulo Santos and Thomas Bashford-Rogers},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27768},
eprint = {2610.27768},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27768},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bidirectional Path Integral Monte Carlo Simulation of Quantum Circuits
AU - Luis Paulo Santos
AU - Thomas Bashford-Rogers
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27768
UR - https://arxiv.org/abs/2610.27768
AB - We propose an approximate classical algorithm for estimating quantum circuits' transition amplitudes, ⟨ Ψ_f | Û | Ψ_0 ⟩, 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, Ψ_0 and Ψ_f, 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.
ER -