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arXiv:2405.01786 · Quantum

On computational complexity and average-case hardness of shallow-depth boson sampling

Byeongseon Go, Changhun Oh, Hyunseok Jeong

Published Mar 13, 2026• 1 citation
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Abstract

Boson Sampling, a computational task believed to be classically hard to simulate, is expected to hold promise for demonstrating quantum computational advantage using near-term quantum devices. However, noise in experimental implementations poses a significant challenge, potentially rendering Boson Sampling classically simulable and compromising its classical intractability. Numerous studies have proposed classical algorithms that can efficiently simulate Boson Sampling under various noise models, particularly as noise rates increase with circuit depth. To address this challenge, we investigate the viability of achieving quantum computational advantage through Boson Sampling implemented with shallow-depth linear optical circuits. In particular, as the average-case hardness of estimating output probabilities of Boson Sampling is a crucial ingredient in demonstrating its classical intractability, we make progress on establishing the average-case hardness of Boson Sampling confined to logarithmic-depth regimes. We also obtain the average-case hardness for logarithmic-depth Fock-state Boson Sampling subject to lossy environments and for the logarithmic-depth Gaussian Boson Sampling. By providing complexity-theoretical backgrounds for the classical simulation hardness of logarithmic-depth Boson Sampling, we expect that our findings will mark a crucial step towards a more noise-tolerant demonstration of quantum advantage with shallow-depth Boson Sampling.

DOI
10.22331/q-2026-03-13-2026

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As a promising candidate for exhibiting quantum computational supremacy, Gaussian boson sampling (GBS) is designed to exploit the ease of experimental preparation of Gaussian states. However, sufficiently large and inevitable experimental noise might render GBS classically simulable. In this work, we formalize this intuition by establishing a sufficient condition for approximate polynomial-time classical simulation of noisy GBS-in the form of an inequality between the input squeezing parameter, the overall transmission rate, and the quality of photon detectors. Our result serves as a nonclassicality test that must be passed by any quantum computational supremacy demonstration based on GBS. We show that, for most linear-optical architectures, where photon loss increases exponentially with the circuit depth, noisy GBS loses its quantum advantage in the asymptotic limit. Our results thus delineate intermediate-sized regimes where GBS devices might considerably outperform classical computers for modest noise levels. Finally, we find that increasing the amount of input squeezing is helpful to evade our classical simulation algorithm, which suggests a potential route to mitigate photon loss.