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Complexity-Theoretic Foundations of BosonSampling with a Linear Number of Modes

Adam Bouland1, Daniel Brod2, Ishaun Datta1, Bill Fefferman3, Daniel Grier4,5, Felipe Hernández6, and Michał Oszmaniec7

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Phys. Rev. X 16, 021059 – Published 24 June, 2026

DOI: https://doi.org/10.1103/xc7b-sjm5

Abstract

BosonSampling is the leading candidate for demonstrating quantum computational advantage in photonic systems. While we have recently seen many impressive experimental demonstrations, there is still a formidable distance between the complexity-theoretic hardness arguments and current experiments. One of the largest gaps involves the ratio of particles to modes—all current hardness evidence assumes a “dilute” regime in which the number of linear optical modes scales at least quadratically in the number of particles. By contrast, current experiments operate in a “saturated” regime with a linear number of modes. In this paper, we bridge this gap, bringing the hardness evidence for experiments in the saturated regime to the same level as had been previously established for the dilute regime. This involves proving a worst-to-average-case reduction for computing the permanent, which is robust both to large numbers of row repetitions and also to distributions over matrices with correlated entries. We also apply similar arguments to give evidence for hardness of Gaussian BosonSampling in the saturated regime.

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Popular Summary

Classical simulation of photonic quantum systems remains a significant challenge, particularly for experiments operating in the “saturated” regime where the number of particles is comparable to the number of optical modes. We bridge the gap between complexity theory and current experiments by proving that BosonSampling in this linear-mode regime is just as hard to simulate classically as the previously established “dilute” regime. This was achieved by developing a new worst-to-average-case reduction for the permanent function that remains robust despite photon collisions and correlated matrix entries. We identify that the number of detector clicks, rather than total photons, serves as the primary measure of computational hardness in these systems. These results reinforce the theoretical foundations for achieving quantum computational advantage and provide realistic benchmarks for scaling up photonic processors in the presence of optical losses.

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Outline

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Phys. Rev. X 16, 021059– Published 24 June, 2026
  • Received 2 April 2025
  • Revised 10 December 2025
  • Accepted 5 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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