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arXiv:2610.06526

Classical Simulation of Lossy Gaussian Boson Sampling beyond Square-Root Regime

Youngrong Lim

quant-phSubmitted Oct 5, 2026• Announced Oct 6, 2026• 0 citations
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Abstract

Photon loss makes Gaussian boson sampling easier to simulate classically. A standard approach replaces the lossy input by classical light, and it stays accurate while the mean number of surviving photons grows no faster than the square root of the number $N$ of squeezed inputs at fixed squeezing and accuracy. We prove that this scale is a limit of replacement by classical light. For explicit interferometers, no state with a nonnegative Glauber--Sudarshan $P$ representation, Gaussian or not and however correlated, keeps the error in the photon count distribution small beyond this scale. We then surpass this limit by approximating the detection instead of the input. The sampler evolves the state exactly between detections and, after each detection, replaces the conditional state by a Gaussian with the same covariance. For every interferometer its total variation error is $O(Nη^3)$ at polynomial expected cost, where $η$ is the photon survival probability. The mean surviving photon number can therefore grow as $N^{2/3}$ at fixed accuracy. A separate pair construction reaches $N^{3/4}$ when every input mode is equally squeezed.

Comments
6+12 pages, 2 figures

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Boson sampling experiments aim to demonstrate quantum computational advantage with increasing numbers of interfering photons. A central question is whether their computational hardness survives under photon loss. We show that even an arbitrarily small nonzero uniform loss rate permits efficient classical sampling in the average case when the rate remains fixed as the system grows. We exploit how loss suppresses correlations involving many photons to approximate the full output distribution using low correlation orders. The resulting samplers run in time polynomial in system size and inverse total variation error, with the error averaged over Haar random interferometers. The results apply to Fock state boson sampling with single-photon states in distinct input modes and arbitrary additional vacuum inputs, and to Gaussian boson sampling with equal squeezing of bounded strength in every input mode. Classical sampling is efficient even when the mean detected photon number grows linearly with system size, showing that large photon counts alone do not ensure computational hardness.
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Gaussian boson sampling (GBS) is a sampling task proposed to demonstrate quantum advantage. We consider Gaussian boson sampling on MMM optical modes, with KKK equally squeezed input modes and NNN observed photon counts. We complete the proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezers KKK, which is a part of the argument for classical hardness of GBS. In particular, we show that for any KKK and N=o(K)N=o(\sqrt{K})N=o(K​), the symmetric product MK−1/2UNKUNKTMK^{-1/2}U_{NK}U_{NK}^TMK−1/2UNK​UNKT​, for UNKU_{NK}UNK​ the top left N×KN\times KN×K submatrix of an M×MM\times MM×M Haar random unitary UUU, is close in total variation distance to both an N×NN\times NN×N symmetric complex Gaussian matrix G\mathbf GG with independent entries, and the symmetric product GGT/KGG^T/\sqrt{K}GGT/K​ for GGG an N×KN\times KN×K matrix of iid standard complex Gaussians. We show however that the density-based instance generating method of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013), Lemma 5.8] used to efficiently implement a hiding procedure fails for Gaussian boson sampling with K=cMK=cMK=cM if c<1/2c<1/2c<1/2. Instead we use approximate instance generating to implement the hiding for the usual classical hardness reduction.
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We analyze the impact of photon loss on the photon-number statistics of Gaussian states. Specifically, we propose and carefully evaluate several methods to mitigate deviations in the photon-number distributions of lossy (displaced) squeezed vacuum states from those of their lossless counterparts. These methods rely on appropriately redefining the parameters of Gaussian states when the loss budget is known in order to recover, as closely as possible, the desired photon-number distribution associated with each target state. While it is intrinsically hard to directly optimize the photon-number distribution of high-dimensional, correlated multimode Gaussian states, the proposed methods are instead based on optimizing specific key properties such as fidelity, phase-space functions, low-order moments of the underlying photon-number statistics, or overlap with the vacuum state. In particular, our results show that optimizing the fidelity between a pure Gaussian target state and a modified Gaussian state that has passed through a loss channel does typically not result in closeness of the corresponding photon-number distributions. Furthermore, we show that correcting for the vacuum overlap minimizes the deviation in the photon-number distribution for large parameter ranges which we explicitly prove for single-mode squeezed vacuum and provide numerical evidence for general (displaced) squeezed vacuum states. As photon loss is a key limitation for Gaussian boson sampling, our results provide insights into the feasibility and limitations of such photonic quantum simulations in lossy environments and offer guidelines for mitigating these imperfections.