Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2604.14323

Boson sampling beyond the dilute regime: second moments and anti-concentration

Hela Mhiri, Hugo Thomas, Léo Monbroussou, Ulysse Chabaud, Zoë Holmes, Elham Kashefi

quant-phSubmitted Apr 15, 2026• Announced Apr 17, 2026• 0 citations
arXivHTMLPDF

Abstract

Boson sampling is a leading candidate for demonstrating quantum advantage in photonic systems. Despite significant experimental and theoretical progress, a characterization of its output statistics remains incomplete. This is especially true beyond the dilute regime, where photon collisions and bunching become significant. The associated saturated regime, characterized by mode number growing linearly with photon number, or more generally sub-quadratically, is precisely the regime of greatest experimental interest. As a consequence, anti-concentration of the output distribution--a key ingredient in hardness arguments--remains poorly understood in boson sampling. In this work, we leverage representation-theoretic tools to address this gap, obtaining closed-form expressions for second moments of generic particle-number-preserving bosonic observables. We express these quantities in terms of Hilbert-Schmidt norms of projections onto irreducible components of the operator space and show that these projection norms admit compact analytical expressions by exploiting the underlying symmetry structure. Focusing on Fock state output probabilities, we further establish anti-concentration beyond the dilute regime. Together with recent complexity-theoretic results, our findings strengthen hardness guarantees for boson sampling in experimentally interesting settings.

Related papers

Transition of Anticoncentration in Gaussian Boson Sampling

Adam Ehrenberg, Joseph T. Iosue, Abhinav Deshpande +2 more

Gaussian boson sampling is a promising method for experimental demonstrations of quantum advantage because it is easier to implement than other comparable schemes. While most of the properties of Gaussian boson sampling are understood to the same degree as for these other schemes, we understand relatively little about the statistical properties of its output distribution. The most relevant statistical property, from the perspective of demonstrating quantum advantage, is the "anticoncentration" of the output distribution as measured by its second moment. The degree of anticoncentration features in arguments for the complexity-theoretic hardness of Gaussian boson sampling. In this Letter, we develop a graph-theoretic framework for analyzing the moments of the Gaussian boson sampling distribution. Using this framework, we show that Gaussian boson sampling undergoes a transition in anticoncentration as a function of the number of modes that are initially squeezed compared to the number of photons measured at the end of the circuit. When the number of initially squeezed modes scales sufficiently slowly with the number of photons, there is a lack of anticoncentration. However, if the number of initially squeezed modes scales quickly enough, the output probabilities anticoncentrate weakly.

arXiv:2312.08433•Physical Review Letters•Apr 8, 2025•3 citations

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

Adam Bouland, Daniel J. Brod, Ishaun Datta +4 more

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.

arXiv:2312.00286•Physical Review X•Mar 5, 2026•5 citations

General framework for anticoncentration and linear cross-entropy benchmarking in photonic quantum advantage experiments

Zoltán Kolarovszki, Ágoston Kaposi, Zoltán Zimborás +1 more

Photonic architectures are one of the leading platforms for demonstrating quantum computational advantage, with Boson Sampling and Gaussian Boson Sampling as the primary schemes. Yet, we lack for these photonic primitives a systematic theoretical understanding of linear cross-entropy benchmarking (LXEB), which is a central tool for testing quantum advantage proposals. In this work, we develop a representation-theoretic framework for the classical computation of average LXEB scores and second moments of output probability distributions, covering a range of quantum advantage experiments based on scattering $n$-photon states through $m$-mode Haar-random interferometers. Our methods apply in any regime, including the saturated regime, where the (expected) number of photons is comparable to the number of optical modes. The same second-moment techniques also allow us to prove anticoncentration for traditional Fock-state Boson Sampling in the saturated regime. Interestingly, for Gaussian Boson Sampling second moments are not sufficient to establish a meaningful anticoncentration statement. The technical core of our approach rests on decomposing two copies of the $n$-particle bosonic space $\mathrm{Sym}^n(\mathbb{C}^m)$ into irreducible representations of $\mathrm{U}(m)$. This reduces two-copy Haar averages to computing purities of initial states after partial traces over particles, highlighting the role that particle entanglement plays for LXEB and anticoncentration.

quant-pharXiv:2604.15258•Apr 17, 2026•0 citations

The Complexity of Bipartite Gaussian Boson Sampling

Daniel Grier, Daniel J. Brod, Juan Miguel Arrazola +2 more

Gaussian boson sampling is a model of photonic quantum computing that has attracted attention as a platform for building quantum devices capable of performing tasks that are out of reach for classical devices. There is therefore significant interest, from the perspective of computational complexity theory, in solidifying the mathematical foundation for the hardness of simulating these devices. We show that, under the standard Anti-Concentration and Permanent-of-Gaussians conjectures, there is no efficient classical algorithm to sample from ideal Gaussian boson sampling distributions (even approximately) unless the polynomial hierarchy collapses. The hardness proof holds in the regime where the number of modes scales quadratically with the number of photons, a setting in which hardness was widely believed to hold but that nevertheless had no definitive proof. Crucial to the proof is a new method for programming a Gaussian boson sampling device so that the output probabilities are proportional to the permanents of submatrices of an arbitrary matrix. This technique is a generalization of Scattershot BosonSampling that we call BipartiteGBS. We also make progress towards the goal of proving hardness in the regime where there are fewer than quadratically more modes than photons (i.e., the high-collision regime) by showing that the ability to approximate permanents of matrices with repeated rows/columns confers the ability to approximate permanents of matrices with no repetitions. The reduction suffices to prove that GBS is hard in the constant-collision regime.

arXiv:2110.06964•Quantum•Nov 28, 2022•41 citations

Threshold and Parity BosonSampling in the Linear-Mode Regime

Byeongseon Go, Hyunseok Jeong, Changhun Oh

BosonSampling is among the most prominent candidates for demonstrating quantum advantage. However, while the hardness of BosonSampling relies on photon-number-resolving detection, many experimentally relevant settings and applications instead use binary readout based on threshold or parity measurements, whose computational complexity has not yet been rigorously characterized. In this work, we investigate the computational complexity of BosonSampling with threshold and parity measurements in the linear-mode regime, where the number of modes scales linearly with the number of photons and is most relevant to current experiments. In particular, we establish average-case #P-hardness of estimating typical output probabilities in threshold and parity BosonSampling, a crucial ingredient in proving the classical hardness of the corresponding sampling problems. The resulting imprecision bounds match those obtained in prior hardness results for standard photon-number-resolving BosonSampling in the linear-mode regime. The key technical ingredient is a Fourier-coefficient extraction method, induced by coherent beam-splitter rotations, that extracts hidden hard components within coarse-grained output probabilities. These results indicate that the hard output-probability structure of photon-number-resolving BosonSampling can persist under natural binary coarse-grainings, even in collision-dominant regimes.

quant-pharXiv:2608.24008•Aug 26, 2026•0 citations

Efficient classical algorithm for estimating linear statistics of Boson Sampling

Benoit Seron, Hugo Thomas, Eduardo Araujo +3 more

Boson Sampling is a prominent candidate for the demonstration of quantum computational advantage but it remains unclear whether a large-scale boson sampler can find useful computational applications. The challenge is that to use a boson sampler to estimate, for example, a physical observable, it is necessary to coarse grain the outcome distribution, owing to the exponential size of the outcome space and the anti-concentration properties of the Boson Sampling distribution. In this work, we analyse the complexity of a specific type of coarse-graining of Boson Sampling distributions based on linear functions of the output photon occupation numbers, which we refer to as linear statistics. We present an efficient classical algorithm for approximating linear statistics of boson samplers within additive error for different kinds of input states, such as Fock states and squeezed states. This allows us to unify in the same framework recent results on efficient quantum-inspired classical algorithms for simulating molecular vibronic spectra, or efficient classical approximations of coarse-grained distributions based on detector binning. Additionally, we show how our algorithm can be used to classically evaluate a proposed one-way function based on Boson Sampling, while other proposals for cryptographic applications escape our classical simulation techniques. We leave open the question of classical simulability of non-linear statistics and connect it to the problem of computing transition amplitudes of linear-optical circuits with one layer of interactions.

quant-pharXiv:2609.35447•Sep 29, 2026•0 citations
Boson sampling is a leading candidate for demonstrating quantum advantage in photonic systems. Despite significant experimental and theoretical progress, a characterization of its output statistics remains incomplete. This is especially true beyond the dilute regime, where photon collisions and bunching become significant. The associated saturated regime, characterized by mode number growing linearly with photon number, or more generally sub-quadratically, is precisely the regime of greatest experimental interest. As a consequence, anti-concentration of the output distribution--a key ingredient in hardness arguments--remains poorly understood in boson sampling. In this work, we leverage representation-theoretic tools to address this gap, obtaining closed-form expressions for second moments of generic particle-number-preserving bosonic observables. We express these quantities in terms of Hilbert-Schmidt norms of projections onto irreducible components of the operator space and show that these projection norms admit compact analytical expressions by exploiting the underlying symmetry structure. Focusing on Fock state output probabilities, we further establish anti-concentration beyond the dilute regime. Together with recent complexity-theoretic results, our findings strengthen hardness guarantees for boson sampling in experimentally interesting settings.
Gaussian boson sampling is a promising method for experimental demonstrations of quantum advantage because it is easier to implement than other comparable schemes. While most of the properties of Gaussian boson sampling are understood to the same degree as for these other schemes, we understand relatively little about the statistical properties of its output distribution. The most relevant statistical property, from the perspective of demonstrating quantum advantage, is the "anticoncentration" of the output distribution as measured by its second moment. The degree of anticoncentration features in arguments for the complexity-theoretic hardness of Gaussian boson sampling. In this Letter, we develop a graph-theoretic framework for analyzing the moments of the Gaussian boson sampling distribution. Using this framework, we show that Gaussian boson sampling undergoes a transition in anticoncentration as a function of the number of modes that are initially squeezed compared to the number of photons measured at the end of the circuit. When the number of initially squeezed modes scales sufficiently slowly with the number of photons, there is a lack of anticoncentration. However, if the number of initially squeezed modes scales quickly enough, the output probabilities anticoncentrate weakly.
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.
Photonic architectures are one of the leading platforms for demonstrating quantum computational advantage, with Boson Sampling and Gaussian Boson Sampling as the primary schemes. Yet, we lack for these photonic primitives a systematic theoretical understanding of linear cross-entropy benchmarking (LXEB), which is a central tool for testing quantum advantage proposals. In this work, we develop a representation-theoretic framework for the classical computation of average LXEB scores and second moments of output probability distributions, covering a range of quantum advantage experiments based on scattering nnn-photon states through mmm-mode Haar-random interferometers. Our methods apply in any regime, including the saturated regime, where the (expected) number of photons is comparable to the number of optical modes. The same second-moment techniques also allow us to prove anticoncentration for traditional Fock-state Boson Sampling in the saturated regime. Interestingly, for Gaussian Boson Sampling second moments are not sufficient to establish a meaningful anticoncentration statement. The technical core of our approach rests on decomposing two copies of the nnn-particle bosonic space Symn(Cm)\mathrm{Sym}^n(\mathbb{C}^m)Symn(Cm) into irreducible representations of U(m)\mathrm{U}(m)U(m). This reduces two-copy Haar averages to computing purities of initial states after partial traces over particles, highlighting the role that particle entanglement plays for LXEB and anticoncentration.
Gaussian boson sampling is a model of photonic quantum computing that has attracted attention as a platform for building quantum devices capable of performing tasks that are out of reach for classical devices. There is therefore significant interest, from the perspective of computational complexity theory, in solidifying the mathematical foundation for the hardness of simulating these devices. We show that, under the standard Anti-Concentration and Permanent-of-Gaussians conjectures, there is no efficient classical algorithm to sample from ideal Gaussian boson sampling distributions (even approximately) unless the polynomial hierarchy collapses. The hardness proof holds in the regime where the number of modes scales quadratically with the number of photons, a setting in which hardness was widely believed to hold but that nevertheless had no definitive proof. Crucial to the proof is a new method for programming a Gaussian boson sampling device so that the output probabilities are proportional to the permanents of submatrices of an arbitrary matrix. This technique is a generalization of Scattershot BosonSampling that we call BipartiteGBS. We also make progress towards the goal of proving hardness in the regime where there are fewer than quadratically more modes than photons (i.e., the high-collision regime) by showing that the ability to approximate permanents of matrices with repeated rows/columns confers the ability to approximate permanents of matrices with no repetitions. The reduction suffices to prove that GBS is hard in the constant-collision regime.
BosonSampling is among the most prominent candidates for demonstrating quantum advantage. However, while the hardness of BosonSampling relies on photon-number-resolving detection, many experimentally relevant settings and applications instead use binary readout based on threshold or parity measurements, whose computational complexity has not yet been rigorously characterized. In this work, we investigate the computational complexity of BosonSampling with threshold and parity measurements in the linear-mode regime, where the number of modes scales linearly with the number of photons and is most relevant to current experiments. In particular, we establish average-case #P-hardness of estimating typical output probabilities in threshold and parity BosonSampling, a crucial ingredient in proving the classical hardness of the corresponding sampling problems. The resulting imprecision bounds match those obtained in prior hardness results for standard photon-number-resolving BosonSampling in the linear-mode regime. The key technical ingredient is a Fourier-coefficient extraction method, induced by coherent beam-splitter rotations, that extracts hidden hard components within coarse-grained output probabilities. These results indicate that the hard output-probability structure of photon-number-resolving BosonSampling can persist under natural binary coarse-grainings, even in collision-dominant regimes.
Boson Sampling is a prominent candidate for the demonstration of quantum computational advantage but it remains unclear whether a large-scale boson sampler can find useful computational applications. The challenge is that to use a boson sampler to estimate, for example, a physical observable, it is necessary to coarse grain the outcome distribution, owing to the exponential size of the outcome space and the anti-concentration properties of the Boson Sampling distribution. In this work, we analyse the complexity of a specific type of coarse-graining of Boson Sampling distributions based on linear functions of the output photon occupation numbers, which we refer to as linear statistics. We present an efficient classical algorithm for approximating linear statistics of boson samplers within additive error for different kinds of input states, such as Fock states and squeezed states. This allows us to unify in the same framework recent results on efficient quantum-inspired classical algorithms for simulating molecular vibronic spectra, or efficient classical approximations of coarse-grained distributions based on detector binning. Additionally, we show how our algorithm can be used to classically evaluate a proposed one-way function based on Boson Sampling, while other proposals for cryptographic applications escape our classical simulation techniques. We leave open the question of classical simulability of non-linear statistics and connect it to the problem of computing transition amplitudes of linear-optical circuits with one layer of interactions.