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.
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.
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 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.
Gaussian boson sampling (GBS) seeks quantum advantage by sampling photon-number patterns generated with squeezed inputs and passive linear optics. Many proposed GBS applications instead target function computation by applying functions to mode-resolved photon-number outcomes---a natural form of experimental postprocessing that produces classical outputs. Sampling hardness alone, however, does not determine the complexity of these tasks. By analyzing the irreducible decomposition of fixed-photon-number operator spaces, we prove that the expectation value of every such function in the average case over passive linear-optical networks can be classically evaluated for inputs with finite squeezing strength. We also provide a classical algorithm that estimates this value to inverse-polynomial additive error. The result provides new theoretical tools for analyzing linear-optical quantum systems, helps clarify the origin of current GBS hardness evidence, and inspires new applications of GBS with genuine quantum advantages.
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.
BosonSampling and Random Circuit Sampling are important both as a theoretical tool for separating quantum and classical computation, and as an experimental means of demonstrating quantum speedups. Prior works have shown that average-case hardness of sampling follows from certain unproven conjectures about the hardness of computing output probabilities, such as the Permanent-of-Gaussians Conjecture (PGC), which states that
e−nlogn−n−O(logn) additive-error estimates to the output probability of most random BosonSampling experiments are #P-hard. Prior works have only shown weaker average-case hardness results that do not imply sampling hardness. Proving these conjectures has become a central question in quantum complexity. In this work, we show that
e−nlogn−n−O(nδ) additive-error estimates to output probabilities of most random BosonSampling experiments are #P-hard for any
δ>0, exponentially improving on prior work. In the process, we circumvent all known barrier results for proving PGC. The remaining hurdle to prove PGC is now “merely” to show that the
O(nδ) in the exponent can be improved to
O(logn). We also obtain an analogous result for Random Circuit Sampling. We then show, for the first time, a hardness of average-case classical sampling result for BosonSampling, under an anticoncentration conjecture. Specifically, we prove the impossibility of multiplicative-error sampling from random BosonSampling experiments with probability
1−2−O~(N1/3) for input size N, unless the Polynomial Hierarchy collapses. This exponentially improves upon the state-of-the-art. To do this, we introduce new proof techniques which tolerate exponential loss in the worst-to-average-case reduction. This opens the possibility to show the hardness of average-case sampling without ever proving PGC.