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Gaussian Boson Sampling as a Calibrated Variance-Reduction Engine: A DSFL Explanation

Camilla Josephson

Published Oct 27, 2025• 0 citations
Publisher

Abstract

Gaussian Boson Sampling (GBS) generates photon-count patterns whose probabilities are governed by hafnian functionals of an encoded covariance. Recent theory shows that GBS samples can approximate Gaussian expectation problems with provable sample-complexity advantages over Monte Carlo (MC) on an open, sizable subset of instances. We give a unifying explanation via the Deterministic Statistical Feedback Law (DSFL). First, GBS acts as a calibrated, admissible map that intertwines target statistics and physical sampling. Second, tuning the mean photon number is a genuine calibration step that minimizes a DSFL residual. Third, principal-angle (Friedrichs) geometry quantifies conditioning and leakage between statistical and sampling features. Fourth, a Lyapunov envelope provides non-asymptotic error decay and design rules. From these ingredients we derive bounds that recover and sharpen known GBS scalings, predict regimes where GBS outperforms MC, and specify a practical hybrid policy that switches by degree. Experiments with simulated distributions and hardware-aligned kernels support the theory and validate the tuning and selection rules.

DOI
10.20944/preprints202510.2042.v1

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Gaussian Boson Sampling (GBS) generates photon-count patterns whose probabilities are governed by hafnian functionals of an encoded covariance. Recent theory shows that GBS samples can approximate Gaussian expectation problems with provable sample-complexity advantages over Monte Carlo (MC) on an open, sizable subset of instances. We give a unifying explanation via the Deterministic Statistical Feedback Law (DSFL). First, GBS acts as a calibrated, admissible map that intertwines target statistics and physical sampling. Second, tuning the mean photon number is a genuine calibration step that minimizes a DSFL residual. Third, principal-angle (Friedrichs) geometry quantifies conditioning and leakage between statistical and sampling features. Fourth, a Lyapunov envelope provides non-asymptotic error decay and design rules. From these ingredients we derive bounds that recover and sharpen known GBS scalings, predict regimes where GBS outperforms MC, and specify a practical hybrid policy that switches by degree. Experiments with simulated distributions and hardware-aligned kernels support the theory and validate the tuning and selection rules.
Gaussian Boson Sampling (GBS), which can be realized with a photonic quantum computing model, perform some special kind of sampling tasks. In [4], we introduced algorithms that use GBS samples to approximate Gaussian expectation problems. We found a non-empty open subset of the problem space where these algorithms achieve exponential speedup over the standard Monte Carlo (MC) method. This speedup is defined in terms of the guaranteed sample size to reach the same accuracy εεε and success probability δδδ under the (ε,δ)(ε, δ)(ε,δ) multiplicative error approximation scheme. In this paper, we enhance our original approach by optimizing the average photon number in the GBS distribution to match the specific Gaussian expectation problem. We provide updated estimates of the guaranteed sample size for these improved algorithms and quantify the proportion of problem space where they outperform MC. Numerical results indicate that the proportion of the problem space where our improved algorithms have an advantage is substantial, and the advantage gained is significant. Notably, for certain special cases, our methods consistently outperform MC across nearly 100\% of the problem space.
Abstract Gaussian Boson Sampling (GBS) is a non-universal model for quantum computing inspired by the original formulation of the Boson Sampling (BS) problem. Nowadays, it represents a paradigmatic quantum platform to reach the quantum advantage regime in a specific computational model. Indeed, thanks to the implementation in photonics-based processors, the latest GBS experiments have reached a level of complexity where the quantum apparatus has solved the task faster than currently up-to-date classical strategies. In addition, recent studies have identified possible applications beyond the inherent sampling task. In particular, a direct connection between photon counting of a genuine GBS device and the number of perfect matchings in a graph has been established. In this work, we propose to exploit such a connection to benchmark GBS experiments. We interpret the properties of the feature vectors of the graph encoded in the device as a signature of correct sampling from the true input state. Within this framework, two approaches are presented. The first method exploits the distributions of graph feature vectors and classification via neural networks. The second approach investigates the distributions of graph kernels. Our results provide a novel approach to the actual need for tailored algorithms to benchmark large-scale Gaussian Boson Samplers.
Quantum computers solve intractable problems which classically require an exponentially long time to compute. With the development of large-scale experiments that claim quantum advantage, a vital issue has now emerged. What are the errors, and how do they affect the complexity of the problem solved? Large-scale Gaussian boson sampling (GBS) experiments give an example in which random numbers are generated. Despite classical hardness, these have computable benchmarks for checking data validity. While there are other quantum computing architectures, Gaussian boson sampling is uniquely testable at all scales. Several large, pioneering quantum computing (QC) experiments have been carried out to investigate quantum advantage. Here, we introduce a highly scalable but classical algorithm that can solve GBS approximately. Our numerical simulation of the output count data is closer to the exact solution than current experiments up to 1152 modes. This algorithm outperforms all previous classical, approximate algorithms and scales efficiently to larger experiments. Our results show that effects beyond losses can cause the errors that allow classical simulability. This work will lead to more precise algorithms and is a step towards understanding how QC quantum advantage is affected by the underlying physics.
Gaussian boson sampling (GBS) is a model of nonuniversal quantum computation that claims to demonstrate quantum supremacy with current technologies. This model entails sampling photocounting events from a multimode Gaussian state at the outputs of a linear interferometer. In this scheme, collision events—those with more than one photon for each mode—are infrequent. However, they are still used for validation purposes. Therefore, the limitation of realistic detectors to perfectly resolve adjacent photon numbers becomes pivotal. We derive a photocounting probability distribution in GBS schemes which is applicable for use with general detectors and photocounting techniques. This probability distribution is expressed in terms of functionals of the field-quadrature covariance matrix, e.g., Hafnian and Torontonian in the well-known special cases of photon-number-resolving and on-off detectors, respectively. Based on our results, we consider a GBS validation technique involving detectors with realistic photon-number resolution. Published by the American Physical Society 2024
Gaussian Boson Sampling (GBS) have shown advantages over classical methods for performing some specific sampling tasks. To fully harness the computational power of GBS, there has been great interest in identifying their practical applications. In this study, we explore the use of GBS samples for computing a numerical approximation to the Gaussian expectation problem, that is to integrate a multivariate function against a Gaussian distribution. We propose two estimators using GBS samples, and show that they both can bring an exponential speedup over the plain Monte Carlo (MC) estimator. Precisely speaking, the exponential speedup is defined in terms of the guaranteed sample size for these estimators to reach the same level of accuracy εεε and the same success probability δδδ in the (ε,δ)(ε, δ)(ε,δ) multiplicative error approximation scheme. We prove that there is an open and nonempty subset of the Gaussian expectation problem space for such computational advantage.
Gaussian boson sampling (GBS), a computational problem conjectured to be hard to simulate on a classical machine, has been at the forefront of recent years’ experimental and theoretical efforts to demonstrate quantum advantage. The classical intractability of the sampling task makes validating these experiments a challenging and essential undertaking. In this paper, we propose binned-detector probability distributions as a suitable quantity to statistically validate GBS experiments employing photon-number-resolving detectors. We develop the theoretical framework to compute such distributions by leveraging their connection with their respective characteristic function. The latter may be efficiently and analytically computed for squeezed input states as well as for relevant classical hypothesis like squashed states. Our theoretical framework encompasses other validation methods based onmarginal distributions and correlation functions. Additionally, it can analytically accommodate various sources of noise, such as losses and partial distinguishability, a feature that has received limited attention within the GBS framework so far. We also derive how binned-detector probability distributions behave when Haar averaged over all possible interferometric networks, extending known results for Fock boson sampling.