Efficient approximation of experimental Gaussian boson sampling

Abstract: Two recent landmark experiments have performed Gaussian boson sampling (GBS) with a non-programmable linear interferometer and threshold detectors on up to 144 output modes (see Refs.~\onlinecite{zhong_quantum_2020,zhong2021phase}). Here we give classical sampling algorithms with better total variation distance and Kullback-Leibler divergence than these experiments and a computational cost quadratic in the number of modes. Our method samples from a distribution that approximates the single-mode and two-mode ideal marginals of the given Gaussian boson sampler, which are calculated efficiently. One implementation sets the parameters of a Boltzmann machine from the calculated marginals using a mean field solution. This is a 2nd order approximation, with the uniform and thermal approximations corresponding to the 0th and 1st order, respectively. The $k$th order approximation reproduces Ursell functions (also known as connected correlations) up to order $k$ with a cost exponential in $k$ and high precision, while the experiment exhibits higher order Ursell functions with lower precision. This methodology, like other polynomial approximations introduced previously, does not apply to random circuit sampling because the $k$th order approximation would simply result in the uniform distribution, in contrast to GBS.

Efficient Approximation of Experimental Gaussian Boson Sampling

The paper titled "Efficient approximation of experimental Gaussian boson sampling" addresses the computational challenges associated with Gaussian Boson Sampling (GBS) tasks. GBS, a variant of boson sampling utilizing Gaussian states, has been a focal point for demonstrating the potential of quantum computation in surpassing classical approaches, particularly in sampling problems.

The core result of the paper is the presentation of classical algorithms that can generate samples with a better approximation to the ideal GBS distributions compared to the existing experimental implementations. These classical algorithms leverage a quadratic computational cost in relation to the number of output modes, which marks a significant advantage in efficiency. The authors compare their methods with the results achieved by two recent experiments that executed GBS using non-programmable linear interferometers with up to 144 modes.

The classical sampling algorithms offered in this study use two primary statistical measures for comparison: total variation distance and Kullback-Leibler divergence. Both these measures indicate that the proposed classical methods better approximate the single-mode and two-mode ideal marginals. These marginals are efficiently computed, facilitating the approximation process. By setting parameters of a Boltzmann machine using a mean-field solution, the authors provide a second-order approximation. This positions their approach as an advancement over thermal and uniform approximations, which are considered first-order and zeroth-order, respectively.

A critical insight is that the $k$th order approximations handle Ursell functions up to order $k$ with high precision but at an exponential cost in $k$. Notably, their method is not designed to work for random circuit sampling, where a $k$th order approximation would reduce to a uniform distribution rather than mirroring GBS results.

Their findings suggest a qualitative difference in the approximation and complexity of simulating GBS when compared to standard circuit sampling tasks, which strengthens the argument for exploring different methodologies in various quantum supremacy frameworks. The broader implications of this work offer pathways for more efficient classical simulations of certain quantum tasks and set a benchmark for developing quantum systems that can achieve fidelity improvements over classical approaches.

The theoretical implication of this research lies in better understanding the boundaries of quantum advantage in the context of Gaussian Boson Sampling and how classical methods can be both a benchmark and a guide in assessing that advantage. Practically, the proposed approaches could be employed to verify or simulate quantum experiments in near-term systems, providing a comprehensive framework for assessing fidelity and experimental outcomes.

Speculating on future developments, the continuity in refining classical algorithms could contribute to more efficient hybrid quantum-classical co-processors where GBS is used. Enhanced method implementations could shape the studies of quantum photonic circuits as technology evolves. Additionally, the insights gathered about the scalability and the precision of classical approximations might inform the design and error mitigation strategies in experimental quantum optics.