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arXiv:2212.09643 · Quantum

Efficient validation of Boson Sampling from binned photon-number distributions

Benoît Seron, Leonardo Novo, Alex Arkhipov, Nicolas J. Cerf

Published Sep 19, 2024• 13 citations
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Abstract

In order to substantiate claims of quantum computational advantage, it is crucial to develop efficient methods for validating the experimental data. We propose a test of the correct functioning of a boson sampler with single-photon inputs that is based on how photons distribute among partitions of the output modes. Our method is versatile and encompasses previous validation tests based on bunching phenomena, marginal distributions, and even some suppression laws. We show via theoretical arguments and numerical simulations that binned-mode photon number distributions can be used in practical scenarios to efficiently distinguish ideal boson samplers from those affected by realistic imperfections, especially partial distinguishability of the photons.

DOI
10.22331/q-2024-09-19-1479

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In order to substantiate claims of quantum computational advantage, it is crucial to develop efficient methods for validating the experimental data. We propose a test of the correct functioning of a boson sampler with single-photon inputs that is based on how photons distribute among partitions of the output modes. Our method is versatile and encompasses previous validation tests based on bunching phenomena, marginal distributions, and even some suppression laws. We show via theoretical arguments and numerical simulations that binned-mode photon number distributions can be used in practical scenarios to efficiently distinguish ideal boson samplers from those affected by realistic imperfections, especially partial distinguishability of the photons.
Abstract We experimentally demonstrate a testing strategy for boson samplers that is based on efficiently computable expressions for the output photon counting distributions binned over multiple optical modes. We apply this method to validate boson sampling experiments with three photons on a reconfigurable photonic chip, which implements a four-mode interferometer, analyzing 50 Haar-random unitary transformations while tuning photon indistinguishability via controlled delays. We show that for (nearly) indistinguishable photons, the experiment accurately reproduces the ideal boson sampling binned-mode distributions, which exhibit variations that depend both on the specific interferometer implemented as well as the choice of bin, confirming the usefulness of the method to diagnose imperfections such as partial distinguishability or imperfect chip control. Finally, we analyze the behavior of Haar-averaged binned-mode distributions in relation to indistinguishability and demonstrate analytically that its variance is proportional to the average of the square of the photons’ indistinguishability parameter. These findings highlight the central role of binning in boson sampling validation, offering a scalable and efficient framework for assessing multiphoton interference and experimental performance.
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Abstract An important challenge with the current generation of noisy, large-scale quantum computers is the question of validation. Does the hardware generate correct answers? If not, what are the errors? This issue is often combined with questions of computational advantage, but it is a fundamentally distinct issue. In current experiments, complete validation of the output statistics is generally not possible because it is exponentially hard to do so. Here, we apply phase-space simulation methods to partially verify recent experiments on Gaussian boson sampling (GBS) implementing photon-number resolving detectors. The positive-P phase-space distribution is employed, as it uses probabilistic sampling to reduce complexity. It is ∼ 10 18 times faster than direct classical simulation for experiments on 288 modes where quantum computational advantage is claimed. When combined with binning and marginalization to improve statistics, multiple validation tests are efficiently computable, of which some tests can be carried out on experimental data. We show that the data as a whole has discrepancies with theoretical predictions for perfect squeezing. However, a modification of the GBS parameters greatly improves agreement for some tests. We suggest that such validation tests could form the basis of feedback methods to improve GBS experiments.
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.
The boson sampler proposed by Aaronson and Arkhipov is a nonuniversal quantum computer, which can serve as evidence against the extended Church-Turing thesis. It samples the probability distribution at the output of a linear unitary optical network with indistinguishable single photons at the input. Four experimental groups have already tested their small-scale prototypes with up to four photons. A boson sampler with a few dozens of single photons is believed to be hard to simulate on a classical computer. For scalability of a realistic boson sampler with current technology it is necessary to know the effect of the photon mode mismatch on its operation. Here a nondeterministic model of the boson sampler is analyzed, which employs partially indistinguishable single photons emitted by identical sources. A sufficient condition on the average mutual fidelity $\ensuremath{\langle}\mathcal{F}\ensuremath{\rangle}$ of the single photons is found, which guarantees that the realistic boson sampler outperforms the classical computer. Moreover, the boson-sampler computer with partially indistinguishable single photons is scalable and has more power than classical computers when the single-photon mode mismatch $1\ensuremath{-}\ensuremath{\langle}\mathcal{F}\ensuremath{\rangle}$ scales as $O({N}^{\ensuremath{-}3/2})$ with the total number of photons NNN.
We demonstrate how boson sampling with photons of partial distinguishability can be expressed in terms of interference of fewer photons. We use this observation to propose a classical algorithm to simulate the output of a boson sampler fed with photons of partial distinguishability. We find conditions for which this algorithm is efficient, which gives a lower limit on the required indistinguishability to demonstrate a quantum advantage. Under these conditions, adding more photons only polynomially increases the computational cost to simulate a boson sampling experiment.