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arXiv:1605.05796 · Physical Review A

Scaling of boson sampling experiments

P. D. Drummond, Bogdan Opanchuk, L. Rosales-Zárate, M. D. Reid, Peter J. Forrester

Published Oct 24, 2016• 14 citations
arXivHTMLPDF

Abstract

Boson sampling is the problem of generating a multiphoton state whose counting probability is the permanent of an $n\ifmmode\times\else\texttimes\fi{}n$ matrix. This is created as the output $n$-photon coincidence rate of a prototype quantum computing device with $n$ input photons. It is a fundamental challenge to verify boson sampling, and therefore the question of how output count rates scale with matrix size $n$ is crucial. Here we apply results from random matrix theory as well as the characteristic function approach from quantum optics to establish analytical scaling laws for average count rates. We treat boson sampling experiments with arbitrary inputs, outputs, and losses. Using the scaling laws we analyze grouping of channel outputs and the count rates for this case.

DOI
10.1103/physreva.94.042339

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Boson sampling is the problem of generating a multiphoton state whose counting probability is the permanent of an n\ifmmode×\else\texttimes\finn\ifmmode\times\else\texttimes\fi{}nn\ifmmode×\else\texttimes\fin matrix. This is created as the output nnn-photon coincidence rate of a prototype quantum computing device with nnn input photons. It is a fundamental challenge to verify boson sampling, and therefore the question of how output count rates scale with matrix size nnn is crucial. Here we apply results from random matrix theory as well as the characteristic function approach from quantum optics to establish analytical scaling laws for average count rates. We treat boson sampling experiments with arbitrary inputs, outputs, and losses. Using the scaling laws we analyze grouping of channel outputs and the count rates for this case.
Multiphoton interference is at the very heart of quantum foundations and applications in quantum sensing and information processing. In particular, boson-sampling experiments have the potential to demonstrate quantum computational supremacy while relying on only multiphoton interference in linear optical interferometers. However, even when photonic losses are negligible, scalable experiments are challenged by the rapid decrease of the probability of the success of current schemes with probabilistic sources for a large number of single photons in each experimental sample. Remarkably, we show a boson sampling scheme in which the probability of success increases instead of decreasing with the number of input photons, eventually approaching a unit value even with nondeterministic sources. It is achieved by sampling measurements not only at the interferometer output but also over all the possible numbers of photons at each input port and, different from scattershot boson sampling, allowing a random number of occupied input ports from one experimental run to another. Therefore, these results provide an exciting route toward future demonstrations of quantum computational supremacy with scalable experimental resources.
Linear-optical platforms provide a promising route to demonstrate quantum computational advantage, particularly through the boson sampling problem. While a boson sampling system should be large enough to show the quantum advantage, the size of the boson sampling problem is often restricted by experimental constraints. In particular, as a boson sampling system is scaled up, experimental constraints can limit its effective problem size to permit efficient classical simulation. In this paper, we investigate the theoretical scalability bounds for boson sampling systems under practical experimental conditions, including photon loss, mode mismatch, and input state error. We employ non-negativity conditions for phase-space quasiprobability distribution functions to analyze these effects. In addition, we introduce two device-level metrics, the boson sampling score which measures the usable capacity of a given device and the boson sampling volume which provides a common hardware-level benchmark for comparing different platforms and assessing their access to an advantage-relevant problem size regime. Our results provide fundamental insights into the performance of boson sampling systems under realistic experimental conditions and offer practical guidelines for scaling up boson sampling systems.
Boson sampling is a well-defined task that is strongly believed to be intractable for classical computers, but can be efficiently solved by a specific quantum simulator. However, an outstanding problem for large-scale experimental boson sampling is the scalability. Here we report an experiment on boson sampling with photon loss, and demonstrate that boson sampling with a few photons lost can increase the sampling rate. Our experiment uses a quantum-dot-micropillar single-photon source demultiplexed into up to seven input ports of a 16×16 mode ultralow-loss photonic circuit, and we detect three-, four- and fivefold coincidence counts. We implement and validate lossy boson sampling with one and two photons lost, and obtain sampling rates of 187, 13.6, and 0.78 kHz for five-, six-, and seven-photon boson sampling with two photons lost, which is 9.4, 13.9, and 18.0 times faster than the standard boson sampling, respectively. Our experiment shows an approach to significantly enhance the sampling rate of multiphoton boson sampling.
We generalize an approach for description of multiphoton experiments with multiport unitary linear optical devices, started in Phys. Rev. A 89, 022333 (2014) with single photons in mixed spectral states, to arbitrary (multiphoton) input and arbitrary photon detectors. We show that output probabilities are always given in terms of the matrix permanents of the Hadamard product of a matrix built from the network matrix and matrices built from the spectral state of photons and spectral sensitivities of detectors. Moreover, in the case of input with up to one photon per mode, the output probabilities are given by a sum (or integral) with each term being the absolute value squared of such a matrix permanent. We conjecture that, for an arbitrary multiphoton input, zero output probability of an output configuration is always the result of an exact cancellation of quantum transition amplitudes of completely indistinguishable photons (a subset of all input photons) and, moreover, is independent of coherence between only partially indistinguishable photons. The conjecture is supported by examples. Furthermore, we propose a measure of partial indistinguishability of photons which generalizes Mandel's observation, and find the law of degradation of quantum coherence in a realistic boson-sampling device with increase of the total number of photons and/or their ``classicality parameter.''
Sampling the distribution of bosons that have undergone a random unitary evolution is strongly believed to be a computationally hard problem. Key to outperforming classical simulations of this task is to increase both the number of input photons and the size of the network. We propose driven boson sampling, in which photons are input within the network itself, as a means to approach this goal. We show that the mean number of photons entering a boson sampling experiment can exceed one photon per input mode, while maintaining the required complexity, potentially leading to less stringent requirements on the input states for such experiments. When using heralded single-photon sources based on parametric down-conversion, this approach offers an ∼e-fold enhancement in the input state generation rate over scattershot boson sampling, reaching the scaling limit for such sources. This approach also offers a dramatic increase in the signal-to-noise ratio with respect to higher-order photon generation from such probabilistic sources, which removes the need for photon number resolution during the heralding process as the size of the system increases.
Boson sampling is a well-defined task that is strongly believed to be intractable for classical computers, but can be efficiently solved by a specific quantum simulator.However, an outstanding problem for largescale experimental boson sampling is the scalability.Here we report an experiment on boson sampling with photon loss, and demonstrate that boson sampling with a few photons lost can increase the sampling rate.Our experiment uses a quantum-dot-micropillar single-photon source demultiplexed into up to seven input ports of a 16 × 16 mode ultralow-loss photonic circuit, and we detect three-, four-and fivefold coincidence counts.We implement and validate lossy boson sampling with one and two photons lost, and obtain sampling rates of 187, 13.6, and 0.78 kHz for five-, six-, and seven-photon boson sampling with two photons lost, which is 9.4, 13.9, and 18.0 times faster than the standard boson sampling, respectively.Our experiment shows an approach to significantly enhance the sampling rate of multiphoton boson sampling.