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arXiv:1311.6796 · Physical Review A

Sufficient condition for the mode mismatch of single photons for scalability of the boson-sampling computer

V. S. Shchesnovich

Published Feb 24, 2014• 72 citations
arXivHTMLPDF

Abstract

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 $N$.

DOI
10.1103/physreva.89.022333

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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⟨F\ensuremath⟩\ensuremath{\langle}\mathcal{F}\ensuremath{\rangle}\ensuremath⟨F\ensuremath⟩ 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⟨F\ensuremath⟩1\ensuremath{-}\ensuremath{\langle}\mathcal{F}\ensuremath{\rangle}1\ensuremath−\ensuremath⟨F\ensuremath⟩ scales as O(N\ensuremath−3/2)O({N}^{\ensuremath{-}3/2})O(N\ensuremath−3/2) with the total number of photons NNN.
Boson-sampling is a highly simplified, but non-universal, approach to implementing optical quantum computation. It was shown by Aaronson and Arkhipov that this protocol cannot be efficiently classically simulated unless the polynomial hierarchy collapses, which would be a shocking result in computational complexity theory. Based on this, numerous authors have made the claim that experimental boson-sampling would provide evidence against, or disprove, the Extended Church-Turing thesis -- that any physically realisable system can be efficiently simulated on a Turing machine. We argue against this claim on the basis that, under a general, physically realistic independent error model, boson-sampling does not implement a provably hard computational problem in the asymptotic limit of large systems.
We give a set of sufficient conditions on the experimental Boson-Sampling computer to satisfy Theorem 1.3 of Aaronson & Arkhipov (Theory of Computing \textbf{9}, 143 (2013)) stating a computational problem whose simulation on a classical computer would collapse the polynomial hierarchy of the computational complexity to the third level. This implies that such an experimental device is in conflict with the Extended Church-Turing thesis. In practical terms, we give a set of sufficient conditions for the scalability of the experimental Boson-Sampling computer beyond the power of the classical computers. The derived conditions can be also used for devising efficient verification tests of the Boson-Sampling computer.
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.
Linear optics quantum computing is a promising approach to implementing scalable quantum computation. However, this approach has very demanding physical resource requirements. Recently, Aaronson and Arkhipov (e-print arXiv:1011.3245) showed that a simplified model, which avoids the requirement for fast feed-forward and postselection, while likely not capable of solving BQP-complete problems efficiently, can solve an interesting sampling problem believed to be classically hard. Loss and mode mismatch are the dominant sources of error in such systems. We provide evidence that even lossy systems or systems with mode mismatch are likely to be classically hard to solve. This is of practical interest to experimentalists wishing to demonstrate such systems since it suggests that, even with errors in their implementation, they are likely implementing an algorithm that is classically hard to solve. Our results also equivalently apply to the multiwalker quantum walk model.
Boson sampling is considered as a strong candidate to demonstrate the quantum computational supremacy over classical computers. However, previous proof-of-principle experiments suffered from small photon number and low sampling rates owing to the inefficiencies of the single-photon sources and multi-port optical interferometers. Here, we develop two central components for high-performance boson sampling: robust multi-photon interferometers with 0.99 transmission rate, and actively demultiplexed single-photon sources from a quantum-dot-micropillar with simultaneously high efficiency, purity and indistinguishability. We implement and validate 3-, 4-, and 5-photon boson sampling, and achieve sampling rates of 4.96 kHz, 151 Hz, and 4 Hz, respectively, which are over 24,000 times faster than the previous experiments, and over 220 times faster than obtaining one sample through calculating the matrices permanent using the first electronic computer (ENIAC) and transistorized computer (TRADIC) in the human history. Our architecture is feasible to be scaled up to larger number of photons and with higher rate to race against classical computers, and might provide experimental evidence against the Extended Church-Turing Thesis.
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.