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arXiv:1510.05245 · Physical Review A

BosonSampling with lost photons

Scott Aaronson, Daniel J. Brod

Published Jan 21, 2016• 78 citations
arXivHTMLPDF

Abstract

BosonSampling is an intermediate model of quantum computation where linear-optical networks are used to solve sampling problems expected to be hard for classical computers. Since these devices are not expected to be universal for quantum computation, it remains an open question of whether any error-correction techniques can be applied to them, and thus it is important to investigate how robust the model is under natural experimental imperfections, such as losses and imperfect control of parameters. Here, we investigate the complexity of BosonSampling under photon losses, more specifically, the case where an unknown subset of the photons is randomly lost at the sources. We show that if $k$ out of $n$ photons are lost, then we cannot sample classically from a distribution that is $1/{n}^{\mathrm{\ensuremath{\Theta}}(k)}$ close (in total variation distance) to the ideal distribution, unless a ${\text{BPP}}^{\text{NP}}$ machine can estimate the permanents of Gaussian matrices in ${n}^{O(k)}$ time. In particular, if $k$ is constant, this implies that simulating lossy BosonSampling is hard for a classical computer, under exactly the same complexity assumption used for the original lossless case.

DOI
10.1103/physreva.93.012335

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Boson sampling is a candidate for quantum protocols to truly realize the quantum computation advantage and to be used in advanced fields where complex computations are needed, such as quantum chemistry. However, this proposal is hard to achieve due to the existence of noise sources such as photon losses. In order to quantificationally analyze the influences of photon losses in optical networks, boson sampling is classically simulated based on the equivalent beam splitter mechanism, where the photon loss happening in optical units is equivalent to the photon transmission into the environmental paths through a virtual beam splitter. In our simulation, networks corresponding to random unitary matrices are made up, considering both the Reck structure and the Clements structure. The photon loss probability in an optical unit is well controlled by adjusting the parameters of the virtual beam splitter. Therefore, to simulate boson sampling with photon losses in optical networks is actually to simulate ideal boson sampling with more modes. It is found that when the photon loss probability is constant, boson sampling with Clements structures distinctly performs much better than that with Reck structures. Furthermore, the photon loss probability is also set to follow the normal distribution, which is thought to be closer to the situation in reality. It is found that when the mean value of photon loss probability is constant, for both network structures, errors of outputs become more obvious with the increase of standard deviation. It can be inferred that the increase of error rate can be explained by the network depth and the conclusion is suitable for larger-scale boson sampling. Finally, the number of output photons is taken into consideration, which is directly related to the classical computation complexity. It is found that with the photon loss probability, the ratio of output combinations without photon losses decreases sharply, implying that photon losses can obviously affect the quantum computation advantage of boson sampling. Our results indicate that photon losses can result in serious errors for boson sampling, even with a stable network structure such as that of Clements. This work is helpful for boson sampling experiments in reality and it is desired to develop a better protocol, for example, a well-designed network or excellent optical units, to well suppress photon losses.

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BosonSampling is an intermediate model of quantum computation where linear-optical networks are used to solve sampling problems expected to be hard for classical computers. Since these devices are not expected to be universal for quantum computation, it remains an open question of whether any error-correction techniques can be applied to them, and thus it is important to investigate how robust the model is under natural experimental imperfections, such as losses and imperfect control of parameters. Here, we investigate the complexity of BosonSampling under photon losses, more specifically, the case where an unknown subset of the photons is randomly lost at the sources. We show that if kkk out of nnn photons are lost, then we cannot sample classically from a distribution that is 1/n\ensuremathΘ(k)1/{n}^{\mathrm{\ensuremath{\Theta}}(k)}1/n\ensuremathΘ(k) close (in total variation distance) to the ideal distribution, unless a BPPNP{\text{BPP}}^{\text{NP}}BPPNP machine can estimate the permanents of Gaussian matrices in nO(k){n}^{O(k)}nO(k) time. In particular, if kkk is constant, this implies that simulating lossy BosonSampling is hard for a classical computer, under exactly the same complexity assumption used for the original lossless case.
Understanding how photon loss changes the classical complexity of BosonSampling is central to assessing the robustness of near-term photonic quantum advantage. In this work, we present a polynomial-time classical simulation algorithm for lossy BosonSampling that applies whenever a constant fraction of the input photons survives. We consider the standard fixed-loss model, where NNN input photons propagate through an MMM-mode interferometer and exactly N−nN-nN−n photons are lost uniformly at random, leaving nnn surviving photons at the output. For any constant surviving fraction n/N∈(0,1)n/N\in(0,1)n/N∈(0,1), the algorithm samples from a distribution within total variation distance ε\varepsilonε of the fixed-loss BosonSampling distribution with probability at least 1−δ1-δ1−δ over Haar-random interferometers, in time poly⁡(M,N,ε−1,δ−1)\operatorname{poly}(M,N,\varepsilon^{-1},δ^{-1})poly(M,N,ε−1,δ−1). We further extend the fixed-loss analysis to the uniform-transmission-loss model and obtain analogous polynomial-time simulability for any fixed transmission rate η∈(0,1)η\in(0,1)η∈(0,1). These results substantially extend the known provably simulable regime of n=O(εN)n=O(\sqrt{\varepsilon N})n=O(εN​) surviving photons or η=O(ε/N)η=O(\sqrt{\varepsilon/N})η=O(ε/N​) transmission rate, and demonstrate that even the experimentally relevant linear-survival regime can admit efficient classical simulation under the conditions considered here.
BosonSampling, which we proposed three years ago, is a scheme for using linear-optical networks to solve sampling problems that appear to be intractable for a classical computer. \ In a recent manuscript, Gogolin et al.\ claimed that even an ideal BosonSampling device's output would be operationally indistinguishable\textquotedblright\ from a uniform random outcome, at least \textquotedblleft without detailed a priori knowledge; or at any rate, that telling the two apart might itself be a hard problem. We first answer these claims---explaining why the first is based on a definition of a priori knowledge such that, were it adopted, almost no quantum algorithm could be distinguished from a pure random-number source; while the second is neither new nor a practical obstacle to interesting BosonSampling experiments.However, we then go further, and address some interesting research questions inspired by Gogolin et al.'s arguments. We prove that, with high probability over a Haar-random matrix AAA, the BosonSampling distribution induced by AAA is far from the uniform distribution in total variation distance. More surprisingly, and counter to Gogolin et al., we give an efficient algorithm that distinguishes these two distributions with constant bias. Finally, we offer three bonus results about BosonSampling. First, we report an observation of Fernando Brandao: that one can efficiently sample a distribution that has large entropy and that's indistinguishable from a BosonSampling distribution by any circuit of fixed polynomial size. Second, we show that BosonSampling distributions can be efficiently distinguished from uniform even with photon losses and for general initial states. Third, we offer the simplest known proof that Fermion Sampling is solvable in classical polynomial time, and we reuse techniques from our Boson Sampling analysis to characterize random FermionSampling distributions.
Gaussian boson sampling (GBS) is a prominent candidate for the experimental demonstration of quantum advantage. However, while the current implementations of GBS are unavoidably subject to noise, the robustness of the classical intractability of GBS against noise remains largely unexplored. In this work, we establish the complexity-theoretic foundations for the classical intractability of noisy GBS under photon loss, which is a dominant source of imperfection in current implementations. We identify the loss threshold below which lossy GBS maintains the same complexity-theoretic level as ideal GBS, and show that this holds when at most a logarithmic fraction of photons is lost. We additionally derive an intractability criterion for the loss rate through a direct quantification of the statistical distance between ideal and lossy GBS. This work presents the first rigorous characterization of classically intractable regimes of lossy GBS, thereby serving as a crucial step toward demonstrating quantum advantage with near-term implementations.
BosonSampling, which we proposed three years ago, is a scheme for using linear-optical networks to solve sampling problems that appear to be intractable for a classical computer. In arXiv:1306.3995, Gogolin et al. claimed that even an ideal BosonSampling device's output would be "operationally indistinguishable" from a uniform random outcome, at least "without detailed a priori knowledge"; or at any rate, that telling the two apart might itself be a hard problem. We first answer these claims---explaining why the first is based on a definition of "a priori knowledge" so strange that, were it adopted, almost no quantum algorithm could be distinguished from a pure random-number source; while the second is neither new nor a practical obstacle to interesting BosonSampling experiments. However, we then go further, and address some interesting research questions inspired by Gogolin et al.'s mistaken arguments. We prove that, with high probability over a Haar-random matrix A, the BosonSampling distribution induced by A is far from the uniform distribution in total variation distance. More surprisingly, and directly counter to Gogolin et al., we give an efficient algorithm that distinguishes these two distributions with constant bias. Finally, we offer three "bonus" results about BosonSampling. First, we report an observation of Fernando Brandao: that one can efficiently sample a distribution that has large entropy and that's indistinguishable from a BosonSampling distribution by any circuit of fixed polynomial size. Second, we show that BosonSampling distributions can be efficiently distinguished from uniform even with photon losses and for general initial states. Third, we offer the simplest known proof that FermionSampling is solvable in classical polynomial time, and we reuse techniques from our BosonSampling analysis to characterize random FermionSampling distributions.
Abstract Boson sampling is the problem of sampling from the same distribution as indistinguishable single photons at the output of a linear optical interferometer. It is an example of a non-universal quantum computation which is believed to be feasible in the near term and cannot be simulated on a classical machine. Like all purported demonstrations of ‘quantum computational supremacy’, this motivates optimising classical simulation schemes for a realistic model of the problem, in this case boson sampling when the implementations experience lost or distinguishable photons. Although current simulation schemes for sufficiently imperfect boson sampling are classically efficient, in principle the polynomial runtime can be infeasibly large. In this work, we develop a scheme for classical simulation of boson sampling under uniform distinguishability and loss, based on the idea of sampling from distributions where at most k photons are indistinguishable. We show that asymptotically this scheme can provide a polynomial improvement in the runtime compared to classically simulating idealised boson sampling. More significantly, we show that in the regime considered experimentally relevant, our approach gives an substantial improvement in runtime over other classical simulation approaches.
Boson sampling is a candidate for quantum protocols to truly realize the quantum computation advantage and to be used in advanced fields where complex computations are needed, such as quantum chemistry. However, this proposal is hard to achieve due to the existence of noise sources such as photon losses. In order to quantificationally analyze the influences of photon losses in optical networks, boson sampling is classically simulated based on the equivalent beam splitter mechanism, where the photon loss happening in optical units is equivalent to the photon transmission into the environmental paths through a virtual beam splitter. In our simulation, networks corresponding to random unitary matrices are made up, considering both the Reck structure and the Clements structure. The photon loss probability in an optical unit is well controlled by adjusting the parameters of the virtual beam splitter. Therefore, to simulate boson sampling with photon losses in optical networks is actually to simulate ideal boson sampling with more modes. It is found that when the photon loss probability is constant, boson sampling with Clements structures distinctly performs much better than that with Reck structures. Furthermore, the photon loss probability is also set to follow the normal distribution, which is thought to be closer to the situation in reality. It is found that when the mean value of photon loss probability is constant, for both network structures, errors of outputs become more obvious with the increase of standard deviation. It can be inferred that the increase of error rate can be explained by the network depth and the conclusion is suitable for larger-scale boson sampling. Finally, the number of output photons is taken into consideration, which is directly related to the classical computation complexity. It is found that with the photon loss probability, the ratio of output combinations without photon losses decreases sharply, implying that photon losses can obviously affect the quantum computation advantage of boson sampling. Our results indicate that photon losses can result in serious errors for boson sampling, even with a stable network structure such as that of Clements. This work is helpful for boson sampling experiments in reality and it is desired to develop a better protocol, for example, a well-designed network or excellent optical units, to well suppress photon losses.