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Optics Express

Using an imperfect photonic network to implement random unitaries

Roel Burgwal, William R. Clements, Devin H. Smith, James C. Gates, W. Steven Kolthammer, Jelmer J. Renema, Ian A. Walmsley

Published Oct 31, 2017• 103 citations
Publisher

Abstract

Boson Sampling has emerged as a tool to explore the advantages of quantum over classical computers as it does not require a universal control over the quantum system, which favours current photonic experimental platforms.Here, we introduce Gaussian Boson Sampling, a classically hard-to-solve problem that uses squeezed states as a non-classical resource. We relate the probability to measure specific photon patterns from a general Gaussian state in the Fock basis to a matrix function called the hafnian, which answers the last remaining question of sampling from Gaussian states. Based on this result, we design Gaussian Boson Sampling, a #P hard problem, using squeezed states. This approach leads to a more efficient photonic boson sampler with significant advantages in generation probability and measurement time over currently existing protocols.

DOI
10.1364/oe.25.028236

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Boson Sampling has emerged as a tool to explore the advantages of quantum over classical computers as it does not require a universal control over the quantum system, which favours current photonic experimental platforms.Here, we introduce Gaussian Boson Sampling, a classically hard-to-solve problem that uses squeezed states as a non-classical resource. We relate the probability to measure specific photon patterns from a general Gaussian state in the Fock basis to a matrix function called the hafnian, which answers the last remaining question of sampling from Gaussian states. Based on this result, we design Gaussian Boson Sampling, a #P hard problem, using squeezed states. This approach leads to a more efficient photonic boson sampler with significant advantages in generation probability and measurement time over currently existing protocols.
Boson sampling has emerged as a tool to explore the advantages of quantum over classical computers as it does not require universal control over the quantum system, which favors current photonic experimental platforms. Here, we introduce Gaussian Boson sampling, a classically hard-to-solve problem that uses squeezed states as a nonclassical resource. We relate the probability to measure specific photon patterns from a general Gaussian state in the Fock basis to a matrix function called the Hafnian, which answers the last remaining question of sampling from Gaussian states. Based on this result, we design Gaussian Boson sampling, a #P hard problem, using squeezed states. This demonstrates that Boson sampling from Gaussian states is possible, with significant advantages in the photon generation probability, compared to existing protocols.
Gaussian boson sampling (GBS) seeks quantum advantage by sampling photon-number patterns generated with squeezed inputs and passive linear optics. Many proposed GBS applications instead target function computation by applying functions to mode-resolved photon-number outcomes---a natural form of experimental postprocessing that produces classical outputs. Sampling hardness alone, however, does not determine the complexity of these tasks. By analyzing the irreducible decomposition of fixed-photon-number operator spaces, we prove that the expectation value of every such function in the average case over passive linear-optical networks can be classically evaluated for inputs with finite squeezing strength. We also provide a classical algorithm that estimates this value to inverse-polynomial additive error. The result provides new theoretical tools for analyzing linear-optical quantum systems, helps clarify the origin of current GBS hardness evidence, and inspires new applications of GBS with genuine quantum advantages.
Since the development of boson sampling, there has been a quest to construct more efficient and experimentally feasible protocols to test the computational complexity of sampling from photonic states. In this paper, we interpret and extend the results presented previously [Phys. Rev. Lett. 119, 170501 (2017)]. We derive an expression that relates the probability to measure a specific photon output pattern from a Gaussian state to the Hafnian matrix function and use it to design a Gaussian boson sampling protocol. Then, we discuss the advantages that this protocol has relative to other photonic protocols and the experimental requirements for Gaussian boson sampling. Finally, we relate it to the previously most general protocol, scattershot boson sampling [Phys. Rev. Lett. 113, 100502 (2014)].
Abstract Boson Sampling is a task that is conjectured to be computationally hard for a classical computer, but which can be efficiently solved by linear-optical interferometers with Fock state inputs. Significant advances have been reported in the last few years, with demonstrations of small- and medium-scale devices, as well as implementations of variants such as Gaussian Boson Sampling. Besides the relevance of this class of computational models in the quest for unambiguous experimental demonstrations of quantum advantage, recent results have also proposed the first applications for hybrid quantum computing. Here, we introduce the adoption of non-linear photon–photon interactions in the Boson Sampling framework, and analyze the enhancement in complexity via an explicit linear-optical simulation scheme. By extending the computational expressivity of Boson Sampling, the introduction of non-linearities promises to disclose novel functionalities for this class of quantum devices. Hence, our results are expected to lead to new applications of near-term, restricted photonic quantum computers.
We pose a randomized boson-sampling problem. Strong evidence exists that such a problem becomes intractable on a classical computer as a function of the number of bosons. We describe a quantum optical processor that can solve this problem efficiently based on a Gaussian input state, a linear optical network, and nonadaptive photon counting measurements. All the elements required to build such a processor currently exist. The demonstration of such a device would provide empirical evidence that quantum computers can, indeed, outperform classical computers and could lead to applications.
Boson sampling has emerged as an important tool to demonstrate the difference between quantum and classical computers and has attracted the interest of experimentalists and theoreticians. In this work we study boson sampling from general, single-mode states using a scheme that can generate any such state by combining Gaussian states and photon number measurements. We derive a formula that can be used to calculate the output photon number probabilities of these states after they travel through a linear interferometer. This extends the Boson sampling protocol to the widest array of possible single-mode states and from this we show that the complexity scaling of all such states is similar and hence there is no complexity advantage of using complex input states over simpler ones.