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What Can Quantum Optics Say About Computational Complexity Theory? week ending PRL 114, 060501 (2015) PHYSICAL REVIEW LETTERS 13 FEBRUARY 2015 What Can Quantum Optics Say about Computational Complexity Theory? Saleh Rahimi-Keshari, Austin P. Lund, and Timothy C. Ralph Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics, University of Queensland, St Lucia, Queensland 4072, Australia (Received 26 August 2014; revised manuscript received 13 November 2014; published 11 February 2015) Considering the problem of sampling from the output photon-counting probability distribution of a linear-optical network for input Gaussian states, we obtain results that are of interest from both quantum theory and the computational complexity theory point of view. We derive a general formula for calculating the output probabilities, and by considering input thermal states, we show that the output probabilities are proportional to permanents of positive-semidefinite Hermitian matrices. It is believed that approximating permanents of complex matrices in general is a #P-hard problem. However, we show that these permanents can be approximated with an algorithm in the BPPNP complexity class, as there exists an efficient classical algorithm for sampling from the output probability distribution. We further consider input squeezed- vacuum states and discuss the complexity of sampling from the probability distribution at the output. DOI: 10.1103/PhysRevLett.114.060501 PACS numbers: 03.67.Ac, 42.50.Ex, 89.70.Eg Introduction.—Boson sampling is an intermediate model general formula for the probabilities of detecting single of quantum computation that seeks to generate random photons at the output of the network. Using this formula we samples from a probability distribution of photon (or, in show that probabilities of single-photon counting for input general, boson) counting events at the output of an M-mode thermal states are proportional to permanents of positive- linear-optical network consisting of passive optical ele- semidefinite Hermitian matrices. However, any classical ments, for an input with N of the modes containing single states can be modeled as a statistical mixture of coherent photons and the rest in the vacuum states [1]. There is great states, and as a result we show that sampling from the interest in this particular computational problem as this output probability distribution can be performed efficiently task, despite its simple physical implementation, is strongly on a classical computer. Thus, by using Stockmeyer’s believed to be a problem that cannot be efficiently approximate counting algorithm [1,13], one can approxi- simulated classically. This has led to several proof of mate permanents of positive-semidefinite Hermitian matri- principle experiments realizing small-scale boson sampling ces in the complexity class BPPNP, which is less – [2 5] and investigations of its characterization [6,7] and computationally complex than #P hard. To the best of implementation [8]. our knowledge this result was not previously known. In boson sampling, the photon-counting probabilities are In addition, we consider squeezed-vacuum states as proportional to the modulus squared of permanents of inputs to a linear-optical network. We find the probabilities complex matrices, which, in the case of single-photon of detecting single photons at the output is proportional to detections, are submatrices of the unitary matrix describing the modulus squared of a quantity ON, which is obtained the linear-optical network [9]. It has been proved that by summing up ðN − 1Þ!! complex terms, with N being the exactly computing the permanent of matrices is difficult number of the detected single photons. It was recently (#P hard in complexity theory) [10,11], and it is in a class shown that a specific case of this problem is equivalent to a that contains the polynomial hierarchy of complexity randomized version of the boson sampling problem that classes [12]. More recently, it was proved that approxi- cannot be efficiently simulated using a classical computer mating squared permanents of real matrices to within a [14]. This implies that, following the results from [1],at 2 multiplicative error is also #P hard, and it is believed this is least for this specific problem approximating jONj ,is#P the case for modulus-squared permanents of arbitrary hard. However, it would be surprising if this problem were complex matrices [1]. Based on this key observation, the only case of the general problem of boson sampling Aaronson and Arkhipov have shown that boson sampling with squeezed-vacuum states, for which approximating 2 cannot be classically simulated unless the polynomial jONj is a #P-hard problem. Such considerations may help hierarchy collapses to the third level, a situation believed a complexity theorist to identify other #P-hard problems. to be highly unlikely. Brief review of previous works.—If the photons behaved In this Letter, we consider the problem of sampling from as classical particles, i.e., there were no interferences (the the photon-counting probability distribution at the output of nonclassical effect) between them as they scattered by a a linear-optical network for input Gaussian states, which is linear-optical network, the output probabilities would be referred to as Gaussian boson sampling. We derive a permanents of matrices with non-negative elements [1]. 0031-9007=15=114(6)=060501(5) 060501-1 © 2015 American Physical Society week ending PRL 114, 060501 (2015) PHYSICAL REVIEW LETTERS 13 FEBRUARY 2015 In this classically simulatable situation, one can use Stockmeyer’s approximate counting algorithm [13] to approximate one particular output probability, even if it is exponentially small, to within a multiplicative error in BPPNP (in the third level of the polynomial hierarchy); for a short description of this algorithm, see the supplementary information of Ref. [14] or theorem 4.1 of Ref. [1]. This algorithm was further improved and it was shown that the approximation can be done in BPP (bounded-error probabilistic polynomial time) that is contained in the second level of the polynomial hierarchy [15]. The prob- ~ ability p is approximated with p to within a multiplicative FIG. 1. In the Gaussian boson sampling problem for a given factor of g,ifp=g ≤ p~ ≤ gp for g ≥ 1 þ 1=hðNÞ, where ρ ⊗M ρ product Gaussian input state, in ¼ s¼1 s, and a unitary matrix hðNÞ is a polynomial function in the size of the problem N describing the network, one samples from the output probability (the number of detected single photons). Throughout this distribution pðnÞ. Letter we refer to this form of approximation only. Aaronson and Arkhipov [1] have shown that if there is a Gaussian boson sampling, inefficiency of detectors will polynomial-time classical algorithm for boson sampling cause errors in distinguishing the events. Note, however, that with single-photon inputs, then one could use Stockmeyer’s the errors can be minimized if the mean-photon number at the approximate counting algorithm to approximate the prob- input is much less than the number of modes. Also, ability of detecting a particular configuration of output for the exact boson sampling case, the detection probabilities photons in BPPNP. This would then approximate the are allowed to be exponentially small. modulus squared of the permanent of a submatrix of a A linear-optical network can also be uniquely repre- unitary matrix. However, on the other hand, it was shown sented by an M × M unitary matrix U that relates the that this approximation is #P hard [1], as the elements of a † creation operators of the output modes bˆ to those of the unitary matrix are, in general, complex numbers, and an † k input modes aˆ , algorithm for this problem can solve all of the problems in j the entire polynomial hierarchy [12]. Therefore, the poly- XM ˆ † U ˆ †U† U ˆ † nomial hierarchy of complexity classes would collapse to bj ¼ aj ¼ jkak: ð2Þ the third level, if there exists a classical algorithm that can k¼1 efficiently simulate boson sampling, a highly implausible For a multimode input coherent state jαi, where situation [1]. It was also shown in Ref. [1] that, modulo two α ¼ðα1; α2; α3; …; αMÞ, the output state is also a multi- conjectures, even sampling from a probability distribution mode coherent state. By using the relation (2), we have that is an approximation of the output probability distri- bution is classically intractable as well. This form of YM YM U α U ˆ †U† α 0 ˆ † β 0 β sampling is referred to as the approximate boson sampling, j i¼ Dð aj ; jÞj i¼ Dðak; kÞj i¼j i; as opposed to the exact boson sampling that is for sampling j¼1 k¼1 from the exact output probability distribution. Here we † † where Dðaˆ ; α Þ¼expðα aˆ − α¯ aˆ Þ is the displacement consider exact boson sampling only. j j j j j j ˆ α¯ Photon-counting probability distribution.—In the operator for mode aj, with j being the complex conjugate α Gaussian boson sampling problem, we consider the photon- of j, and the output amplitudes are counting probability distribution at the output of an XM M-mode linear-optical network for an input multimode β ¼ α U : ð3Þ ρ k j jk Gaussian quantum state in, which is a product state of j the individual states fρsg in each mode; see Fig. 1. We are then interested in the output probabilities of detecting N Using this equation the probability distribution (1) is then single photons, given by pðnÞ¼Tr½ρ jnihnj; ð1Þ YM out − 2 P pðnÞ¼e I jβ j nk ; ð4Þ n … ∈ 0 1 k where ¼ðn1;n2;n3; ;nMÞ, ns f ; g, sns ¼ N, k¼1 ρ Uρ U† U and out ¼ in with being the unitary operator that P P M β 2 M α 2 describes the linear-optical network.
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