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].

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 distribution 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. In practice, one must where I ¼ k j kj ¼ j j jj . This probability distri- use photon-number-resolving detectors in order to distin- bution can be efficiently calculated using a classical guish the single-photon events from events in which a computer. This implies that there exists an efficient detector registers more than one photon. Hence, in classical algorithm for boson sampling with coherent states.

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Note, however, that coherent states are useful for efficiently where characterizing linear-optical networks that are indispen- YM sable for the classical verification of boson sampling in 2 jα j ns ns 2 Pnn α e s ∂α ∂α¯ δ α 9 practice [16]. ð Þ¼ s s ð sÞðÞ In deriving a general formula for calculating the prob- s¼1 ability distribution (1), without loss of generality, we make is the P function of the number state jnihnj, ns ∈ f0; 1g, two assumptions about input Gaussian states for Gaussian n n n 2 with ∂α ≔ ∂ =∂α and δ α ≡ δ Re α δ Im α [18]. boson sampling. First, we assume that the input states have ð Þ ð ð ÞÞ ð ð ÞÞ Integration by parts yields zero first order moments. This is because any displacement operations before the linear-optical network are equivalent YM to some displacement operations at the output, which will ns ns Fðα;α¯Þ p n K ∂α ∂ ¯ e ; 10 ð Þ¼ s αs jαs¼0 ð Þ not change the correlations between output states [17]. s¼1 Second, we assume the covariance matrices of the Gaussian states ρs are diagonal with the variance in the x quadrature where

Vxs being larger than or equal to the variance in the p quadrature V . The reason is that, in general, any local DC~ ps F α; α¯ α~ α~†; phase-shift operation before the linear-optical network can ð Þ¼ C¯ 0 ð11Þ be absorbed into the unitary operation describing the network. We use the Q function to represent each input with D~ ¼ 1 − D, 1 being the M × M identity matrix. In the ρ Gaussian state s, above expression, we have to take 2N derivatives with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi respect to independent variables fα ; α¯ jn ≠ 0g at α ¼ 0; μ2 − 4λ2 s s s α s s λ α2 α¯ 2 − μ α 2 hence, that expression can be written as Qsð sÞ¼ π exp ½ sð s þ s Þ sj sj ; ð5Þ X∞ where pðnÞ¼K Lð2N; F; rÞ; ð12Þ r¼1 1 1 1 1 λ − μ s ¼ ; s ¼ þ ; L 2N; F; r 2V þ 2 2V þ 2 V þ 1 V þ 1 where ð Þ, analogous to distributing distinguish- ps xs xs ps able balls into indistinguishable boxes, can be understood as a sum over all possible ways to distribute 2N derivatives and for the vacuum state V V 1. The parameter λ is i1 ir x ¼ p ¼ s (balls)P among r functions (boxes), ∂ F; …; ∂ F, such that between zero (when Vp ¼ Vx ) and infinity (for infinite r 2 ≠ 0 α α¯ s s s¼1 is ¼ N and is .AsFð ; Þ is a second order squeezing), and μ is between zero (for infinite variances) i s polynomial in α and α¯, and ∂ s Fjα 0 ¼ 0 for i ≠ 2, only and one (for pure states). The Q function of the output state ¼ s Lð2N; F; NÞ for is ¼ 2 is nonzero. Therefore, we obtain the using Eq. (3) can be calculated as desired formula for calculating the probabilities of N 1 1 single-photon detections as α α Uρ U† α η ρ η Qoutð Þ¼ M h j in j i¼ M h j inj i π π 2 −1 !! ð NXÞ YN ∂2F YM XM pðnÞ¼K ; ð13Þ α U¯ ∂Xi ∂Xi ¼ Qs j js ; ð6Þ i l¼1 2l−1 2l s¼1 j¼1 2N − 1 !! † ¯ where the sum is over ð Þ possible ways of where jηi¼U jαi¼jαUi is an M-mode coherent state. i i 2N distributing 2N balls (∂=∂X , where fX g 1 ¼fα ; By using the expression for the input Q function (5), the l l l¼ s α¯ sjns ≠ 0g) into N boxes (F’s) such that each box contains output Q function can be written in this compact form: two balls. In the following, by using this new formula, we consider two cases of thermal states and squeezed-vacuum K −DC Q α exp α~ α~† ; 7 states as inputs. outð Þ¼πM C¯ 0 ð Þ Boson sampling with thermal states.—If one subjects M Q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi thermal states with the same temperatures, i.e., μs ¼ M 2 2 with α~ ≔ α1; …; α ; α¯ 1; …; α¯ , K μ − 4λ , ð M MÞ ¼ s¼1 s s 2=ðVs þ 1Þ¼μ and λs ¼ 0 for all s, to a linear-optical C UλUT D UμU† λ λ … λ ¼ , ¼ , where ¼ diagð 1; ; MÞ and network, we have D ¼ μ1 and C ¼ 0 in the output Q μ μ … μ ¼ diagð 1; ; MÞ. Now, by using this Q function, the function (7). In this case the output Q function is identical probability distribution (1) is then given by to the input Q function and no correlation is created. Here Z we assume the input thermal states have different temper- n π M 2Mα α α atures such that the matrix D is not diagonal, in general. pð Þ¼ð Þ d Qoutð ÞPnnð Þ; ð8Þ CM In this case, the formula (13) becomes

060501-3 week ending PRL 114, 060501 (2015) PHYSICAL REVIEW LETTERS 13 FEBRUARY 2015 YM XN! YN ∂2 also be calculated by using the output probabilities for input pðnÞ¼ μ ½αD~ α¯ T; ð14Þ s ∂Xi ∂Xi coherent state (4) and the P functions of the input states: s¼1 i l¼1 2l−1 2l Z YM XM 2n 2 2 2 − α 2 k i N α 1 i N α¯ 1 n Mα α j kj α where fX2l−1gl¼1 ¼f sjns ¼ g and fX2lgl¼1 ¼f sjns ¼ g. pð Þ¼ d Pkð kÞe jUjk : ð17Þ CM By comparing this equation with the definition of a k¼1 j permanent [1], it can be seen by inspection that Therefore, according to the above argument, for all of the P YM functions that are valid probability density functions, the n μ D~ NP pð Þ¼ s Perð½ N×NÞ: ð15Þ above integral can be approximated in BPP . s¼1 Boson sampling with squeezed-vacuum states.—Let us now consider squeezed-vacuum states whose variances in Thus, the probabilities of having N simultaneous single- 2rs −2rs the x and p quadratures are Vx ¼ e and Vp ¼ e , photon detections at the output are proportional to perma- s s ~ respectively, where rs is the squeezing parameter for input nents of N × N submatrices of the Hermitian matrix D, μ 1 D~ 0 mode s. In this case, weQ have s ¼ for all s, ¼ , D~ −1 denoted by ½ N×N. The submatrices are obtained by λ ¼ðtanh r Þ=2 and K ¼ M ðcosh r Þ . − − s s s¼1 s removing M N rows and the same M N columns As the function (11) becomes Fðα; α¯Þ¼F1ðαÞþ T corresponding to those output modes from which no F1ðα¯Þ, F1ðαÞ¼αCα ,wehave∂α ∂α¯ Fjα 0 ¼ 0, for ~ † j j ¼ photon was detected. Notice that we have D ¼ Uμ~U , any i and j. Thus, by using the formula (13), the probability ~ where the elements of matrix μ are ð1 − μjÞδij ≥ 0; hence, distribution for detecting N single photons at the output is D~ D~ given by and its principal submatrices ½ N×N are positive- semidefinite Hermitian matrices. 2 YM ðNX−1Þ!! YN=2 2 We now see whether boson sampling with thermal states 1 ∂ F1ðαÞ p n ; 18 can be efficiently simulated classically. Each input thermal ð Þ¼ ∂ i ∂ i ð Þ 1 cosh rs i 1 X2l−1 X2l state can be expressed as a Gaussian statistical mixture of s¼ l¼ coherent states due to the Glauber-Sudarshan representa- i N α 1 where fXlgl¼1 ¼f sjns ¼ g. One can immediately see tion [19,20] from this distribution that, independent of what the linear- Z optical network is, the probability of detecting an odd ρth 2α th α α α number of single photons at the output is always zero, as j ¼ d jPj ð jÞj jih jj; ð16Þ C expected from squeezed-vacuum inputs. The probabilities (18) are proportional to the modulus squared of this th α where Pj ð jÞ is a Gaussian P function for the thermal state quantity: to input mode j. By choosing a random set of input α M ðNX−1Þ!! YN=2 2 coherent states with amplitudes f jgj¼1 from the proba- ∂ F1ðαÞ th M ON ¼ i i ; ð19Þ bility distributions fP ðα Þg 1, one can efficiently find ∂X ∂X j j j¼ i l¼1 2l−1 2l β M the amplitudes of output coherent states f kgk¼1 and the probability distribution from Eq. (4). This implies that there which depends on the off-diagonal elements of the matrix exists an efficient classical algorithm for boson sampling C and the number of detected single photons. Notice that with thermal states. Hence, using Stockmeyer’s approxi- quantity ON is not a permanent, but it is a sum of ðN − 1Þ!! mate counting algorithm [13], the probability (15) for a complex numbers. Considering that the matrix C is NP ∂ ∂ α 2 specific n can be approximated in BPP . As any arbitrary symmetric, cij ¼ cji,wehave α α F1ð Þ¼ cij, with ≠ i j positive-semidefinite Hermitian matrix D~ 0 can be written i j. Hence, the above quantity can be written as D~ 0 U μ~U† ≥ 1 D~ 0 as ¼ q with q , we then have Perð½ N×NÞ¼ X X X N D~ O ¼ c c ðc …c Þ q Perð½ N×NÞ, which is proportional to the output prob- N i1i2 i3i4 i2k−1i2k iN−1iN ≠ ≠ ≠ ability (15). Therefore, using Stockmeyer’s algorithm, the i1 i2 i3 i4 i2k−1 i2k permanent of any arbitrary positive-semidefinite Hermitian × 2N=2; ð20Þ matrix, despite having complex number elements, can be NP approximated in BPP , which is in the third level of the where i1 ¼ 1, il ≠ i1; …;il−1 for 2 ≤ l ≤ N. polynomial hierarchy. Unless the polynomial hierarchy For a particular case of boson sampling with squeezed- collapses to this level, this problem is not #P hard. vacuum states, it has been shown that sampling cannot be Based on the above argument, boson sampling with any simulated classically [14]. Consider an M-mode linear- classical input states, i.e., quantum states with non-negative optical network, which consists of M=2 beam splitters with P functions, can be efficiently simulated with a classical a π=2-phase shifter at one of the input ports and an M=2- computer as well. Notice that the output probabilities can mode linear-optical network that acts only on half of the

060501-4 week ending PRL 114, 060501 (2015) PHYSICAL REVIEW LETTERS 13 FEBRUARY 2015 output modes of the beam splitters. By feeding this This research was conducted by the Australian Research M-mode network with M squeezed-vacuum states, the Council Centre of Excellence for Quantum Computation beam splitters generate M=2 two-mode entangled and Communication Technology (Project No. CE110 (two-mode squeezed-vacuum) states. Then, conditional 001027). on detecting N=2 single photons from one particular configuration of the output modes of beam splitters, N=2 single photons in the corresponding other modes 9 are subjected to the M=2-mode network, and the problem [1] S. Aaronson and A. Arkhipov, Theory Comput. , 143 reduces to that of the original boson sampling. This implies (2013). [2] M. A. Broome, A. Fedrizzi, S. Rahimi-Keshari, J. Dove, S. that sampling from the single-photon-counting probability Aaronson, T. C. Ralph, and A. G. White, Science 339, 794 distribution at the output of the M-mode network cannot be (2013). simulated classically and thus, following the Aaronson and [3] J. B. Spring, B. J. Metcalf, P. C. Humphreys, W. S. Arkhipov results [1], for at least this type of configuration, Kolthammer, X. Jin, M. Barbieri, A. Datta, N. Thomas-Peter, 2 approximating jONj is a #P-hard problem. It would be N. K. Langford, D. Kundys, J. C. Gates, B. J. Smith, P. G. R. surprising if this were the only configuration for which Smith, and I. A. Walmsley, Science 339, 798 (2013). 2 ć approximating jONj was #P hard, as the squeezed-vacuum [4] M. Tillmann, B. Daki , R. Heilmann, S. Nolte, A. Szameit, states are highly nonclassical with a highly singular P and P. Walther, Nat. Photonics 7, 540 (2013). function and the output is almost always an entangled state [5] A. Crespi, R. Osellame, R. Ramponi, D. J. Brod, E. F. Galvão, N. Spagnolo, C. Vitelli, E. Maiorino, P. Mataloni, [17]. This result may be of interest to computational 7 complexity theory as a way of identifying other classically and F. Sciarrino, Nat. Photonics , 545 (2013). [6] N. Spagnolo, C. Vitelli, M. Bentivegna, D. J. Brod, A. hard problems besides the computing of permanents. — Crespi, F. Flamini, S. Giacomini, G. Milani, R. Ramponi, P. Conclusion. We have presented new results that are Mataloni, R. Osellame, E. F. Galvão, and F. Sciarrino, Nat. interesting from quantum computation, computational Photonics 8, 615 (2014). complexity theory, and optics perspectives, by considering [7] M. C. Tichy, K. Mayer, A. Buchleitner, and K. Mølmer, the problem of sampling from the output probability Phys. Rev. Lett. 113, 020502 (2014). distribution of a linear-optical network for input [8] K. R. Motes, J. P. Dowling, and P. P. Rohde, Phys. Rev. A Gaussian states. Our results show that the consideration 88, 063822 (2013). of problems in quantum optics can help to classify and [9] S. Scheel, Permanents in linear optical networks, arXiv: identify new problems in computational complexity theory. quant-ph/0406127. [10] L. Valiant, Theor. Comput. Sci. 8, 189 (1979). There are two interesting open questions. The first question 467 is whether permanents of positive-semidefinite Hermitian [11] S. Aaronson, Proc. R. Soc. A , 3393 (2011). [12] S. Toda, SIAM J. Comput. 20, 865 (1991). matrices can be approximated with an algorithm similar to [13] L. J. Stockmeyer, SIAM J. Comput. 14, 849 (1985). the algorithm for matrices with non-negative entries [15] in [14] A. P. Lund, A. Laing, S. Rahimi-Keshari, T. Rudolph, J. L. BPP. Note that the probabilities (15) for input thermal states O’Brien, and T. C. Ralph, Phys. Rev. Lett. 113, 100502 and (18) for squeezed-vacuum states are special cases of the (2014). formula (13) for general squeezed thermal input states. By [15] M. Jerrum, A. Sinclair, and E. Vigoda, J. Assoc. Comput. adding sufficient thermal noise to input squeezed-vacuum Mach. 51, 671 (2004). states, they will become classical with positive P function [16] S. Rahimi-Keshari, M. A. Broome, R. Fickler, A. Fedrizzi, and, as shown, sampling can be simulated classically. T. C. Ralph, and A. G. White, Opt. Express 21, 13450 Hence, the second question is, as we add thermal noise (2013). 88 to pure squeezed-vacuum input states, at what point does [17] Z. Jiang, M. D. Lang, and C. M. Caves, Phys. Rev. A , sampling become classically simulatable; does entangle- 044301 (2013). [18] L. Mandel and E. Wolf, Optical Coherence and Quantum ment play any role? Optics (Cambridge University Press, New York, 1995). We thank Howard Wiseman for the discussions and [19] R. J. Glauber, Phys. Rev. Lett. 10, 84 (1963). 10 Scott Aaronson and Alex Arkhipov for their comments. [20] E. C. G. Sudarshan, Phys. Rev. Lett. , 277 (1963).

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