Boson Sampling with efficient scaling and efficient verification

Raphael A. Abrahao1, 2, ∗ and Austin P. Lund3, †

1Centre for Engineered Quantum Systems, School of Mathematics and Physics,University of Queensland, St Lucia, Brisbane, Queensland 4072, Australia

2Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario, Canada K1N 6N53Centre for Quantum Computation and Communication Technology, School of Mathematics and Physics,

University of Queensland, St Lucia, Brisbane, Queensland 4072, Australia(Dated: April 14, 2020)

A universal quantum computer of moderate scale is not available yet, however intermediate modelsof quantum computation would still permit demonstrations of a quantum computational advantageover classical computing and could challenge the Extended Church-Turing Thesis. One of these mod-els based on single photons interacting via linear optics is called Boson Sampling. Proof-of-principleBoson Sampling has been demonstrated, but the number of photons used for these demonstrationsis below the level required to claim quantum computational advantage. To make progress with thisproblem, here we conclude that the most practically achievable pathway to scale Boson Sampling ex-periments with current technologies is by combining continuous-variables quantum information andtemporal encoding. We propose the use of switchable dual-homodyne and single-photon detections,the temporal loop technique and scattershot based Boson Sampling. This proposal gives details as towhat the required assumptions are and a pathway for a quantum optical demonstration of quantumcomputational advantage. Furthermore, this particular combination of techniques permits a singleefficient implementation of Boson Sampling and efficient verification in a single experimental setup.

I. INTRODUCTION

Boson Sampling is a model of intermediate—as op-posed to universal—quantum computation initially pro-posed to confront the limits of classical computation com-pared to quantum computation [1]. An efficient classicalcomputation of the Boson Sampling protocol would sup-port the Extended Church-Turing Thesis “which assertsthat classical computers can simulate any physical pro-cess with polynomial overhead”[2], i.e., polynomial timeand memory requirements. But an efficient classical al-gorithm for Boson Sampling would also imply that thePolynomial Hierarchy (PH) of complexity classes, whichis believed to have an infinite number of discrete levels,would reduce (or “collapse”) to just three levels. Con-sequently, a computer scientist could not simultaneouslysupport the Extended Church-Turing Thesis and an in-finite structure of the PH. Hence one is cornered intoa position that either a fundamental change in compu-tational complexity is needed or quantum enabled algo-rithms must be able to perform some tasks efficiently thatcannot be performed efficiently on a classical computer.An example of such a task is the quantum Shor’s algo-rithm [3] for factorization which is an extremely impor-tant result due to the role that factoring prime numbershas in cryptography. Even in the absence of a full-scalequantum computer, a physically constructed Boson Sam-pling device could outperform a classical device samplingfrom the same distribution, and therefore, it is one ofthe leading candidates in the quest for a quantum opti-cal demonstration of quantum computational advantage

∗ raphael.abrahao@gmail.com† a.lund@uq.edu.au

Phase-Shifter 50:50 Beamsplitter

(a)

(b) DET

DET

DET

DET

Single Photon Vacuum state

DET

DET

DET

DET

Um x m

Figure 1. (a) Conceptual schematics for the Boson Samplingprotocol: n photons, eg. n = 2, enter a linear optical networkwhose action is represented by a m x m unitary matrix (U),eg. m = 4, which relates the input amplitudes to the outputamplitudes. The m − n remaining inputs are considered tobe in a vacuum state. The outputs are recorded using single-photon detectors (DET). (b) Conceptual implementation ofBoson Sampling. Here the unitary U is implemented in spatialmodes using phase-shifters and 50:50 beamsplitters.

[2, 4–6]. The first experimental demonstration of quan-tum computational advantage was published in October2019 for a sampling problem using 53 qubits in a super-conducting circuit architecture[7].

In a simplified view, an implementation of the BosonSampling protocol can be summarized as following: n in-distinguishable single photons are inputs into the ports of

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0

2

50:50

Modulator

HomodyneDetection

movable mirrorDET

DET

X1

X2

X1

X2

HomodyneDetection

ττ

τN

movable mirror

τ

Figure 2. (a) Schematics for Continuous-Variable Boson Sampling using temporal encoding. Input states are pulsed-Gaussian-squeezed states on orthogonal basis, whose temporal difference is given by τ . The 50:50 beamsplitter interfere those 2 inputs.The photons at the upper arm enter a “Modulator”, while photons at the bottom arm propagate freely. At each arm, movablemirrors are placed to direct light in a characterization stage using Homodyne Detection. After the characterization, the movablemirrors must be removed and let light go to the corresponding single-photon detectors (DET) to generate the output (samples)for the Boson Sampling protocol, this last stage similarly to the scattershot case [8].

a m modes linear optical network, represented by a Uni-tary matrix (U), and at the output ports single-photondetection is performed (Fig.1). Any alleged Boson Sam-pling device must give samples from this output distri-bution for any given U . Proof-of-principle implementa-tions of Boson Sampling have been successfully demon-strated, initially using single-photon pairs from Sponta-neous Parametric Down Conversion (SPDC) and laterusing single photons from Quantum Dots (QD) [9–17].However, even the current world record of 20 photonsin 60 modes [18] is well below any threshold of quan-tum computational advantage. Three factors are cur-rently contributing against quantum demonstrations ofBoson Sampling: (a) better classical algorithms whichmove the threshold of quantum computational advantageto greater number of input single photons [19, 20], e.g.the classical algorithm of Neville et al. [19] solved theBoson Sampling problem with 30 photons in a standardcomputer efficiently; (b) difficulties on the scaling of thepreparation of manifold single photons (n |1〉); and (c)scaling of photon losses in the linear optical network [21–23]. Therefore, at this point in time Boson Samplingfaces an unclear future with difficult perspectives. Moti-vated by these constraints, we propose a method to scaleBoson Sampling experiments using continuous-variablequantum information and time-bin encoding. Our pro-posal also takes into account finite squeezing and givensome reasonable assumptions hold, operational perfor-mance can be characterized efficiently.

Our proposal presents an efficient way to implementBoson Sampling experiments, and also an efficient wayto verify it using the same experimental setup. Due to

the exponential sized sampling space at the output, it is adifficult problem in general to find all the pre-conditionsrequired to be satisfied on the output distribution, giventhat this distribution is inevitability going to be non-ideal. The condition presented in [1], was on the totalvariation distance for the output distribution comparedwith the ideal distribution. For the exponentially sizedsample space, an exponentially sized ensemble is requiredto directly estimate this distance without any prior as-sumptions. Here, by changing the detection type andmaking the assumption that the state before detection isGaussian, we show that estimating the total variation ispossible via the state covariance matrix. This is a quan-tity that has a polynomial dependence on the input sizeand hence this statistical distance can be efficiently esti-mated.

II. APPLICATIONS OF BOSON SAMPLING

It is noteworthy that the current search for applica-tions of Boson Sampling goes beyond the scope of com-putational complexity. For instance, Boson Sampling hasbeen adapted to simulate molecular vibrational spectra[24, 25] and may be used as a tool for quantum simulation[26, 27]. Other Boson Sampling-inspired applications arethe verification of NP-complete problems [28], quantummetrology sensitivity improvements [29], and a quantumcryptography protocol [30].

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III. PROPOSAL

Here we present a proposal to scale Boson Sampling ex-periments based on continuous variables (CV) and tem-poral encoding. In the CV case, the information is en-coded on the quantum modes of light, specifically, on theeigenstates of operators with continuum spectrum [31].Continuous-variables quantum information has achievedimpressive results. An initial report of 10,000 entangledmodes in a continuous-variable cluster mode [32] was lat-ter upgraded to one million modes [33]. Some of thesesystems were conceived to perform measurement-basedquantum computation (MBQC), and here we show theycan be adapted to Boson Sampling. Moreover, whilesome of the theoretical work for MBQC assume unre-alistically infinite squeezing, here we require only finitesqueezing. The world record for detected squeezed lightis 15db [34], while it is estimated a 20.5db threshold ofsqueezing is needed for fault-tolerant quantum compu-tation using Gottesman-Kitaev-Preskill (GKP) encoding[35, 36].

The work of Lund et al. [8] (a.k.a. Scattershot BosonSampling) demonstrated Gaussian states can be used asinputs in Boson Sampling experiments and only boundedsqueezing is necessary, provided each output is projectedin the number basis by single-photon detection. It is im-portant at this point to emphasise that the specific taskof Scattershot Boson Sampling requires the generationof a different distribution to standard Boson Sampling.One result presented in [8] was to show that samplingfrom this alternative distribution is also a hard prob-lem when employing classical computing resources. Thesingle photon Boson Sampling distribution is containedwithin the Scattershot Boson Sampling distribution, butthis is merely used as part of the proof for computationalhardness. The requisite task is the efficient generation ofsamples from a Gaussian state measured in the Fock basiswithout any further processing.

A detailed discussion of how much squeezing is neces-sary for Scattershot Boson Sampling experiments can befound on [8]. Interestingly, the authors [8] showed thatfor a two-mode squeezer, like SPDC, there is a trade-offbetween the strength of the SPDC (linked to χ) and themost likely number of photons detected represented bythe variable n. This indicates that Scattershot BosonSampling experiments that are done with less photonsrequire higher χ levels:

P (n) =

(m

n

)χ2n(1− χ2)m︸ ︷︷ ︸

standard Boson Sampling︸ ︷︷ ︸scattershot Boson Sampling

, (1)

where this probability is locally maximised when

χ =

√n

m+ n(2)

and this maximum probability is lower bounded by 1/√n

if m ≥ n2 and n > 1. In this regime, χ decreases as n

increases and when taking m = n2 at n = 8 only 3dB ofsqueezing is required to achieve this optimal probability.See [37] for clarification of the notation on the ScattershotBoson Sampling.

More recently, a new variant of Boson Sampling wasproposed in which the Boson Sampling protocol is for-mulated in terms of the Hafnian of a matrix, a more gen-eral function belonging to the #P class, and was provenfor the case of exact sampling. The case of approxi-mate sampling is not proven, but it seems very plausi-ble. This protocol is called Gaussian Boson Sampling[38, 39]. In this present work, we use the terminologyscattershot Boson Sampling and Gaussian input BosonSampling as synonyms. A demonstration of scattershot,Gaussian, and standard Boson Sampling using integratedoptics and single photons from Spontaneous Four WaveMixing (SFWM) sources can be found in Reference [40].

A. Scaling

Now, we will analyse the scaling of our Boson Samplingexperimental proposal. Consider two pulsed-squeezed-light sources, with τ time interval between subsequentpulses, where these two states interfere in a 50:50 beam-splitter, followed by a controllable delay, where a pulsecan be delayed by Nτ before being released, Fig.(2). Thismay be a loop architecture [41] or a quantum memory,provided it returns the given input state with enoughhigh fidelity. Here we will use the loop architecture. Themodulator should implement the desired unitary U op-eration by interfering delayed pulses. At the end of eachspatial path, there are two possible measurement schemesthat can be performed. Either, the light can be sent toa single-photon detector to record the samples (output)for Boson Sampling, or the light is directed towards ahomodyne detection [42, 43] setup that is used to char-acterize the output state, including the output state fromthe optical network.

The use of time-bin (loop architecture) for Boson Sam-plings has been proposed and experimentally demon-strated [15, 41]. This technique converts somewhat ex-pensive spatial modes into cheaper temporal modes. Thisgives a quadratic reduction in required spatial modes andprovides a significant benefit in scaling to a large numberof single photons as input states for the Boson Samplingprotocol. However, the original time-bin proposal wasconceived for the discrete case, i.e., manifold single pho-tons. The verification in the discrete case, as addressedin the article [44] requires the tested state to be generatedO(nmn) times when verifying a process with n photonsin m modes. So, for m = n2, this is worse than an expo-nential growth. Therefore, the discrete case constitutesa not scalable approach for Boson Sampling when alsoconsidering verification. The verification of the BosonSampling protocol is an important aspect of our proposaland is discussed in section Verification below.

A significant benefit to our approach is that, under

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some reasonable assumptions, the operation of the sam-pling device can be characterized using the samplingstate itself without the need for other probe input states.To achieve this, the following assumptions are needed: (i)the output state received by the single-photon detectionis the same as that received by the homodyne detection,which is achievable by movable mirrors, for example asin the procedure given by [45]; (ii) the two squeezed in-put states are Gaussian and that the modulation networkchanges the states but leaves the output still in a Gaus-sian form, a standard Gaussian optics property; (iii) theoutput is fully characterized by a multi-mode covariancematrix, and finally (iv) the choice of when to make asampling run and when to made a characterization runis irrelevant. In other words, the experimental setup isassumed stable and the output will not change over thetime as one changes between the two different measure-ment schemes.

A Gaussian output state can be fully characterized bythe mean vector (which we will assume zero) and covari-ance matrix. For a m mode state and n detected pho-tons, the number of possible photon number detectionevents scales as mn. However, to describe a Gaussianstate before the detection has occurred, only the numberof entries in a covariance matrix for a m modes state isrequired and this scales as 4m2. For the case of Gaussianinput Boson Sampling (a.k.a. scattershot) where thereare two groups of m modes and n photon detections, thesize of the Fock basis detection sample space is m2n, butthe full covariance matrix for the state prior to detectionwill require 16m2 entries.

Performing the characterization involves reconstruct-ing the covariance matrix from the CV measurementsamples. The measurements chosen must be sufficientin number to estimate all elements of the covariance ma-trix, including terms involving the correlations betweenX and P in the same mode. To avoid repeated changesto measurement settings, we propose performing this bymeans of dual homodyne. In a dual-homodyne arrange-ment, the signal mode is split at a 50:50 beamsplitterand both modes undergo a CV homodyne detection, onemeasured in X and the other in P. This permits a simul-taneous measurement of the X and P quadratures at thecost of adding 1/2 a unit of vacuum noise to the diagonalelements of the state covariance matrix. So, if Σ is thestate covariance matrix, then the dual homodyne modeswill see Gaussian statistics with a covariance matrix of(Σ + I)/2 (under units where the variance of vacuumnoise is unity), where I is the identity operator. Thiscovariance matrix can then be estimated by constructingmatrix-valued samples from each sampling run. Let

si = (x1,i, p1,i, x2,i, p2,i, . . . , xm,i, pm,i)T (3)

be a 2m-dimensional real vector representing the ith datasample from the dual-homodyne measurement with thefirst subscript representing the mode to which the cor-responding homodyne detector is attached. From thissample vector, a sample matrix can be formed from the

outer product of the si

ξi = sisTi . (4)

This sample matrix is then a positive semi-definite matrixfor all i. The expectation value for each sample ξi overthe incoming Gaussian distribution is then

〈ξi〉 = (Σ + I)/2 (5)

and so a sample average over K samples

ξ =1

K

K∑i=1

ξi (6)

will be an unbiased estimator for (Σ + I)/2.To see how close the sample average is to the true

average, we apply the operator Chernoff bound [46, 47](following the notation of Wilde [46], Section 16.3). Thisgives the probability that the sample average deviates sig-nificantly from the expected value. Let K be the numberof sample matrices and ξ the sample average of K sam-ples as defined in Eq. 6. The input state covariancematrix Σ is positive definite, and we have

(Σ + I)/2 ≥ I/2 (7)

which is the expectation of each operator forming thesum in Eq. 6. For this situation, the operator Chernoffbound for any 0 < η < 1/2 is given by

Pr{(1− η)(Σ + I)/2 ≤ ξ ≤ (1 + η)(Σ + I)/2}

≥ 1− 8me−Kη2/(8 ln 2). (8)

To then bound the probability for making a multiplica-tive estimate of Σ, the spectrum of Σ needs to be boundedaway from zero.

Let the parameter b represent the variance of thequadrature for the maximum possible squeezing for thestate being estimated. This means that

Σ ≥ bI. (9)

The Chernoff bound for the estimator ξ can be rewrit-ten as

Pr{(1− η)Σ− ηI ≤ 2ξ − I ≤ (1 + η)Σ + ηI}

≥ 1− 8me−Kη2/(8 ln 2). (10)

Then using the inequality in Eq. 9, this can be writtenas

Pr{(1− η(1 + b−1)Σ) ≤ 2ξ − I ≤ (1 + η(1 + b−1)Σ)}

≥ 1− 8me−Kη2/(8 ln 2). (11)

This means the rewritten estimate 2ξ − I gives a multi-plicative estimate of the covariance matrix Σ.

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The interpretation of Equation 11 is that the chancethat the finite sample estimate of the covariance ma-trix deviates from the true value decays exponentiallyin the number of covariance matrix samples K and thesquare of deviation permitted η, but depends linearly onm, the number of modes. In our application of Gaus-sian input Boson Sampling, the value of b is fixed as toomuch squeezing can actually degrade performance. Sofor fixed η, as the number of modes m increases, thenumber of samples K required to achieve the same prob-ability bound in the operator Chernoff bound only growslogarithmically O(lnm).

B. Verification

Finally, one would like to verify if the generated stateis sufficient to perform the task at hand, that is BosonSampling. For approximate Boson Sampling, one doesnot need to generate the state ideally but within sometrace distance bound ε. Using the Fuchs-van de Graffinequality, a trace distance is upper bounded by the fi-delity by 1 − F < ε. A robust certification strategy isgiven by Aolita et al. [44], which tests if a fidelity lowerbound (or equivalently maximum trace distance) holdsbetween a pure Gaussian target state and a potentiallymixed preparation state. Here, we argue that this resultcan also be employed for verification. In order to performthe verification, the Gaussian covariance matrix elementsneed to be estimated and manipulated with knowledge ofthe target pure state. This produces a bound of the fi-delity which can be used to test for appropriateness ofthe apparatus to perform Gaussian input Boson Sam-pling. The samples needed to achieve a fixed fidelitybound (or fixed trace distance) is higher than the Cher-noff bound and scales as O(m4) times, where m is thenumber of modes in the state being verified. This verifi-cation process can require considerable amounts of datato scale, but the scaling with system size is polynomial,making the process feasible. This is as opposed to theverification in the discrete variables model as addressedin the same article [44] which requires the tested state tobe generated O(nmn) samples when verifying a processwith n photons in m modes.

C. Sampling

After the stage of verification is finished, the movablemirrors must be removed and direct the light toward thesingle-photon detectors. Doing so, one is projecting theGaussian states into a Fock basis, and thus obtaining theoutput of the Boson Sampling experiment, similarly asin the Gaussian input Boson Sampling (a.k.a. scatter-shot) [8]. Our proposed method greatly simplifies thenumbers of required resources for scaling Boson Sam-pling experiments. Here we benefit from having well veri-fied states, with verification growing polynomially as dis-

cussed above, and from having only two squeezed lightsources, and thus simplifying the preparation of inputstates. Our method also requires less detectors. For in-stance, if one wishes to implement a 20 input single pho-tons Boson Sampling, then it requires a 400× 400 linearoptical network, and therefore 400 single-photon detec-tors. Not obeying m � n, at least m = n2, violatesthe mathematical assumptions upon which the approx-imate Boson Sampling problem is currently formulated,and therefore can only the interpreted as an experimen-tal proof-of-principle. In our proposal, due to the time-bin implementation of the linear optical network, only 2single-photon detectors are required.

D. The role of imperfections

An important consideration in the performance of anysampling device is the role that imperfections, originat-ing from any process, have on the ability one has to makeconclusions about the classical easiness or hardness ofcomputing random samples. Crucially, the dominant im-perfection using current technology is photon losses. Theeffects of photon losses are included within the approxi-mate sampling requirements, i.e., how much one can devi-ate from the perfect sampling. Unfortunately, this doesnot mean that losses can be neglected, as they will inmany cases give rise to exponential scaling in the totalvariation distance between the lossless distribution andlossy distribution. For example, a constant loss rate foreach mode will induce an exponential scaling in the totalvariation distance as a function of the number of pho-tons to be detected. It would seem that this impliesthat any level of loss would render classically hard Bo-son Sampling impossible, but this is not the case. Thegoal to demonstrate the quantum computational advan-tage only requires producing samples that are close tosome distribution, given some ε tolerance, that is hardfor a classical computer to reproduce. Aaronson andBrod [48] showed that the distribution generated fromlosing a fixed number of photons (i.e. not scaling withthe number of photons) is hard for a classical computerto sample. A lossy Boson Sampling device will be closeto this distribution at some scale. Unfortunately, the to-tal variation distance for constant loss per mode will stillasymptotically scale exponentially against this distribu-tion with a fixed number of lost photons. Oszmaniec andBrod [22] showed that if the total number of photonsthat remained after loss scaled as

√n of the number in-

put photons n, then a simple distribution from samplingdistinguishable bosons would satisfy the total variationdistance requirement for approximate sampling. Hencean efficient classical computation could reproduce sam-ples close to the required Boson sampling distribution,nullifying the quantum computational advantage. In alater article, Brod and Oszmaniec studied the case ofnonuniform losses. [49].

Despite advances in understanding the effect of losses

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[21–23, 48–53], there remains a gap between the neces-sary and sufficient criterion for the hardness of approxi-mate sampling with lossy Boson Sampling devices, whichmakes this an important open question. All implementa-tions will be subject to imperfections, including the pro-posed implementation presented here. However, in thispresent work, we are primarily concerned with the exper-imental implications of scaling and verifying continuous-variables Boson Sampling. We expect that future resultson the hardness of lossy Boson Sampling would be ableto be incorporated into the proposal we present.

IV. DISCUSSION

Continuous-variables (CV) quantum information, par-ticularly in the context of optical Gaussian states [54],has been put forward as an alternative for quantum com-putation. Due to the scaling factors discussed in the pre-vious sections, we point out that Boson Sampling cangreatly benefit from the current optical CV technology[32–34, 55–57]. In this sense, all the building blocks forthis proposal have been successfully demonstrated andhave good performance for scaling. The threshold forquantum computational advantage in the CV regime iscurrently uncertain and the subject of further investiga-tion, but our Boson Sampling proposal certainly does notsuffer from the scaling issues of discrete Boson Sampling,in particular the issue of preparing a large number ofindistinguishable single photons.

In summary, we revisited the motivation behind Bo-

son Sampling and the experimental challenges currentlyfaced. Despite impressive improvements towards demon-strating a quantum computational advantage using Bo-son Sampling, the current number of input single pho-tons and modes are considerably below what is necessary.Photon losses and scaling of many input single photonsare factors working against quantum implementations ofBoson Sampling. These facts pose great challenges andmake evident a new scalable approach is necessary. Herewe presented a new method to do so based on continu-ous variables and temporal encoding. Our method as-sumes finite squeezing and also provides a feasible wayto perform the characterization of the input states andthe verification of the Boson Sampling protocol, provid-ing viable scaling as the system size increases. With thisapproach, the quest for a quantum optical demonstra-tion of quantum computational advantage moves closerto experimental reality.

V. ACKNOWLEDGMENTS

We thank Andrew G. White, T. C. Ralph, T. Wein-hold, and Nathan Walk for helpful discussions. Thiswork was supported by the Australian Research Coun-cil Centre of Excellence for Engineered Quantum Sys-tems (Grants No. CE170100009) and the Australian Re-search Council Centre of Excellence for Quantum Com-putation and Communication Technology (Grant No.CE170100012).

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