Continuous-Variables Boson Sampling: Scaling and Verification
A universal quantum computer of a moderate scale is not available yet, however intermediate models of quantum computation would still permit demonstrations of a quantum computational advantage over classical computing. One of these models, based on single photons interacting via linear optics, is called Boson Sampling. Proof-of-principle Boson Sampling has been demonstrated, but the number of photons used for these demonstrations is well below any claim of quantum computational advantage. To circumvent this problem, here we present a new pathway to scale Boson Sampling experiments by combining continuous-variables quantum information and temporal encoding. In our proposal, we use dual-homodyne and single-photon detections. By simply switching detection methods the performance of the device can be verified with a number of measurement samples growing polynomially in the number of photons. All building blocks of our proposal have been successfully demonstrated and have shown good performance for scaling. This proposal is within the reach of current technology.
Boson Sampling is a model of intermediate—as opposed to universal—quantum computation initially proposed to confront the limits of classical computation compared to quantum computation AA (). An efficient classical computation of the Boson Sampling protocol would support the Extended Church-Turing Thesis ”which asserts that classical computers can simulate any physical process with polynomial overhead”harrow17 (), i.e., polynomial time and memory requirements. But an efficient classical algorithm for Boson Sampling would also imply that the Polynomial Hierarchy (PH) of complexity classes, which is believed to have an infinite number of discrete levels, would reduce (or ”collapse”) to just three levels. Consequently, a computer scientist cannot simultaneously support the Extended Church-Turing Thesis and an infinite structure of the PH. Hence one is cornered into a position that either a fundamental change in computational complexity is needed or quantum enabled algorithms must be able to perform some tasks efficiently that cannot be performed efficiently on a classical computer. An example of such a task is the quantum Shor’s algorithm shor1994 () for factorization which is an extremely important result due to the role that factoring prime numbers has in cryptography. Even in the absence of a full-scale quantum computer, a physically constructed Boson Sampling device could outperform a classical device producing the same output, and therefore, it is one of the leading candidates for the quest for an initial demonstration of quantum computational advantage harrow17 (); lund2017 (); review_road_QCS ().
In a simplified view, an implementation of the Boson Sampling protocol can be summarized as following: indistinguishable single photons are inputs into the ports of an mode linear optical network, represented by a Unitary matrix (), and at the output ports single-photon detection is performed (Fig.1). Any alleged Boson Sampling device must give samples from this output distribution for any given . Proof-of-principle implementations of Boson Sampling have been successfully demonstrated, initially using single photons from Spontaneous Parametric Down Conversion (SPDC) and latter using Quantum Dots (QD)BroomeBS (); SpringOX-BS (); tillmannVienna (); CrespiRome (); ScattershotRome (); LoredoBS (); JWP_BS_2017 (); JWPan_BS_2018_PRL (); flamini2018photonic (). For a more detailed discussion on Boson Sampling and on the the experimental challenges, refer to the Supplementary Material. However, even the current world record of 7 photons in 16 modes JWPan_BS_2018_PRL () is well below any threshold of quantum computational advantage. Three factors are currently contributing against quantum demonstrations of Boson Sampling: (a) better classical algorithms which move the threshold of quantum computational advantage to greater number of input single photons nevilleBristol2017 (); Clifford&Clifford (), e.g. the classical algorithm of Neville et al. nevilleBristol2017 () solved the Boson Sampling problem with 30 photons in a standard computer efficiently; (b) difficulties on the scaling of the preparation of manifold single photons (); and (c) scaling of photon losses in the linear optical network Patron_Renema (); Dan_Brod_err (). Therefore, at this point in time Boson Sampling faces an unclear future with difficult perspectives. Motivated by these constraints, we present a new method to scale Boson Sampling experiments using continuous-variable quantum information and time-bin encoding. Our proposal also takes into account finite squeezing and given some reasonable assumptions hold, operational performance can be characterized efficiently.
Ii Continuous-variables and time-bin Boson Sampling
Here we present an alternative way to scale Boson Sampling experiments based on continuous variables (CV) and temporal encoding. In the CV case, the information is encoded on the quantum modes of light, specifically, on the eigenstates of operators with continuum spectrum kok2010introduction (). Continuous-variable quantum information has achieved impressive results. An initial report of 10,000 entangled modes in a continuous-variable cluster mode Furusawa_10000 () was latter upgraded to one million modes Furusawa_one_million (). Some of these systems were conceived to perform measurement-based quantum computation (MBQC), and here we show they can be adapted to Boson Sampling. Moreover, while some of the theoretical work for MBQC assume unrealistically infinite squeezing, here we require only finite squeezing. The world record for detected squeezed light is 15db Roman_15db (), while it is estimated a 20.5db threshold of squeezing is needed for fault-tolerant quantum computation using Gottesman-Kitaev-Preskill (GKP) encoding Menicicci2014 (); GKP_encoding (). The work of Lund et al. Lund2014 () (a.k.a. Scattershot Boson Sampling) demonstrated Gaussian states can be used as inputs in Boson Sampling experiments and only bounded squeezing is necessary, provided each output is projected in the number basis by single-photon detection. The discussion of how much squeezing is necessary for Scattershot Boson Sampling experiments can be found on Lund2014 () and it is commented in detail on the ”Experimental Demonstrations” section in the Supplementary Material of this article. But for the sake of exemplification, take the number of modes to be the square of the number of input single photons, i.e., , then at only dB of squeezing is required to achieve the optimal operational probability in ideal Scattershot Boson Sampling, not considering losses.
Consider two pulsed-squeezed-light sources, with time interval between subsequent pulses, where these two states are mixed by a 50:50 beamsplitter, followed by a controllable delay, where a pulse can be delayed by before being released, Fig.(2). This may be a loop architecture Time-Bin_BS_motes () or a quantum memory. The modulator should implement the desired unitary operation by interfering delayed pulses. At the end of each spatial path, there are two possible measurement schemes that can be performed. Either, the light can be sent to a single-photon detector to record the samples (output) for Boson Sampling, or the light is directed towards a homodyne detection leonhardt1997measuring (); Bachor&Ralph () setup that is used to characterize the output state, including the output state from the optical network.
A significant benefit to this approach is that, under some reasonable assumptions, the operation of the sampling device can be characterized using the sampling state 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 detection is the same as that received by the homodyne detection, which is achievable by movable mirrors, for example as in the procedure given by webb2006homodyne (); (ii) the two squeezed input states are Gaussian and that the modulation network changes the states but leaves the output still in a Gaussian form, a standard Gaussian optics property; (iii) the output is fully characterized by a multi-mode covariance matrix, and finally (iv) the choice of when to make a sampling run and when to made a characterization run is irrelevant. In order words, the experimental setup is assumed stable and the output will not change over the time as one changes between the two different measurement schemes.
A Gaussian output state can be fully characterized by the mean vector (which we will assume zero) and covariance matrix. For an mode state and detected photons, the number of possible photon number detection events scales as . However, to describe a Gaussian state before the detection has occurred, only the number of entries in a covariance matrix for an modes state is required and this scales as . For the case of Gaussian input Boson Sampling (a.k.a. scattershot) where there are two groups of modes and photon detections, the size of the Fock basis detection sample space is , but the full covariance matrix for the state prior to detection will require entries.
Performing the characterization involves reconstructing the covariance matrix from the CV measurement samples. The measurements chosen must be sufficient in number to estimate all elements of the covariance matrix, including terms involving the correlations between X and P in the same mode. To avoid repeated changes to measurement settings, we propose performing this by means of dual homodyne. In a dual-homodyne arrangement, the signal mode is split at a 50:50 beamsplitter and both modes undergo a CV homodyne detection, one measured in X and the other in P. This permits a simultaneous measurement of the X and P quadratures at the cost of adding a unit of vacuum noise to the diagonal elements of the state covariance matrix. So if is the state covariance matrix, then the dual homodyne modes will see Gaussian statistics with a covariance matrix of (under units where the variance of vacuum noise is unity), where is the identity operator. This covariance matrix can then be estimated by constructing matrix-valued samples from each sampling run. Let
be a -dimensional real vector representing the data sample from the dual-homodyne measurement with the first subscript representing the mode to which the corresponding homodyne detector is attached. From this sample vector, a sample matrix can be formed from the outer product of the
This sample matrix is then a positive semi-definite matrix for all . The expectation value for each sample over the incoming Gaussian distribution is then
and so a sample average over samples
will be an unbiased estimator for .
To see how close the sample average is to the true average, we apply the operator Chernoff bound (following the notation of Wilde wilde_2013 (), Section 16.3). This gives the probability that the sample average deviates significantly from the expected value. Let be the number of sample matrices, the sample average of K samples, the expectation value defined in Eq. 3. Also, suppose the expectations value is greater that the identity scaled by a number , where represents the minimum variance in the most squeezed quadrature, and is a parameter representing the allowable error level satisfying , and . The parameter can also be interpreted as representing the variance of the quadrature for the maximum possible squeezing for the state being estimated. The parameter forms a multiplicative error rate which bounds the probability of deviation (See Supplementary Material). Therefore one would usually desire to be small, but for any value of away from zero, the final inequality puts an upper bound on for which the Chernoff bound holds. Then the operator Chernoff bound for this estimate is:
The interpretation of Equation (5) is that the chance that the finite sample estimate of the covariance matrix deviates from the true value decays exponentially in the number of covariance matrix samples , the deviation permitted and the smallest squeezing variance but depends linearly on , the number of modes. In our application of Gaussian input Boson Sampling, the value of is fixed as too much squeezing can actually degrade performance. So for fixed , as the the number of modes increases, the number of samples required to achieve the same probability bound in the operator Chernoff bound only grows logarithmically .
Finally, one would like to verify if the generated state is sufficient to perform the task at hand, that is Boson Sampling. For approximate Boson Sampling, one does not need to generate the state ideally but within some trace distance bound . Using the Fuchs-van de Graff inequality, a trace distance is upper bounded by the fidelity by . A robust certification strategy is given by Aolita et al. Aolita2015 (), which tests if a fidelity lower bound (or equivalently maximum trace distance) holds between a pure Gaussian target state and a potentially mixed preparation state. In order to perform the verification, the Gaussian covariance matrix elements need to be estimated and manipulated with knowledge of the target pure state. This produces a bound of the fidelity which can be used to test for appropriateness of the apparatus to perform Gaussian input Boson Sampling. The samples needed to achieve a fixed fidelity bound (or fixed trace distance) is higher than the Chernoff bound and scales as times, where is the number of modes in the state being verified. This verification process can require considerable amounts of data to scale, but the scaling with system size is polynomial making the process feasible. This is as opposed to the verification in the discrete variables model as addressed in the same paper Aolita2015 () which requires the tested state to be generated samples when verifying a process with photons in modes.
After the stage of verification is finished, the movable mirrors must be removed and direct the light toward the single-photon detectors. Doing so, one is projecting the Gaussian states into a Fock basis, and thus obtaining the output of the Boson Sampling experiment, similarly as in the Gaussian input Boson Sampling (a.k.a. scattershot) Lund2014 (). Our proposed method greatly simplifies the numbers of required resources for scaling Boson Sampling experiments. Here we benefit from having well verified states, with verification growing polynomially as discussed above, and from having only two squeezed light sources, and thus simplifying the preparation of input states. Our method also requires less detectors. For instance, if one wishes to implement a 20 input single photons Boson Sampling, then it requires a x linear optical network, and therefore 400 single-photon detectors. Not obeying , at least , violates the mathematical assumptions upon which the approximate Boson Sampling problem is currently formulated, and therefore can only the interpreted as an experimental proof-of-principle. In our proposal, due to the time-bin implementation of the linear optical network, only 2 single-photon detectors are required.
Continuous Variables (CV) quantum information, particularly in the context of optical Gaussian states weedbrook2012gaussian () has been put forward as an alternative for quantum computation. Due to the scaling factors discussed in the previous section, we point out that Boson Sampling can greatly benefit from the current optical CV technology Furusawa_10000 (); Furusawa_one_million (); takeda2017universal (); spie_travelling_wave_of_light (); Roman_15db (). In this sense, all the building blocks for this proposal have been successfully demonstrated and have good performance for scaling. The threshold for quantum computational advantage in the CV regime is currently uncertain and object of further investigation, but our Boson Sampling proposal certainly does not suffer from the scaling issues as the discrete Boson Sampling, specially preparing a large number of indistinguishable single photons.
In addition to that, it is noteworthy that the current search for applications of Boson Sampling goes beyond the scope of computational complexity. For instance, Boson Sampling has been adapted to simulate molecular vibrational spectra molecularBS_theory (); molecularBS_exp () and may be used as a tool for quantum simulation aspuru2012photonic (); georgescu2014quantum (). Other Boson Sampling-inspired applications are the verification of NP-complete problems verifying_NP () and quantum metrology sensitivity improvements MORDOR (). Moreover, new platforms are being proposed for the quest of quantum computational advantage boixo2018characterizing () including superconducting qubits neill2018blueprint ().
In summary, we revisited the motivation behind Boson Sampling and the experimental challenges currently faced. Despite notorious improvements to demonstrate quantum computational advantage using Boson Sampling, the current number of input single photons and modes are considerably below what is necessary. Photon losses and scaling of many input single photons are factors working against quantum implementations of Boson Sampling. These facts pose great challenges and makes evident a new scalable approach is necessary. Here we presented a new method to do so based on continuous variables and temporal encoding. Our method assumes finite squeezing and also provides a feasible way to perform the characterization of the input states and the verification of the Boson Sampling protocol, providing viable scaling as the system size increases. With this approach, the quest for quantum computational advantage moves closer to experimental reality.
We thank Andrew G. White, T. C. Ralph, and T. Weinhold for helpful discussions. This work was supported by the Centre for Engineered Quantum Systems (Grants No. CE170100009) and the Centre for Quantum Computation and Communication Technology (Grant No. CE170100012).
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Appendix A Introduction to Boson Sampling
Linear optical circuits with post-selection are known to be able to perform universal quantum computation kok_review_rmp_2007 (). In particular the Knill-Laflamme-Milburn (KLM) KLM () protocol shows how this can be implemented using ”beam splitters, phase shifters, single photon sources, photo-detectors and quantum memory” by exploiting post-selection and error correction to achieve arbitrarily high probabilities of success. Boson sampling is a restricted model of quantum computation, which means that not all state transformations can be achieved and hence it is not universal. The original proposal for Boson Sampling by Aaronson and Arkhipov AA () is based on the manipulation of single photons using linear optics and is sometimes referred to as a discrete variable approach. Consider indistinguishable single photons injected on a network of linear optics with modes, where , at least . The Boson Sampling protocol assumes that the linear network is described by a large unitary matrix. But one can take advantage of a decomposition where any unitary network in linear optics can be decomposed in a network of beamsplitters and phase-shifters whose number is quadratically related to the unitary matrix dimension (Fig.1). The most commonly used decomposition of this form is the Reck decomposition Reck94 (), while a recent symmetric and optimized version can be found in Clem2016 (). The output of a Boson Sampling device is not deterministic and hence is mathematically represented by a probability distribution over the outputs of detecting the photons. What a Boson Sampling device does is not to compute this probability distribution but to output samples from this distribution. The form of the underlying probability distribution for Boson Sampling is used to argue about the computational hardness of this sampling problem. The distribution can be expressed in terms of the matrix permanent from a submatrix of the unitary describing the linear optical network. The matrix permanent is a quantity computed similarly to the matrix determinant, but without the alternations of ”+” and ”–” signs. Permanents are, from a computational perspective, hard to calculate valiant1979complexity (), i.e. time to compute scales asymptotically to an exponential function.
Aaronson and Arkhipov AA () proved that Boson Sampling works for the exact case, and provided a reasonable argument for accepting that the conclusions of Boson Sampling are still valid for a more realistic case of approximate sampling, i.e. when one must assume a small divergence between the actual measured state and the desired one .
Appendix B Experimental Demonstrations
The early experiments of Boson Sampling provided proof-of-principle demonstrations, limited to 2 to 4 input photons in 5 to 6 modes BroomeBS (); SpringOX-BS (); CrespiRome (); tillmannVienna (). All these demonstrations used single photons produced by Spontaneous Parametric Down Conversion (SPDC), a nonlinear and intrinsically probabilistic but fundamentally quantum coherent process. In SPDC, a pump laser is directed through a nonlinear crystal, and when the phase-matching and energy conservation conditions are fulfilled, a pair of single-photons in the down-conversion output modes can be probabilistically produced. As the process is coherent, the photon number (Fock basis ) representation is given by:
where is the photon number on the mode and is a parameter representing the strength of the SPDC. The higher order terms with are generally undesirable and can generate errors in most quantum information processing protocols Till_higher_order_terms_errors (). Therefore, SPDC sources are not typically run at high powers, what decreases the probability of producing the higher order terms. This state is also a Gaussian state with covariance matrix
(using scaled and to vacuum noise) where we have written the squeeze parameter such that . Another way to write this is in terms of squeezing relative to the vacuum state in decibels. This is computed by comparing the smallest covariance eigenvalue, which is , to the vacuum and computing the logarithmic decibel quantity,
An important step towards scalability is a variant of the Boson Sampling protocol called ”Boson Sampling from a Gaussian State” Lund2014 () (a.k.a. scattershot) (Fig.3), a protocol already demonstrated at small scales ScattershotRome (). In this alternative version Lund et al. Lund2014 () proved that Gaussian states can be injected as inputs while the outputs are projected on the numbers basis by using single-photon detectors. Typical single-photon detectors are Avalanche Photodiodes (APDs) and Superconducting Nanowire Detectors (SNDs). Note that APDs and SNDs are not number-resolving single-photon detectors, i.e. they operate on a ”click/no-click” basis, and a ”click” event means that at least one photon was detected. However, as explained before, the probability of producing the high order terms in the SPDC is minimized by reducing the input power that drives the SPDC process. Therefore, APDs and SNDs have been commonly used on Boson Sampling experiments. The scattershot Boson Sampling takes advantage of the vast multitude of combinations from multiple independent SPDC sources generating a particular total number of photons in the output modes as Eq. (9), irrespective of which mode each pair is probabilistically generated in. Interestingly, the authors Lund2014 () showed that for a two-mode squeezer, like SPDC, there is a trade-off between the strength of the SPDC (linked to ) and the number of total detected photons, which indicates Boson Sampling experiments could be done with less photons at the cost of higher levels.
This probability is locally maximised when
and this maximum probability is lower bounded by if and . In this regime, decreases as increases and when taking at only dB of squeezing is required to achieve this optimal probability.
Alternatively, solid-state single-photon sources based on quantum dots (QD) senellartReview (); lodahl2015interfacing () have achieved good rates of single-photon production. Demonstrations of Boson Sampling using QD started in 2017 LoredoBS (); JWP_BS_2017 () and achieved the current world-record with 7 photons in a 16-mode photonic circuit JWPan_BS_2018_PRL ().
From the experimental point of view, it is imperative to establish how tolerant Boson Sampling is to photons losses. This question is commonly rephrased as determining the scaling of the rate of photon loss that permits an efficient classical simulation of Boson Sampling, hence nullifying any computational advantage Arkhipov_error_analysis (); rohde2012error (); Aaronson_error (); L_Patron_error (); rahimi2016sufficient (); Dan_Brod_err (); Patron_Renema (). In particular, let us revisit three results on photon loss: (i) Arkhipov Arkhipov_error_analysis () demonstrated the tolerance in error of each element of the optical network should scale as for the requirements of the Boson Sampling protocol to remain valid; (ii) Oszmaniec et al. Dan_Brod_err () argue ”the output statistics can be well approximated by an efficient classical simulation, provided that the number of photons that is left grows slower than ”; (iii) García-Patrón et al. argue ”all current architectures that suffer from an exponential decay of the transmission with the depth of the circuit […] can be efficiently simulated classically” and ”either the depth of the circuit is large enough that it can be simulated by thermal noise with an algorithm running in polynomial time, or it is short enough that a tensor network simulation runs in quasipolynomial time”, and explicitly state that a new paradigm for implementation of Boson Sampling is needed in order to reach quantum computational advantage. One implementation demonstrated tolerance for few photons losses JWPan_BS_2018_PRL ().
Appendix C Two Paths to scale Boson Sampling experiments
c.1 Using Spatial Encoding
Quantum Dots produce single photons on the same spatial mode, but different temporal modes. To produce manifold single photons (), one must be able to move photons into multiple spatial modes. Attempts based on bulk optics rely on using fast Electrical-Optical-Modulators (EOMs) to change a photon polarization to select a particular path as used in JWPan_BS_2018_PRL (). It was also demonstrated that this demultiplexing stage can be achieved using photonics chips integrated waveguides MuChOS ().
c.2 Using Temporal Encoding
Time-bin encoding uses a single optical path, but manipulates each photon at a different time Time-Bin_Oxford_2013 (); Photon_Temporal_Modes_2015 (). It circumvents the necessity of having a demultiplexing stage as in the previous case and simplifies the requirements to mechanically stabilize many optical paths, specially when interference effects are involved. Motes et al. Time-Bin_BS_motes () proposed to use time-bin encoding in a loop-architecture to scale Boson Sampling experiments and their approach is still based on discrete-variables Boson Sampling. The first demonstration of time-bin Boson Sampling was given in 2017 JWP_BS_2017 () with 4 photons.
c.3 Estimated Covariance Matrix Confidence Interval
The operator Chernoff bound assumes the operator inequality where lower bounds the spectrum of eigenvalues from . From this we have and . The Chernoff bound for the estimator is
which can be rewritten as
then using the inequalities between and this can be written as
This means the rewritten estimate gives a multiplicative estimate of the covariance matrix .