We introduce an exact classical algorithm for simulating Gaussian Boson Sampling (GBS). The complexity of the algorithm is exponential in the number of photons detected, which is itself a random variable. For a fixed number of modes, the complexity is in fact equivalent to that of calculating output probabilities, up to constant prefactors. The simulation algorithm can be extended to other models such as GBS with threshold detectors, GBS with displacements, and sampling linear combinations of Gaussian states. In the specific case of encoding non-negative matrices into a GBS device, our method leads to an approximate sampling algorithm with polynomial runtime. We implement the algorithm, making the code publicly available as part of Xanadu's The Walrus library, and benchmark its performance on GBS with random Haar interferometers and with encoded Erdős-Renyi graphs.
We study the hardness of classically simulating Gaussian boson sampling at nonzero photon distinguishability. We find that similar to regular boson sampling, distinguishability causes exponential attenuation of the many-photon interference terms in Gaussian boson sampling. Barring an open problem in the theory of matrix permanents, this leads to an efficient classical algorithm to simulate Gaussian boson sampling in the presence of distinguishability. We also study a new form of boson sampling based on photon number superposition states, for which we also show noise sensivity. The fact that su…
Gaussian boson sampling is a promising model for demonstrating quantum computational supremacy, which eases the experimental challenge of the standard boson-sampling proposal. Here by analyzing the computational costs of classical simulation of Gaussian boson sampling,we establish a lower bound for achieving quantum computational supremacy for a class of Gaussian boson-sampling problems, where squeezed states are injected into every input mode. Specifically, we propose a method for simplifying the brute-force calculations for the transition probabilities in Gaussian boson sampling, leading to …
We introduce the notion of "generalized bosons" whose exchange statistics resemble those of bosons, but the local bosonic commutator $[a_i,a_i^\dagger]=1$ is replaced by an arbitrary single-mode operator that is diagonal in the generalized Fock basis. Examples of generalized bosons include boson pairs and spins. We consider the analogue of the boson sampling task for these particles and observe that its output probabilities are still given by permanents, so that the results regarding hardness of sampling directly carry over. Finally, we propose implementations of generalized boson sampling in …
Boson sampling solves a classically intractable problem by sampling from a probability distribution given by matrix permanents. We propose a scalable implementation of Boson sampling using local transverse phonon modes of trapped ions to encode the Bosons. The proposed scheme allows deterministic preparation and high-efficiency readout of the Bosons in the Fock states and universal mode mixing. With the state-of-the-art trapped ion technology, it is feasible to realize Boson sampling with tens of Bosons by this scheme, which would outperform the most powerful classical computers and constitute…
Boson-sampling is a simplified model for quantum computing that may hold the key to implementing the first ever post-classical quantum computer. Boson-sampling is a non-universal quantum computer that is significantly more straightforward to build than any universal quantum computer proposed so far. We begin this chapter by motivating boson-sampling and discussing the history of linear optics quantum computing. We then summarize the boson-sampling formalism, discuss what a sampling problem is, explain why boson-sampling is easier than linear optics quantum computing, and discuss the Extended C…
Efficiently simulating the work distribution of multiple identical bosons with boson sampling
Center for Quantum Technology Research and Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurements (MOE), School of Physics, Beijing Institute of Technology（北京理工大学物理学院量子技术研究中心及先进光电子量子架构与测量教育部重点实验室）
It is predicted that quantum computers will dramatically outperform their conventional counterparts. However, large-scale universal quantum computers are yet to be built. Boson sampling is a rudimentary quantum algorithm tailored to the platform of photons in linear optics, which has sparked interest as a rapid way to demonstrate this quantum supremacy. Photon statistics are governed by intractable matrix functions known as permanents, which suggests that sampling from the distribution obtained by injecting photons into a linear-optical network could be solved more quickly by a photonic experi…
Boson sampling is a well-defined task that is strongly believed to be intractable for classical computers, but can be efficiently solved by a specific quantum simulator. However, an outstanding problem for large-scale experimental boson sampling is the scalability. Here we report an experiment on boson sampling with photon loss, and demonstrate that boson sampling with a few photons lost can increase the sampling rate. Our experiment uses a quantum-dot-micropillar single-photon source demultiplexed into up to seven input ports of a 16*16 mode ultra-low-loss photonic circuit, and we detect thre…
Boson-Sampling holds the potential to experimentally falsify the Extended Church Turing thesis. The computational hardness of Boson-Sampling, however, complicates the certification that an experimental device yields correct results in the regime in which it outmatches classical computers. To certify a boson-sampler, one needs to verify quantum predictions and rule out models that yield these predictions without true many-boson interference. We show that a semiclassical model for many-boson propagation reproduces coarse-grained observables that were proposed as witnesses of Boson-Sampling. A te…
On computational complexity and average-case hardness of shallow-depth boson sampling
NextQuantum Innovation Research Center, Department of Physics and Astronomy, Seoul National University（首尔大学物理与天文学系Next量子创新研究中心）；Department of Physics, Korea Advanced Institute of Science and Technology（韩国科学技术院物理系）
A boson sampling device could efficiently sample from the output probability distribution of noninteracting bosons undergoing many-body interference. This problem is not only classically intractable, but its solution is also believed to be classically unverifiable. Hence, a major difficulty in experiment is to ensure a boson sampling device performs correctly. We present an experimental friendly scheme to extract useful and robust information from the quantum boson samplers based on coarse-grained measurements. The procedure can be applied to certify the equivalence of boson sampling devices w…
Boson sampling can simulate physical problems for which classical simulations are inefficient. However, not all problems simulated by boson sampling are classically intractable. We consider a situation in which it is known that the outcome from boson sampling is sparse. It can be determined from a few marginal distributions which are classically calculable. Still, recovering of the joint distribution can be of high complexity. We show classically efficient methods of the recovery assuming high sparsity of the joint distribution. Various extensions are discussed including a version involving qu…
It has been proven that Boson sampling is a much promising model of optical quantum computation, which has been applied to designing quantum computer successfully, such as "Jiuzhang". However, the meaningful randomness of Boson sampling results, whose correctness and significance were proved from a specific quantum mechanical distribution, has not been utilized or exploited. In this research, Boson sampling is applied to design a novel Quantum Random Number Generator (QRNG) by fully exploiting the randomness of Boson sampling results, and its prototype system is constructed with the programmab…
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 in [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 \textit{hafnian} matrix function and us it to design a Gaussian Boson sampling protocol. Then, we discuss the advantages that this protocol has relative to other photonic …
Boson-sampling has attracted much interest as a simplified approach to implementing a subset of optical quantum computing. Boson-sampling requires indistinguishable photons, but far fewer of them than universal optical quantum computing architectures. In reality, photons are never indistinguishable, and exhibit a rich spectral structure. Here we consider the operation of boson-sampling with photons of arbitrary spectral structure and relate the sampling statistics of the device to matrix permanents. This sheds light on the computational complexity of different regimes of the photons' spectral …
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 res…
Boson sampling can provide strong evidence that the computational power of a quantum computer outperforms a classical one via currently feasible linear optics experiments. However, how to identify an actual boson sampling device against any classical computing imposters is an ambiguous problem due to the computational complexity class in which boson sampling lies. The certification protocol based on bosonic bunching fails to rule out the so-called mean-field sampling. We propose a certification scheme to distinguish the boson sampling from the mean-field sampling for any random scattering matr…
We demonstrate how boson sampling with photons of partial distinguishability can be expressed in terms of interference of fewer photons. We use this observation to propose a classical algorithm to simulate the output of a boson sampler fed with photons of partial distinguishability. We find conditions for which this algorithm is efficient, which gives a lower limit on the required indistinguishability to demonstrate a quantum advantage. Under these conditions, adding more photons only polynomially increases the computational cost to simulate a boson sampling experiment.
Boson sampling is the problem of generating a quantum bit stream whose average is the permanent of a $n\times n$ matrix. The bitstream is created as the output of a prototype quantum computing device with $n$ input photons. It is a fundamental challenge to verify boson sampling, and the question of how output count rates scale with matrix size $n$ is crucial. Here we apply results from random matrix theory to establish scaling laws for average count rates in boson sampling experiments with arbitrary inputs and losses. The results show that, even with losses included, verification of nonclassic…
Boson Sampling is the problem of sampling from the same distribution as indistinguishable single photons at the output of a linear optical interferometer. It is an example of a non-universal quantum computation which is believed to be feasible in the near term and cannot be simulated on a classical machine. Like all purported demonstrations of "quantum supremacy", this motivates optimizing classical simulation schemes for a realistic model of the problem, in this case Boson Sampling when the implementations experience lost or distinguishable photons. Although current simulation schemes for suf…
The first post-classical computation will most probably be performed not on a universal quantum computer, but rather on a dedicated quantum hardware. A strong candidate for achieving this is represented by the task of sampling from the output distribution of linear quantum optical networks. This problem, known as boson sampling, has recently been shown to be intractable for any classical computer, but it is naturally carried out by running the corresponding experiment. However, only small scale realizations of boson sampling experiments have been demonstrated to date. Their main limitation is …
The boson sampling problem has brought a lot of attention in the quantum information field because it is not efficiently solvable with a classical computer; nonetheless it can be implemented with linear optical interferometers with single-boson sources. Recently, we introduced a more general problem, the multi-boson correlation sampling problem, which allows us to take advantage of the multi-mode spectral distribution of the bosonic sources together with time-correlated measurements in order to achieve sampling not only over the output ports of the interferometer but also over the joint detect…
Boson sampling is a simple model for non-universal linear optics quantum computing using far fewer physical resources than universal schemes. An input state comprising vacuum and single photon states is fed through a Haar-random linear optics network and sampled at the output using coincidence photodetection. This problem is strongly believed to be classically hard to simulate. We show that an analogous procedure implements the same problem, using photon-added or -subtracted squeezed vacuum states (with arbitrary squeezing), where sampling at the output is performed via parity measurements. Th…
We present methods which allow orders of magnitude increase in the number of modes in linear optics experiments by moving from spatial encoding to temporal encoding and using dispersion. This enables significant practical advantages for linear quantum optics and Boson Sampling experiments. Passing consecutively heralded photons through time-independent dispersion and measuring the output time of the photons is equivalent to a Boson Sampling experiment for which no efficient classical algorithm is reported, to our knowledge. With time-dependent dispersion, it is possible to implement arbitrary …
The boson sampling problem has triggered a lot of interest in the scientific community because of its potential of demonstrating the computational power of quantum interference without the need of non-linear processes. However, the intractability of such a problem with any classical device relies on the realization of single photons approximately identical in their spectra. In this paper we discuss the physics of boson sampling with non-identical single photon sources, which is strongly relevant in view of scalable experimental realizations and triggers fascinating questions in complexity theo…
Optical losses are the main obstacle to demonstrating a quantum advantage via boson sampling without leaving open the possibility of classical spoofing. We propose a method for generating low-depth optical circuits suitable for boson sampling with very high efficiencies. Our circuits require only a constant number of optical components (namely three) to implement an optical transformation suitable for demonstrating a quantum advantage. Consequently, our proposal has a constant optical loss regardless of the number of optical modes. We argue that sampling from our family of circuits is computat…
We show how phase-space simulations of Gaussian quantum states in a photonic network permit verification of measurable correlations of Gaussian boson sampling (GBS) quantum computers. Our results agree with experiments for up to 100-th order correlations, provided decoherence is included. We extend this to more than 16,000 modes, and describe how to simulate genuine multipartite entanglement.
Boson sampling is a promising candidate for quantum supremacy. It requires to sample from a complicated distribution, and is trusted to be intractable on classical computers. Among the various classical sampling methods, the Markov chain Monte Carlo method is an important approach to the simulation and validation of boson sampling. This method however suffers from the severe sample loss issue caused by the autocorrelation of the sample sequence. Addressing this, we propose the sample caching Markov chain Monte Carlo method that eliminates the correlations among the samples, and prevents the sa…
Boson-sampling is a highly simplified, but non-universal, approach to implementing optical quantum computation. It was shown by Aaronson and Arkhipov that this protocol cannot be efficiently classically simulated unless the polynomial hierarchy collapses, which would be a shocking result in computational complexity theory. Based on this, numerous authors have made the claim that experimental boson-sampling would provide evidence against, or disprove, the Extended Church-Turing thesis -- that any physically realisable system can be efficiently simulated on a Turing machine. We argue against thi…
The determination of the computational complexity of the boson sampling problem with single boson sources has opened a novel research direction in the quantum computing field. Some research effort has also been devoted towards the use of different input sources where the sampling from the output distribution of a linear interferometer is still a complex task not achievable by a classical device. In this letter, we discuss the use of multi-mode thermal sources, based on a generalization of the boson sampling problem which relies on time-correlation measurements. We derive, for any possible samp…
BosonSampling is a quantum mechanical task involving Fock basis state preparation and detection and evolution using only linear interactions. A classical algorithm for producing samples from this quantum task cannot be efficient unless the polynomial hierarchy of complexity classes collapses, a situation believe to be highly implausible. We present method for constructing a device which uses Fock state preparations, linear interactions and Gaussian continuous-variable measurements for which one can show exact sampling would be hard for a classical algorithm in the same way as Boson Sampling. T…
Boson Sampling is the problem of sampling from the same output probability distribution as a collection of indistinguishable single photons input into a linear interferometer. It has been shown that, subject to certain computational complexity conjectures, in general the problem is difficult to solve classically, motivating optical experiments aimed at demonstrating quantum computational "supremacy". There are a number of challenges faced by such experiments, including the generation of indistinguishable single photons. We provide a quantum circuit that simulates bosonic sampling with arbitrar…
An important step for photonic quantum technologies is the demonstration of a quantum advantage through boson sampling. In order to prevent classical simulability of boson sampling, the photons need to be almost perfectly identical and almost without losses. These two requirements are connected through spectral filtering, improving one leads to a decrease of the other. A proven method of generating single photons is spontaneous parametric downconversion (SPDC). We show that an optimal trade-off between indistinguishability and losses can always be found for SPDC. We conclude that a 50-photon s…
Hybrid boson sampling-neural network architecture for enhanced classification
Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China（电子科技大学基础与前沿研究院）；Key Laboratory of Quantum Physics and Photonic Quantum Information, Ministry of Education, University of Electronic Science and Technology of China（电子科技大学量子物理与光子量子信息教育部重点实验室）；Shimmer Center, Tianfu Jiangxi Laboratory（天府江西实验室 shimmer 中心）
We develop an alternative boson sampling model operating on single-photon states followed by linear interferometry and Gaussian measurements. The hardness proof for simulating such continuous-variable measurements is established in two main steps, making use of the symmetry of quantum evolution under time reversal. Namely, we first construct a twofold version of scattershot boson sampling in which, as opposed to the original proposal, both legs of a collection of two-mode squeezed vacuum states undergo parallel linear-optical transformations. This twofold scattershot model yields, as a corolla…
Certification of Gaussian Boson Sampling via graph theory
Sapienza Università di Roma（罗马大学）；Dipartimento di Fisica, Sapienza Università di Roma（罗马大学物理系）；Dipartimento di Informatica, Sapienza Università di Roma（罗马大学计算机系）
We study what is arguably the most experimentally appealing Boson Sampling architecture: Gaussian states sampled with threshold detectors. We show that in this setting, the probability of observing a given outcome is related to a matrix function that we name the Torontonian, which plays an analogous role to the permanent or the Hafnian in other models. We also prove that, provided that the probability of observing two or more photons in a single output mode is sufficiently small, our model remains intractable to simulate classically under standard complexity-theoretic conjectures. Finally, we …
We give a full description of the problem of Multi-Boson Correlation Sampling (MBCS) at the output of a random interferometer for single input photons in arbitrary multimode pure states. The MBCS prob- lem is the task of sampling at the interferometer output from the probability distribution associated with polarization- and time-resolved detections. We discuss the richness of the physics and the complexity of the MBCS problem for nonidentical input photons. We also compare the MBCS problem with the standard bo- son sampling problem, where the input photons are assumed to be identical and the …
Sampling the distribution of bosons that have undergone a random unitary evolution is strongly believed to be a computationally hard problem. Key to outperforming classical simulations of this task is to increase both the number of input photons and the size of the network. We propose driven boson sampling, in which photons are input within the network itself, as a means to approach this goal. When using heralded single-photon sources based on parametric down-conversion, this approach offers an $\sim e$-fold enhancement in the input state generation rate over scattershot boson sampling, reachi…
Boson sampling devices are a prime candidate for exhibiting quantum supremacy, yet their application for solving problems of practical interest is less well understood. Here we show that Gaussian boson sampling (GBS) can be used for dense subgraph identification. Focusing on the NP-hard densest k-subgraph problem, we find that stochastic algorithms are enhanced through GBS, which selects dense subgraphs with high probability. These findings rely on a link between graph density and the number of perfect matchings -- enumerated by the Hafnian -- which is the relevant quantity determining samplin…
We pose a generalized 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 Gaussian input states, a linear optical network and non-adaptive photon counting measurements. All the elements required to build such a processor currently exist. The demonstration of such a device would provide the first empirical evidence that quantum computers can indeed outperform classical computers and could lead to applications.
Scientists in quantum technology aspire to quantum advantage: a computational result unattainable with classical computers. Gaussian boson sampling experiment has been already claimed to achieve this goal. In this setup squeezed light states interfere in a mid-sized linear optical network, where multi-photon collisions take place. The exact simulation of the counting statistics of $n$ threshold detectors is far beyond the possibilities of modern supercomputers once $n$ exceeds $100$. Here we challenge quantum advantage for a mid-sized Gaussian boson sampling setup and propose the approximate a…
Random point patterns are ubiquitous in nature, and statistical models such as point processes, i.e., algorithms that generate stochastic collections of points, are commonly used to simulate and interpret them. We propose an application of quantum computing to statistical modeling by establishing a connection between point processes and Gaussian Boson Sampling, an algorithm for special-purpose photonic quantum computers. We show that Gaussian Boson Sampling can be used to implement a class of point processes based on hard-to-compute matrix functions which, in general, are intractable to simula…
We present an architecture for arbitrarily scalable boson-sampling using two nested fiber loops. The architecture has fixed experimental complexity, irrespective of the size of the desired interferometer, whose scale is limited only by fiber and switch loss rates. The architecture employs time-bin encoding, whereby the incident photons form a pulse train, which enters the loops. Dynamically controlled loop coupling ratios allow the construction of the arbitrary linear optics interferometers required for boson-sampling. The architecture employs only a single point of interference and may thus b…
Boson sampling is a specific quantum computation, which is likely hard to implement efficiently on a classical computer. The task is to sample the output photon number distribution of a linear optical interferometric network, which is fed with single-photon Fock state inputs. A question that has been asked is if the sampling problems associated with any other input quantum states of light (other than the Fock states) to a linear optical network and suitable output detection strategies are also of similar computational complexity as boson sampling. We consider the states that differ from the Fo…
Evaluating noises of fast-simulated boson sampling with statistical benchmark methods
Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；The 32nd Research Institute of China Electronics Technology Group Corporation（中国电子科技集团公司第三十二研究所）
Boson-sampling has emerged as a promising avenue towards post-classical optical quantum computation, and numerous elementary demonstrations have recently been performed. Spontaneous parametric down-conversion (SPDC) is the mainstay for single-photon state preparation, the technique employed in most optical quantum information processing implementations to-date. Here we present a simple architecture for boson-sampling based on multiplexed SPDC sources and demonstrate that the architecture is limited only by the post-selection detection efficiency assuming that other errors, such as spectral imp…
