An asymptotic analytical approach is proposed for bosonic probability amplitudes in unitary linear networks, such as the optical multiport devices for photons. The asymptotic approach applies for large number of bosons N ≫ M in the M-mode network, where M is finite. The probability amplitudes of N bosons unitarily transformed from the input modes to the output modes of a unitary network are approximated by a multidimensional integral with the integrand containing a large parameter (N) in the exponent. The integral representation allows an asymptotic estimate of bosonic probability amplitudes up to a multiplicative error of order 1/N by the saddle point method. The estimate depends on solution of the scaling problem for the M × M-dimensional unitary network matrix: to find the left and right diagonal matrices which scale the unitary matrix to a matrix which has specified row and column sums (equal, respectively, to the distributions of bosons in the input and output modes). The scaled matrices give the saddle points of the integral. For simple saddle points, an explicit formula giving the asymptotic estimate of bosonic probability amplitudes is derived. Performance of the approximation and the scaling of the relative error with N are studied for two-mode network (the beam splitter), where the saddle-points are roots of a quadratic and an exact analytical formula for the probability amplitudes is available, and for three-mode network (the tritter).
An asymptotic analytical approach is proposed for bosonic probability amplitudes in unitary linear networks, such as the optical multiport devices for photons. The asymptotic approach applies for large number of bosons N ≫ M in the M-mode network, where M is finite. The probability amplitudes of N bosons unitarily transformed from the input modes to the output modes of a unitary network are approximated by a multidimensional integral with the integrand containing a large parameter (N) in the exponent. The integral representation allows an asymptotic estimate of bosonic probability amplitudes up to a multiplicative error of order 1/N by the saddle point method. The estimate depends on solution of the scaling problem for the M × M-dimensional unitary network matrix: to find the left and right diagonal matrices which scale the unitary matrix to a matrix which has specified row and column sums (equal, respectively, to the distributions of bosons in the input and output modes). The scaled matrices give the saddle points of the integral. For simple saddle points, an explicit formula giving the asymptotic estimate of bosonic probability amplitudes is derived. Performance of the approximation and the scaling of the relative error with N are studied for two-mode network (the beam splitter), where the saddle-points are roots of a quadratic and an exact analytical formula for the probability amplitudes is available, and for three-mode network (the tritter).
Probabilities of photon counts at the output of a multiport optical device are generalised for optical sources of arbitrary quantum states in partially distinguishable optical modes. For the single-mode photon sources, the generating function for the probabilities is a linear combination of the matrix permanents of positive semi-definite Hermitian matrices, where each Hermitian matrix is a Hadamard product of a submatrix of the multiport matrix and a Hermitian matrix describing partial distinguishability. For the multi-mode sources the generating function is given by an integral of the Husimi functions of the sources. When each photon source outputs exactly a Fock state, the obtained expression reduces to the probability formula derived for partially distinguishable photons, \textit{Physical Review A \textbf{91}, 013844 (2015)}. The derived probability formula can be useful in analysing experiments with partially distinguishable sources and error bounds of experimental Boson Sampling devices.
We generalize an approach for description of multiphoton experiments with multiport unitary linear optical devices, started in Phys. Rev. A 89, 022333 (2014) with single photons in mixed spectral states, to arbitrary (multiphoton) input and arbitrary photon detectors. We show that output probabilities are always given in terms of the matrix permanents of the Hadamard product of a matrix built from the network matrix and matrices built from the spectral state of photons and spectral sensitivities of detectors. Moreover, in the case of input with up to one photon per mode, the output probabilities are given by a sum (or integral) with each term being the absolute value squared of such a matrix permanent. We conjecture that, for an arbitrary multiphoton input, zero output probability of an output configuration is always the result of an exact cancellation of quantum transition amplitudes of completely indistinguishable photons (a subset of all input photons) and, moreover, is independent of coherence between only partially indistinguishable photons. The conjecture is supported by examples. Furthermore, we propose a measure of partial indistinguishability of photons which generalizes Mandel's observation, and find the law of degradation of quantum coherence in a realistic boson-sampling device with increase of the total number of photons and/or their ``classicality parameter.''
ABSTRACT In linear optics, photons are scattered in a network through passive optical elements including beam splitters and phase shifters, leading to many intriguing applications in physics, such as Mach–Zehnder interferometry, the Hong–Ou–Mandel effect, and tests of fundamental quantum mechanics. Here we present the fundamental limit in the transition amplitudes of bosons, applicable to all physical linear optical networks. Apart from boson sampling, this transition bound results in many other interesting applications, including behaviors of Bose–Einstein condensates (BEC) in optical networks, counterparts of Hong–Ou–Mandel effects for multiple photons, and approximating permanents of matrices. In addition, this general bound implies the existence of a polynomial-time randomized algorithm for estimating the transition amplitudes of bosons, which represents a solution to an open problem raised by Aaronson and Hance (Quantum Inf Comput 2012; 14: 541–59). Consequently, this bound implies that computational decision problems encoded in linear optics, prepared and detected in the Fock basis, can be solved efficiently by classical computers within additive errors. Furthermore, our result also leads to a classical sampling algorithm that can be applied to calculate the many-body wave functions and the S-matrix of bosonic particles.
Boson sampling is a candidate for quantum protocols to truly realize the quantum computation advantage and to be used in advanced fields where complex computations are needed, such as quantum chemistry. However, this proposal is hard to achieve due to the existence of noise sources such as photon losses. In order to quantificationally analyze the influences of photon losses in optical networks, boson sampling is classically simulated based on the equivalent beam splitter mechanism, where the photon loss happening in optical units is equivalent to the photon transmission into the environmental paths through a virtual beam splitter. In our simulation, networks corresponding to random unitary matrices are made up, considering both the Reck structure and the Clements structure. The photon loss probability in an optical unit is well controlled by adjusting the parameters of the virtual beam splitter. Therefore, to simulate boson sampling with photon losses in optical networks is actually to simulate ideal boson sampling with more modes. It is found that when the photon loss probability is constant, boson sampling with Clements structures distinctly performs much better than that with Reck structures. Furthermore, the photon loss probability is also set to follow the normal distribution, which is thought to be closer to the situation in reality. It is found that when the mean value of photon loss probability is constant, for both network structures, errors of outputs become more obvious with the increase of standard deviation. It can be inferred that the increase of error rate can be explained by the network depth and the conclusion is suitable for larger-scale boson sampling. Finally, the number of output photons is taken into consideration, which is directly related to the classical computation complexity. It is found that with the photon loss probability, the ratio of output combinations without photon losses decreases sharply, implying that photon losses can obviously affect the quantum computation advantage of boson sampling. Our results indicate that photon losses can result in serious errors for boson sampling, even with a stable network structure such as that of Clements. This work is helpful for boson sampling experiments in reality and it is desired to develop a better protocol, for example, a well-designed network or excellent optical units, to well suppress photon losses.
Abstract This work shows that the majorization of photon distributions is related to the runtime of classically simulating multimode passive linear optics, which explains one aspect of the boson sampling hardness. A Shur-concave quantity which we name the Boltzmann entropy of elementary quantum complexity ( ) is introduced to present some quantitative analysis of the relation between the majorization and the classical runtime for simulating linear optics. We compare with two quantities that are important criteria for understanding the computational cost of the photon scattering process, (the runtime for the classical simulation of linear optics) and (the additive error bound for an approximated amplitude estimator). First, for all the known algorithms for computing the permanents of matrices with repeated rows and columns, the runtime becomes shorter as the input and output distribution vectors are more majorized. Second, the error bound decreases as the majorization difference of input and output states increases. We expect that our current results would help in understanding the feature of linear optical networks from the perspective of quantum computation.
Sampling from probability distributions of quantum circuits is a fundamentally and practically important task which can be used to demonstrate quantum supremacy using noisy intermediate-scale quantum devices. In the present work, we examine classical simulability of sampling from the output photon-number distribution of linear-optical circuits composed of random beam splitters with equally distributed squeezed vacuum states and single-photon states input. We provide efficient classical algorithms to simulate linear-optical random circuits and show that the algorithms' error is exponentially small up to a depth less than quadratic in the distance between sources using a classical random walk behavior of random linear-optical circuits. Notably, the average-case depth allowing an efficient classical simulation is larger than the worst-case depth limit, which is linear in the distance. Besides, our results together with the hardness of boson sampling give a lower-bound on the depth for constituting global Haar-random unitary circuits.