Generalized concurrence in boson sampling

Abstract

A fundamental question in linear optical quantum computing is to understand the origin of the quantum supremacy in the physical system. It is found that the multimode linear optical transition amplitudes are calculated through the permanents of transition operator matrices, which is a hard problem for classical simulations (boson sampling problem). We can understand this problem by considering a quantum measure that directly determines the runtime for computing the transition amplitudes. In this paper, we suggest a quantum measure named “Fock state concurrence sum” CS, which is the summation over all the members of “the generalized Fock state concurrence” (a measure analogous to the generalized concurrences of entanglement and coherence). By introducing generalized algorithms for computing the transition amplitudes of the Fock state boson sampling with an arbitrary number of photons per mode, we show that the minimal classical runtime for all the known algorithms directly depends on CS. Therefore, we can state that the Fock state concurrence sum CSbehaves as a collective measure that controls the computational complexity of Fock state BS. We expect that our observation on the role of the Fock state concurrence in the generalized algorithm for permanents would provide a unified viewpoint to interpret the quantum computing power of linear optics.

Universal bound on sampling bosons in linear optics and its computational implications

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Classical simulation of linear optics subject to nonuniform losses

“…Concretely, we provide a strong simulation of boson sampling (i.e. computing outcome probabilities) by showing how an expression for the permanent function from [36] can be computed efficiently for inputs of types A and B (though a similar scaling follows from an Appendix of [37]). We then generalize a result of [17] for weak classical simulation (i.e.…”

“…We split the proof into two parts. First, we recall the results of [36] to give bounds on the computation cost on individual probability amplitudes p U S → T . Then we show, by generalizing the ideas from [17], how to sample from the output probability distribution by sampling the evolution of the individual particles one at the time, sidestepping the need to compute exponentially many individual probabilities.…”

“…Then we show, by generalizing the ideas from [17], how to sample from the output probability distribution by sampling the evolution of the individual particles one at the time, sidestepping the need to compute exponentially many individual probabilities. We start by reviewing results of [36]. According to Eq.…”

Unification of the Nature’s Complexities via a Matrix Permanent—Critical Phenomena, Fractals, Quantum Computing, ♯P-Complexity

“…First of all, the permanent gives a result of the boson sampling in a multichannel quantum-optical network [5][6][7][8][9][10][11][12][13][14][15][16]-a simple prototype of a many-body quantum simulator. The latter amounts to evaluating the permanent of a unitary matrix of the channel couplings that could be addressed, for example, by an interesting method developed recently in Reference [11] on the basis of a complex phase-space representation.…”