Generalized concurrence in boson sampling

TL;DR

This paper introduces the Fock state concurrence sum as a quantum measure that directly influences the classical computational complexity of boson sampling, offering insights into the quantum advantage in linear optical systems.

Contribution

It proposes the Fock state concurrence sum as a new measure that links the complexity of boson sampling to quantum coherence properties, unifying the understanding of quantum computational power in linear optics.

Findings

The classical runtime depends on the Fock state concurrence sum.

The measure controls the complexity of computing transition amplitudes.

Provides a unified framework for interpreting quantum advantage in linear optics.

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" $C_S$, 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 $C_S$. Therefore, we can state that \emph{the Fock state concurrence sum $C_S$ behaves 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.