Effect of partial distinguishability on quantum supremacy in Gaussian Boson sampling

Abstract

Gaussian boson sampling (GBS) allows for a way to demonstrate quantum supremacy with the relatively modest experimental resources of squeezed light sources, linear optics, and photon detection. In a realistic experimental setting, numerous effects can modify the complexity of the sampling, in particular loss, partial distinguishability of the photons, and the use of threshold detectors rather than photon counting detectors. In this paper, we investigate GBS with partial distinguishability using an approach based on virtual modes and indistinguishability efficiency. We develop a model using these concepts and derive the probabilities of measuring a specific output pattern from partially distinguishable and lossy GBS for both types of detectors. In the case of threshold detectors, the probability as calculated by the Torontonian is a special case under our framework. By analyzing the expressions of these probabilities we propose an efficient classical simulation algorithm which can be used to calculate the probabilities. Our model and algorithm provide foundations for an approximate method for calculating probabilities. It also allows for a way to design sampling algorithms that are not only compatible with existing algorithms for ideal GBS, but can also reduce their complexity exponentially, depending on the indistinguishability efficiency. Using this we show how the boundary of quantum supremacy in GBS can be affected by partial distinguishability.