# Boson sampling with Gaussian measurements

## TL;DR

This paper introduces a new boson sampling model using Gaussian measurements on single-photon states, demonstrating the computational hardness of simulating such continuous-variable measurements through symmetry and time-reversal arguments.

## Contribution

It presents a novel boson sampling framework with Gaussian measurements and proves its computational hardness using symmetry under time reversal.

## Key findings

- Gaussian measurement-based boson sampling is computationally hard to simulate.
- Time-reversal symmetry helps analyze the complexity of quantum optical models.
- The model extends the understanding of quantum computational complexity with continuous variables.

## Abstract

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 corollary, an instance of boson sampling from Gaussian states where photon counting is hard to simulate. Then, a time-reversed setup is used to exhibit a boson sampling model in which the simulation of Gaussian measurements -- namely the outcome of eight-port homodyne detection -- is proven to be computationally hard. These results illustrate how the symmetry of quantum evolution under time reversal may serve as a tool for analyzing the computational complexity of novel physically-motivated computational problems.