Complexity of Gaussian quantum optics with a limited number of non-linearities

TL;DR

This paper investigates the computational complexity of Gaussian quantum optics processes with limited non-linearities, showing that even a single layer of non-linearities can be classically hard to simulate, with implications for quantum advantage.

Contribution

It demonstrates the classical hardness of computing transition amplitudes in Gaussian processes with limited non-linearities and introduces a Hadamard test for continuous-variable systems.

Findings

Single-layer non-linear Gaussian processes are classically hard to simulate.

Efficient algorithms for these processes could approximate Gaussian boson sampling outcomes.

Develops a Hadamard test for continuous-variable systems, extending complexity results.

Abstract

It is well known in quantum optics that any process involving the preparation of a multimode gaussian state, followed by a gaussian operation and gaussian measurements, can be efficiently simulated by classical computers. Here, we provide evidence that computing transition amplitudes of Gaussian processes with a single-layer of non-linearities is hard for classical computers. To do so, we show how an efficient algorithm to solve this problem could be used to efficiently approximate outcome probabilities of a Gaussian boson sampling experiment. We also extend this complexity result to the problem of computing transition probabilities of Gaussian processes with two layers of non-linearities, by developing a Hadamard test for continuous-variable systems that may be of independent interest. Given recent experimental developments in the implementation of photon-photon interactions, our results may inspire new schemes showing quantum computational advantage or algorithmic applications of non-linear quantum optical systems realizable in the near-term.