Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian circuits

TL;DR

This paper develops a method to evaluate measurement probabilities of GKP states under various operations, identifying new classes of multimode circuits that can be efficiently simulated classically, beyond previously known sets.

Contribution

It introduces a novel approach to compute measurement probabilities for GKP states and identifies new classically simulatable circuits outside the GKP Clifford group.

Findings

Identified two large classes of multimode circuits that are classically efficiently simulatable.

Developed a method involving transformed Jacobi theta functions for probability density evaluation.

Extended the set of circuits known to be classically simulatable beyond existing theorems.

Abstract

We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements. For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems such as the Gottesman-Knill theorem can be employed to assess the simulatability. We first develop a method to evaluate the probability density function corresponding to measuring a single GKP state in the position basis following arbitrary squeezing and a large set of rotations. This method involves evaluating a transformed Jacobi theta function using techniques from analytic number theory. We then use this result to identify two large classes of multimode circuits which are classically efficiently simulatable and are not contained by the GKP encoded Clifford group. Our results extend the set of circuits previously known to be classically efficiently simulatable.