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

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.

Resources for Bosonic Quantum Computational Advantage

“…The time complexity in Theorem 1 is a worst-case complexity, based on the fastest known classical algorithm for computing the hafnian [50], and may be reduced for particular instances. On the other hand, due to its broad applicability, our simulation technique may be outperformed by classical simulation algorithms targeting specific classes of bosonic circuits [51][52][53][54][55]. Nonetheless, Theorem 1 may be used primarily as a tool for identifying necessary resources for bosonic quantum computational advantage: it establishes the stellar rank as a necessary non-Gaussian property.…”