Phase-space methods for representing, manipulating, and correcting Gottesman-Kitaev-Preskill qubits

TL;DR

This paper develops a phase-space toolkit using Wigner functions for representing, manipulating, and correcting Gottesman-Kitaev-Preskill (GKP) qubits, facilitating noise-tolerant quantum computation with Gaussian operations.

Contribution

It introduces a comprehensive phase-space framework for GKP states and operations, including error correction, applicable to various lattice structures.

Findings

Wigner functions effectively represent ideal and approximate GKP states.

Gaussian unitaries correspond to simple phase-space transformations.

The framework enables phase-space-based GKP error correction and magic-state preparation.

Abstract

The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit into a bosonic mode is a promising bosonic code for quantum computation due to its tolerance for noise and all-Gaussian gate set. We present a toolkit for phase-space description and manipulation of GKP encodings that includes Wigner functions for ideal and approximate GKP states, for various types of mixed GKP states, and for GKP-encoded operators. One advantage of a phase-space approach is that Gaussian unitaries, required for computation with GKP codes, correspond to simple transformations on the arguments of Wigner functions. We use this fact and our toolkit to describe GKP error correction, including magic-state preparation, entirely in phase space using operations on Wigner functions. While our focus here is on the square-lattice GKP code, we provide a general framework for GKP codes defined on any lattice.