Equivalence of approximate Gottesman-Kitaev-Preskill codes

TL;DR

This paper rigorously establishes the equivalence among various approximate GKP codes, providing explicit parameter relations and a standard form to facilitate analysis in quantum error correction and communication.

Contribution

It proves the equivalence of different approximate GKP codes with explicit parameter mappings and introduces a standard form for their states in position space.

Findings

Explicit relations among approximate GKP codes are derived.

A standard form for approximate GKP states is proposed.

Closed-form expressions for Wigner functions and inner products are obtained.

Abstract

The Gottesman-Kitaev-Preskill (GKP) quantum error correcting code attracts much attention in continuous variable (CV) quantum computation and CV quantum communication due to the simplicity of error correcting routines and the high tolerance against Gaussian errors. Since the GKP code state should be regarded as a limit of physically meaningful approximate ones, various approximations have been developed until today, but explicit relations among them are still unclear. In this paper, we rigorously prove the equivalence of these approximate GKP codes with an explicit correspondence of the parameters. We also propose a standard form of the approximate code states in the position representation, which enables us to derive closed-from expressions for the Wigner functions, the inner products, and the average photon numbers in terms of the theta functions. Our results serve as fundamental tools for further analyses of fault-tolerant quantum computation and channel coding using approximate GKP codes.