TL;DR
This paper analyzes GKP quantum error correction codes using lattice theory, deriving bounds, exploring decoding strategies, and proposing new code constructions with resource efficiency improvements.
Contribution
It introduces a lattice-theoretic framework for GKP codes, derives formal bounds, relates decoding methods, and proposes new code constructions with resource savings.
Findings
Derived bounds on GKP code parameters
Established relationships between decoding strategies
Proposed new GKP code constructions using lattices
Abstract
We examine general Gottesman-Kitaev-Preskill (GKP) codes for continuous-variable quantum error correction, including concatenated GKP codes, through the lens of lattice theory, in order to better understand the structure of this class of stabilizer codes. We derive formal bounds on code parameters, show how different decoding strategies are precisely related, propose new ways to obtain GKP codes by means of glued lattices and the tensor product of lattices and point to natural resource savings that have remained hidden in recent approaches. We present general results that we illustrate through examples taken from different classes of codes, including scaled self-dual GKP codes and the concatenated surface-GKP code.
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