Stabilizer codes over general phase spaces

2026-10-03

AI Summary

This study addresses the encoding and decoding challenges in hybrid quantum systems by constructing a unified stabilizer code theory based on generalized phase space. Methodologically, it integrates noncommutative geometry, symplectic dual lattices, and concatenated GKP codes to develop a stabilizer framework encompassing bosonic, rotor, and qubit modes. By exploiting phase-space lattice structures, indecomposable composite codes are constructed, yielding novel oscillator-rotor encodings and partially lifted Golay code constructions. The project establishes an analytical relationship between code dimension and covolume, derives logical operators alongside optimal decoding strategies, and constructs finite-energy codewords. Furthermore, it recovers the MacWilliams identity and weight enumerator polynomials, thereby achieving a unification of encoding theories across diverse quantum systems.

Abstract

We develop a theory of stabilizer codes whose stabilizer groups consist of commuting qudit Paulis, oscillator displacements, and/or planar-rotor displacements. We consider codes whose stabilizer groups form generalized lattices in quantum phase space, a condition that guarantees a finite logical dimension. We construct oscillator-rotor, rotor-qudit, and oscillator-rotor-qubit codes that cannot be decomposed into separate subsystems by generalized Clifford transformations. We also introduce an oscillator-qubit code arising from a partially lifted Golay code, where Construction A is applied to half of the Golay code coordinates. We derive analytical forms of logical operators from the symplectic dual lattice, determine Clifford-Gaussian gates from lattice symmetries, show that the code dimension equals the covolume of the stabilizer lattice, and organize the syndrome subspaces into a vector bundle. Our technique builds on earlier non-commutative geometric results by Rieffel, but it can often be interpreted simply as concatenating a given stabilizer code with a Gottesman-Kitaev-Preskill (GKP) code, applying conventional lattice-theoretic results, and unconcatenating. We formulate distances, optimal decoding, and quantum weight enumerators, recovering the Pauli, qudit, and GKP versions together with known MacWilliams identities. We construct finite-energy codewords whose encoding is an isometry up to an error exponentially small in the inverse damping parameter.

Problem

• stabilizer codes
• quantum phase space
• continuous-variable
• quantum error correction
• hybrid quantum systems

Innovation

• stabilizer codes
• quantum phase space
• generalized lattices
• GKP code
• quantum weight enumerators