Symmetry-Based Quantum Codes Beyond the Pauli Group

TL;DR

This paper introduces a generalized, symmetry-aware framework for quantum error correction that extends stabilizer codes by incorporating group representation theory, enabling passive error mitigation and syndrome measurement beyond the Pauli group.

Contribution

It provides a unifying, representation-theoretic framework for quantum codes that includes stabilizer codes as a special case and enables symmetry-based error correction tailored to specific systems.

Findings

All stabilizer codes are encompassed within this framework.

A natural one-qubit code associated with the dihedral group is constructed.

The framework allows passive error mitigation and symmetry-resolved syndrome measurements.

Abstract

Typical stabilizer codes aim to solve the general problem of fault-tolerance without regard for the structure of a specific system. By incorporating a broader representation-theoretic perspective, we provide a generalized framework that allows the code designer to take this structure into account. For any representation of a finite group, we produce a quantum code with a code space invariant under the group action, providing passive error mitigation against errors belonging to the image of the representation. Furthermore, errors outside this scope are detected and diagnosed by performing a projective measurement onto the isotypic components corresponding to irreducible representations of the chosen group, effectively generalizing syndrome extraction to symmetry-resolved quantum measurements. We show that all stabilizer codes are a special case of this construction, including qudit stabilizer codes, and show that there is a natural one logical qubit code associated to the dihedral group. Thus we provide a unifying framework for existing codes while simultaneously facilitating symmetry-aware codes tailored to specific systems.