Intrinsic Quantum Codes

TL;DR

This paper develops an intrinsic, representation-theoretic framework for quantum error correction, enabling certification of error-protection properties across multiple physical realizations and introducing a new notion of code distance called depth.

Contribution

It introduces an intrinsic formulation of quantum codes based on group representations, proving the Schur bootstrap and an intrinsic Eastin--Knill theorem, and provides new insights into code distance and symmetries.

Findings

Intrinsic codes satisfy error-correction conditions across all realizations.

Depth generalizes conventional code distance, refining error-protection measures.

Examples include SU(2) codes unifying permutation-invariant and bosonic codes.

Abstract

We introduce an intrinsic formulation of quantum error correction based on representation theory, in which error-protection structure is encoded directly in a unitary group representation, rather than being tied to a particular embedding into a larger Hilbert space. In this framework, error models are classified according to the isotypic decomposition of the conjugation action on the operator algebra. Our main result, the \emph{Schur bootstrap}, shows that if an intrinsic code satisfies the Knill--Laflamme conditions on a given symmetry sector, then the same error-protection relations hold for every extrinsic realization obtained from a group-equivariant isometric embedding into a larger Hilbert space. Thus a single intrinsic verification certifies the corresponding symmetry-resolved error-correction conditions across an entire family of physical realizations. We further introduce an intrinsic notion of distance, called depth, defined via adjoint order. For standard multi-qudit systems this coincides with conventional code distance, while for more general representations it refines the usual weight-based notion. We also prove an intrinsic Eastin--Knill theorem: any intrinsic code of depth at least two has a discrete logical symmetry group, with the obstruction to continuous covariant gates arising from the representation-theoretic structure of the adjoint action. We illustrate the framework with several examples, including a minimal $\mathrm{SU}(2)$ construction that unifies permutation-invariant qubit codes and bosonic codes, and higher-dimensional constructions exhibiting transversal Clifford symmetries and realizations beyond qubit systems.