Intrinsic Quantum Codes
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arXiv
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| Format: | Preprint |
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2025
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| author | Kubischta, Eric Teixeira, Ian |
| author_facet | Kubischta, Eric Teixeira, Ian |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_14840 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Intrinsic Quantum Codes Kubischta, Eric Teixeira, Ian Quantum Physics 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. |
| title | Intrinsic Quantum Codes |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2511.14840 |