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arXiv:2608.06304

Time-Reversal Selection Rules for Quantum Error Correction

Eric Kubischta, Ian Teixeira

quant-phSubmitted Aug 6, 2026• Announced Aug 7, 2026• 0 citations
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Abstract

We apply time-reversal symmetry to quantum codes and show that it imposes parity selection rules on the physical error algebra. A time-reversal-invariant logical qubit on an odd number of spins is a Kramers doublet, forcing every even-weight Pauli to act as a scalar. Consequently, all even-weight Knill--Laflamme conditions hold automatically, so single-qubit error detection implies correction. We then reinterpret the Rains shadow enumerator through time reversal: each coefficient is a sum of error-resolved overlaps between a code and its time-reversed image.

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We apply time-reversal symmetry to quantum codes and show that it imposes parity selection rules on the physical error algebra. A time-reversal-invariant logical qubit on an odd number of spins is a Kramers doublet, forcing every even-weight Pauli to act as a scalar. Consequently, all even-weight Knill--Laflamme conditions hold automatically, so single-qubit error detection implies correction. We then reinterpret the Rains shadow enumerator through time reversal: each coefficient is a sum of error-resolved overlaps between a code and its time-reversed image.
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A bstract We investigate the relationship between superselection rules and quantum error correcting codes. We demonstrate that the existence of a superselection rule implies the Knill-Laflamme condition in quantum error correction. As an example, we examine the code built from quantum chromodynamics, where the proton and neutron states in the model are explored as different superselection sectors that protect logical information. Finally we comment on topological quantum error correcting codes and supersymmetric quantum field theory within this framework.
We derive simple necessary and sufficient conditions under which a quantum channel obtained from an arbitrary perturbation from the identity can be reversed on a given code to the lowest order in fidelity. We find the usual Knill-Laflamme conditions applied to a certain operator subspace which, for a generic perturbation, is generated by the Lindblad operators. For a weak interaction with an environment, the error space to be corrected is a subspace of that spanned by the interaction operators, selected by the environment's initial state.
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 SU(2)\mathrm{SU}(2)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.
A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground space contains all permutation-invariant states. We use techniques from combinatorics and operator theory to construct families of permutation-invariant quantum codes. These codes have length proportional to t2{t}^{2}t2; one family of codes perfectly corrects arbitrary weight ttt errors, while the other family of codes approximately correct ttt spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.