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arXiv:2607.02164 · Physical Review Research

Phase-Space Representations of Quantum Error-Correcting Codes

Enrico Bozzetto, Jonte R. Hance

quant-phSubmitted Jul 2, 2026• Announced Jul 3, 2026• 0 citations• Revised Sep 29, 2026 (v5)
v1 · Jul 2, 2026v2 · Aug 30, 2026v3 · Sep 4, 2026v4 · Sep 8, 2026v5 · Sep 29, 2026
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Abstract

We connect the structure theorem for quasiprobability representations to quantum error correction. For a code space invariant under a subgroup of an extended Weyl-Heisenberg group, the Brif-Mann construction gives a semi-functorial representation whenever its kernel satisfies the Stratonovich-Weyl axioms, and composes on channels covering full error-correction cycles. With phase-space parity available, a kernel other than the ordinary Wigner one exists iff the code space is invariant under a lattice of displacements: among compact-phase-space bosonic codes, this is only the ideal Gottesman-Kitaev-Preskill family; in finite dimension, this is every qudit stabiliser code, whose negative volume for odd prime dimension is the syndrome-averaged logical magic. For rotation-symmetric, one-photon, non-Pauli qudit and spin codes only the ordinary kernel remains, its negativity reflecting the physical carrier, not logical content; a sector-graded representation built from recovery isometries restores a resource reading. We add a channel atlas, a closed-form magic lifetime under displacement noise, a Kirkwood-Dirac layer, and finite-energy corrections.

Journal reference
Phys. Rev. Res. 8, 043076. (2026)
DOI
10.1103/7my9-zgfz
Comments
32+12 pages, no figures. Accepted for publication in Phys Rev Research - matches accepted version

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We connect the structure theorem for quasiprobability representations to quantum error correction. For a code space invariant under a subgroup of an extended Weyl-Heisenberg group, the Brif-Mann construction gives a semi-functorial representation whenever its kernel satisfies the Stratonovich-Weyl axioms, and composes on channels covering full error-correction cycles. With phase-space parity available, a kernel other than the ordinary Wigner one exists iff the code space is invariant under a lattice of displacements: among compact-phase-space bosonic codes, this is only the ideal Gottesman-Kitaev-Preskill family; in finite dimension, this is every qudit stabiliser code, whose negative volume for odd prime dimension is the syndrome-averaged logical magic. For rotation-symmetric, one-photon, non-Pauli qudit and spin codes only the ordinary kernel remains, its negativity reflecting the physical carrier, not logical content; a sector-graded representation built from recovery isometries restores a resource reading. We add a channel atlas, a closed-form magic lifetime under displacement noise, a Kirkwood-Dirac layer, and finite-energy corrections.
We study a class of quantum error-correcting codes through the geometry of complex abelian varieties. These codes, introduced by Gottesman--Kitaev--Preskill, are built from symplectically integral lattices and therefore naturally define polarized complex abelian varieties. We give a precise mathematical formulation of this relationship and extend it to a dictionary between the main structures of GKP code theory and classical objects in the theory of abelian varieties. For instance, under this dictionary, the finite-dimensional code space becomes the space of theta functions H0(X,L)H^0(X, L)H0(X,L), logical Pauli gates arise from the theta group, passive logical Clifford gates correspond to automorphisms of the polarized abelian variety, and concatenation with stabilizer codes corresponds to isogeny. We also prove several key results that give precise mathematical formulations of statements about these codes that often appear in heuristic form in the physics literature. In particular, we prove that the encoding is asymptotically isometric, that every logical Clifford gate is realized by a Gaussian unitary, and that, for noise of small variance, the failure probability is governed to first order by the shortest nontrivial displacement in the kernel of the polarization isogeny, a systolic invariant of the underlying polarization. This leads naturally to optimization problems on the moduli space of polarized abelian varieties.
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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.
Bosonic codes offer a promising approach towards hardware-efficient fault tolerance, with many leading examples such as the cat-code and the GKP code organized by an underlying symmetry. We build on a representation-theoretic framework of quantum error correction to formalize the construction and analysis of multimode symmetric bosonic codes, focusing on finite groups. Here, a physical, logical representation and an initial seed state together define the codewords via a covariant encoding. A central observation is that QEC matrices can be viewed as morphisms of group representations, so that Schur's lemma forces entire irreducible representations blocks to vanish. We recover several known single- and two-mode bosonic codes and build and analyze the error protection capabilities of three codes based on finite subgroups of SU(2). We analyze the impact of reducibility of the logical representation and optimize the seed state using the near-optimal fidelity.