Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes

• URL: http://arxiv.org/abs/2607.26214v1
• Date: Tue, 28 Jul 2026 19:31:39 GMT
• Title: Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes
• Abstract summary: The MacWilliams extension theorem fails for module alphabets with non-cyclic socle.<n>The label alphabet of qudit stabilizer codes, $F_q2$ over $F_q$, is such an alphabet.<n> Quantum error correction, however, only ever sees emphself-orthogonal additive codes.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The MacWilliams extension theorem fails for module alphabets with non-cyclic socle, and the label alphabet of qudit stabilizer codes, $\F_{q^2}$ over $\F_q$, is such an alphabet. Quantum error correction, however, only ever sees \emph{self-orthogonal} additive codes, and whether that rigidity rescues the theorem---equivalently, whether every weight-preserving isomorphism of stabilizer groups is implemented by local Cliffords and a qudit permutation---was asked by Gluesing-Luerssen and Pllaha and answered negatively by Pllaha for particular qubit codes. We develop the negative answer systematically and at the smallest possible scales. For every prime power $q$ we construct a pair of $[[q+1,q-1]]_q$ stabilizer codes and a weight-preserving isomorphism between them extending to no monomial transformation; the codespaces are inequivalent even under arbitrary local unitaries combined with permutations, though they share Shor--Laflamme enumerators. Self-orthogonality is automatic here, by two elementary lemmas which also show that Dyshko's threshold-length counterexamples were already self-orthogonal, unremarked. For qubits we prove by exhaustive search that length $3$ is minimal and the counterexample essentially unique. Dropping the ``idle qudit'' invariant that detects these, we find the minimal full-support lengths: $4$ for a non-extendable isometry, $5$ for a weight-isometric pair that is not monomially equivalent, realized by explicit $[[5,2]]$ codes; at length $6$ all nontrivial stabilizer elements can have weight $\ge 4$. Whether these codespaces are locally unitarily equivalent is posed as an open problem, connecting the extension problem to the LU--LC circle of questions.

Related papers

• Sharp Mixed Spectral Barron Regularity of Coulombic Many-Electron Wave Functions [5.361030050051251]
  We establish sharp mixed spectral Barron regularity for eigenfunctions of molecular Coulomb Hamiltonians.<n>For fixed-spin components with two occupied spin blocks, it reduces to $s++1$.<n>For a fully spin-polarized state, $mathcal I_=1,ldots,N$.
  arXiv  Detail & Related papers  (2026-09-01T08:05:47Z)
• The Condition-Number Barrier in Sparse Least Squares [77.64108812086542]
  In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in convex optimization cannot be improved by a sparse-time algorithm.<n>We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis.
  arXiv  Detail & Related papers  (2026-08-03T17:57:01Z)
• Quantum Codes from $r$-Nearly Self-Orthogonal Linear Codes via Jordan Canonical Form over $\mathbb{F}_{q^2}$ [5.6342405886627]
  We introduce a Jordan-canonical-form framework for constructing $q$-ary quantum stabilizer codes from arbitrary classical linear codes over $F_q2$.<n>The framework does not require the classical linear code $mathcalC$ to satisfy the dual-containing condition (i.e., self-orthogonality)<n>Applying this construction to classical codes produces several record quantum codes that improve or supplement the best-known parameters in Grassl's tables.
  arXiv  Detail & Related papers  (2026-07-14T00:58:42Z)
• Minimal Permutation-Invariant Qudit Codes from Edge-Colorings of Complete Graphs [4.297070083645049]
  We study permutation-invariant quantum codes in the symmetric subspace $mathrmSymn(mathbbCq) $ of $n$ qudits of local dimension $q$.<n>Four physical qudits suffice to encode one logical qudit with distance two in the symmetric sector for every local dimension.
  arXiv  Detail & Related papers  (2026-05-21T13:08:38Z)
• The Collapse of Unentangled Stoquastic Merlin-Arthur Proof Systems [46.657699628769336]
  Entanglement and interference are among the most fundamental properties of quantum mechanics.<n>We prove that entanglement gives no additional power to stoquastic verification.
  arXiv  Detail & Related papers  (2026-05-15T17:52:33Z)
• Hardness of High-Dimensional Linear Classification [58.29089693778071]
  We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem.<n>Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms.
  arXiv  Detail & Related papers  (2026-03-19T15:53:41Z)
• Quantum Mechanics from Finite Graded Equality [0.40611352512781856]
  We propose that quantum mechanics follows from a single hypothesis: equality has finite resolution.<n>We formalize the first two axioms as axioms; a structural Leibniz condition within the saturation axiom forces permutation invariance of basis elements.<n>Local tomography follows from $mathbbF = mathbbC$ with tensor product composition.
  arXiv  Detail & Related papers  (2026-03-12T13:14:48Z)
• Discrete symmetries in classical and quantum oscillators [51.56484100374058]
  We show the eigenfunctions $_n=zn$ of the quantum Hamiltonian in the complex Bargmann-Fock-Segal representation.<n>The superposition $=sum_n c_n_n$ arises only with incomplete knowledge of the initial data for solving the Schrdinger equation.
  arXiv  Detail & Related papers  (2026-01-05T10:04:39Z)
• Burau representation, Squier's form, and non-Abelian anyons [53.92822954974537]
  We introduce a frequency-tunable, two-dimensional non-Abelian control of operation order constructed from the Burau representation of the braid group $B_3$.<n>The sign change of $Delta(omega)$ across the Squier positivity window reveals alternating constructive and destructive interference of causal orders.<n> Numerical simulations confirm both enhancement and suppression regimes, establishing a minimal $B_3$ braid control.
  arXiv  Detail & Related papers  (2025-10-21T00:25:21Z)
• Quantum error correction beyond $SU(2)$: spin, bosonic, and permutation-invariant codes from convex geometry [48.254879700836376]
  We develop a framework for constructing quantum error-correcting codes and logical gates for three types of spaces.<n>We prove that many codes and their gates in $SU(q)$ can be inter-converted between the three state spaces.<n>We present explicit constructions of codes with shorter length or lower total spin/excitation than known codes with similar parameters.
  arXiv  Detail & Related papers  (2025-09-24T20:21:30Z)
• Non-representable quantum measures [55.2480439325792]
  Grade-$d$ measures on a $sigma$-algebra $mathcalAsubseteq 2X$ over a set $X$ are generalizations of measures satisfying one of a hierarchy of weak additivity-type conditions.<n>Every signed polymeasure $lambda$ on $(X,mathcalA)d$ produces a grade-$d$ measure as its diagonal $widetildelambda(A):=lambda(A,cdots,A)$.
  arXiv  Detail & Related papers  (2025-08-20T00:47:24Z)
• Constructions of $k$-uniform states in heterogeneous systems [65.63939256159891]
  We present two general methods to construct $k$-uniform states in the heterogeneous systems for general $k$. We can produce many new $k$-uniform states such that the local dimension of each subsystem can be a prime power.
  arXiv  Detail & Related papers  (2023-05-22T06:58:16Z)
• Linear programming with unitary-equivariant constraints [2.0305676256390934]
  Unitary equivariance is a natural symmetry that occurs in many contexts in physics and mathematics. We show that under additional symmetry assumptions, this problem reduces to a linear program that can be solved in time that does not scale in $d$. We also outline a possible route for extending our method to general unitary-equivariant semidefinite programs.
  arXiv  Detail & Related papers  (2022-07-12T17:37:04Z)
• Monogamy of entanglement between cones [43.57338639836868]
  We show that monogamy is not only a feature of quantum theory, but that it characterizes the minimal tensor product of general pairs of convex cones.<n>Our proof makes use of a new characterization of products of simplices up to affine equivalence.
  arXiv  Detail & Related papers  (2022-06-23T16:23:59Z)
• Quantum double aspects of surface code models [77.34726150561087]
  We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry. We show how our constructions generalise to $D(H)$ models based on a finite-dimensional Hopf algebra $H$.
  arXiv  Detail & Related papers  (2021-06-25T17:03:38Z)
• Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers [2.4851820343103035]
  Physical platforms such as trapped ions suffer from coherent noise where errors manifest as rotations about a particular axis and can accumulate over time. We investigate passive mitigation through decoherence free subspaces, requiring the noise to preserve the code space of a stabilizer code. By adjusting the size of these components, we are able to construct a large family of QECC codes, oblivious to coherent noise.
  arXiv  Detail & Related papers  (2020-10-31T06:09:40Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.