Variance Geometry of Exact Pauli-Detecting Codes: Continuous Landscapes Beyond Stabilizers

• URL: http://arxiv.org/abs/2604.21800v1
• Date: Thu, 23 Apr 2026 15:59:05 GMT
• Title: Variance Geometry of Exact Pauli-Detecting Codes: Continuous Landscapes Beyond Stabilizers
• Abstract summary: We show that exact quantum codes often form connected continuous families rather than collections of disjoint solution regions.<n>Within these continuous landscapes, stabilizer codes occupy only discrete, measure-zero subsets of the attainable $*$-spectrum.<n>These results place stabilizer, symmetric, and nonadditive code families within a unified higher-rank variance framework.
• Score: 1.3599288060332324
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Exact quantum codes detecting a prescribed set of Pauli errors are approached through algebraic constructions--stabilizer, codeword-stabilized, permutation-invariant, topological, and related families. Geometrically, exact Pauli detection is governed by joint higher-rank numerical ranges of these Pauli operators, whose structure for rank $\geq 2$ is largely uncharted. From this viewpoint, we show that such codes often form connected continuous families rather than collections of disjoint solution regions. These families are characterized by a single scalar derived from the Knill-Laflamme conditions: denoted $λ^*$, it is the Euclidean norm of the signature vector of Pauli expectation values on the maximally mixed code state, and provides a one-parameter summary of the code's joint Pauli variance profile. Within these continuous landscapes, stabilizer codes occupy only discrete, measure-zero subsets of the attainable $λ^*$-spectrum, exposing a largely unexplored continuum of genuinely nonadditive exact codes. We establish this picture by analyzing the geometry of higher-rank operator compressions, and extend it to symmetry-restricted settings where cyclic and permutation symmetries are imposed on both the error model and the code projector. Small-system cases reveal interval, singleton, and empty regimes through eigenvalue interlacing and symmetry-sector decompositions; larger systems are treated numerically via Stiefel-manifold optimization and symmetry-adapted parameterizations. In every unrestricted and symmetry-compatible case analyzed, the attainable $λ^*$-spectrum forms a single closed interval whenever nonempty--although a general proof remains open. These results place stabilizer, symmetric, and nonadditive code families within a unified higher-rank variance framework, suggesting a continuous geometric perspective on the landscape of exact quantum codes.

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