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

TL;DR

This paper explores the geometric structure of exact quantum codes detecting Pauli errors, revealing they form continuous families characterized by a scalar parameter, with stabilizer codes being a measure-zero subset.

Contribution

It introduces a unified geometric framework based on higher-rank numerical ranges and variance profiles, extending understanding beyond stabilizer codes to nonadditive and symmetry-restricted codes.

Findings

Codes often form connected continuous families rather than isolated solutions.

The scalar $oldsymbol{ extlambda^*}$ characterizes the code's variance profile and forms a spectrum often as a single interval.

Stabilizer codes are measure-zero within the continuous landscape of possible codes.

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 $\lambda^*$, 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 $\lambda^*$-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 $\lambda^*$-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.