Variance Geometry of Exact Pauli-Detecting Codes: Continuous Landscapes Beyond Stabilizers
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866915952263692288 |
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| author | Gupta, Arunaday Sun, Baisong He, Xi Zeng, Bei |
| author_facet | Gupta, Arunaday Sun, Baisong He, Xi Zeng, Bei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21800 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Variance Geometry of Exact Pauli-Detecting Codes: Continuous Landscapes Beyond Stabilizers Gupta, Arunaday Sun, Baisong He, Xi Zeng, Bei Quantum Physics Information Theory Mathematical Physics 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. |
| title | Variance Geometry of Exact Pauli-Detecting Codes: Continuous Landscapes Beyond Stabilizers |
| topic | Quantum Physics Information Theory Mathematical Physics |
| url | https://arxiv.org/abs/2604.21800 |