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arXiv:2604.21800

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

Arunaday Gupta, Baisong Sun, Xi He, Bei Zeng

quant-phcs.ITmath-phSubmitted Apr 23, 2026• Announced Apr 24, 2026• 0 citations
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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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30 pages, 1 figure

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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 ≥2\geq 2≥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∗λ^*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∗λ^*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∗λ^*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.
Knill-Laflamme coefficients characterize exact quantum error detection through scalar compressions of error operators to the code space. For prescribed Pauli observables and a fixed code dimension, attainable coefficient vectors form a joint higher-rank numerical range, whose Euclidean norm image is the signature spectrum. We study how operator representation, code dimension, and detection constraints govern this geometry. Our main theorem identifies representation multiplicity as sufficient to lift active expectation data to exact scalar compressions: a lift exists whenever each occupied block has multiplicity at least the code dimension times the rank of the corresponding active state. When multiplicity suffices to realize all active states, the coefficient range is the full active expectation body and is therefore convex and connected, with a closed interval as its signature spectrum. This framework unifies commuting Pauli families, where sufficient common-eigenspace degeneracy yields polytopes, and subsystem stabilizer codes, where protected logical subsystems supply multiplicity spaces for noncommuting gauge observables. Explicit examples show how increasing the code dimension can make a range shrink, collapse, or become empty, while increasing multiplicity can turn a Bloch sphere into a filled ball. In the subsystem setting, parent-Hamiltonian constructions realize exact detecting codes as degenerate ground spaces, selecting distinguished vectors and continuous paths within these ranges. A final three-qubit example shows additional noncommuting detection constraints reduce a tetrahedral range to its barycenter and four vertices, producing a disconnected signature spectrum. The results provide a structural framework for Knill-Laflamme coefficient geometry and motivate sharper criteria for connected coefficient ranges and interval signature spectra under structured error models.
Quantum error correction (QEC) is essential for protecting quantum information against noise, yet understanding the structure of the Knill-Laflamme (KL) coefficients λijλ_{ij}λij​ from the condition PEi†EjP=λijPPE_i^\dagger E_j P = λ_{ij} PPEi†​Ej​P=λij​P remains challenging, particularly for nonadditive codes. In this work, we introduce the signature vector λ⃗(P)\vecλ(P)λ(P), composed of the off-diagonal KL coefficients λijλ_{ij}λij​, where each coefficient corresponds to equivalence classes of errors counted only once. We define its Euclidean norm λ∗(P)λ^*(P)λ∗(P) as a scalar measure representing the total strength of error correlations within the code subspace defined by the projector PPP. We parameterize PPP on a Stiefel manifold and formulate an optimization problem based on the KL conditions to systematically explore possible values of λ∗λ^*λ∗. Moreover, we show that, for ((n,K,d))((n,K,d))((n,K,d)) codes, λ∗λ^*λ∗ is invariant under local unitary transformations. Applying our approach to the ((6,2,3))((6, 2, 3))((6,2,3)) quantum code, we find that λmin∗=0.6λ^*_{\text{min}} = \sqrt{0.6}λmin∗​=0.6​ and λmax∗=1λ^*_{\text{max}} = 1λmax∗​=1, with λ∗=1λ^* = 1λ∗=1 corresponding to a known degenerate stabilizer code. We construct continuous families of new nonadditive codes parameterized by vectors in R5\mathbb{R}^5R5, with λ∗λ^*λ∗ varying over the interval [0.6,1][\sqrt{0.6}, 1][0.6​,1]. For the ((7,2,3))((7, 2, 3))((7,2,3)) code, we identify λmin∗=0λ^*_{\text{min}} = 0λmin∗​=0 (corresponding to the non-degenerate Steane code) and λmax∗=7λ^*_{\text{max}} = \sqrt{7}λmax∗​=7​ (corresponding to the permutation-invariant code by Pollatsek and Ruskai), and we demonstrate continuous paths connecting these extremes via cyclic codes characterized solely by λ∗λ^*λ∗. Our findings provide new insights into the structure of quantum codes, advance the theoretical foundations of QEC, and open new avenues for investigating intricate relationships between code subspaces and error correlations.
We present a new geometric perspective on quantum error correction based on spectral triples in noncommutative geometry. In this approach, quantum error correcting codes are reformulated as low energy spectral projections of Dirac type operators that separate global logical degrees of freedom from local, correctable errors. Locality, code distance, and the Knill Laflamme condition acquire a unified spectral and geometric interpretation in terms of the induced metric and spectrum of the Dirac operator. Within this framework, a wide range of known error correcting codes including classical linear codes, stabilizer codes, GKP type codes, and topological codes are recovered from a single construction. This demonstrates that classical and quantum codes can be organized within a common geometric language. A central advantage of the spectral triple perspective is that the performance of error correction can be directly related to spectral properties. We show that leakage out of the code space is controlled by the spectral gap of the Dirac operator, and that code preserving internal perturbations can systematically increase this gap without altering the encoded logical subspace. This yields a geometric mechanism for enhancing error correction thresholds, which we illustrate explicitly for a stabilizer code. We further interpret Berezin Toeplitz quantization as a mixed spectral code and briefly discuss implications for holographic quantum error correction. Overall, our results suggest that quantum error correction can be viewed as a universal low energy phenomenon governed by spectral geometry.
Quantum error correction (QEC) is essential for protecting quantum information against noise, yet understanding the structure of the Knill-Laflamme (KL) coefficients {λij}\left\{{\lambda }_{ij}\right\}{λij​} from the condition PEi†EjP=λijPP{E}_{i}^{\dagger }{E}_{j}P={\lambda }_{ij}PPEi†​Ej​P=λij​P remains challenging, particularly for nonadditive codes. In this work, we introduce the signature vector λ→(P)\overrightarrow{\lambda }(P)λ(P) , composed of the off-diagonal KL coefficients {λij}\left\{{\lambda }_{ij}\right\}{λij​} , where each coefficient corresponds to equivalence classes of errors counted only once. We define its Euclidean norm λ * ( P ) as a scalar measure representing the total strength of error correlations within the code subspace defined by the projector P . We parameterize P on a Stiefel manifold and formulate an optimization problem based on the KL conditions to systematically explore possible values of λ * . Moreover, we show that, for (( n , K , d )) codes, λ * is invariant under local unitary transformations. Applying our approach to the ((6, 2, 3)) quantum code, we find that λmin⁡∗=0.6{\lambda }_{\min }^{* }=\sqrt{0.6}λmin∗​=0.6​ and λmax⁡∗=1{\lambda }_{\max }^{* }=1λmax∗​=1 , with λ * = 1 corresponding to a known degenerate stabilizer code. We construct continuous families of new nonadditive codes parameterized by vectors in R5{{\mathbb{R}}}^{5}R5 , with λ * varying over the interval [0.6,1][\sqrt{0.6},1][0.6​,1] . For the ((7, 2, 3)) code, we identify λmin⁡∗=0{\lambda }_{\min }^{* }=0λmin∗​=0 (corresponding to the non-degenerate Steane code) and λmax⁡∗=7{\lambda }_{\max }^{* }=\sqrt{7}λmax∗​=7​ (corresponding to the permutation-invariant code by Pollatsek and Ruskai), and we demonstrate continuous paths connecting these extremes via cyclic codes characterized solely by λ * . Our findings provide new insights into the structure of quantum codes, advance the theoretical foundations of QEC, and open new avenues for investigating intricate relationships between code subspaces and error correlations.
The Pauli-ball discrepancy, a quadratic discrepancy summed over all error radii, measures how uniformly the Pauli error images of a KKK-dimensional quantum code cover the ddd-dimensional Hilbert space. At parameters attaining equality in the quantum Singleton bound with minimum code distance ≥3\geq3≥3, or in the one-error Hamming bound with prime-power local dimension, the discrepancy of every projector of the prescribed rank equals an explicit benchmark plus an exact nonnegative remainder, which vanishes precisely for pure quantum maximum-distance-separable codes and nondegenerate perfect codes, respectively. For a Pauli ball of radius eee below half the target distance and size VeV_eVe​, this remainder linearly bounds the squared Hilbert--Schmidt distance of the error-image Gram matrix from the identity. When KVe≤dKV_e\leq dKVe​≤d, it linearly bounds the diamond error and entanglement infidelity of a noise-independent polar recovery under any noise in that ball's span. For five qubits, we obtain the sharp rank-two Gram constant and exact logical channels of rotated codes attaining this linear order in diamond error. For pure states, the discrepancy equals a pure-state benchmark plus a positive weighted sum of marginal-purity defects, so absolutely maximally entangled states, when they exist, are exactly the minimizers.
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.