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.

翻译：精确检测指定泡利误差集合的量子码可通过代数构造方法实现——包括稳定子码、码字稳定码、置换不变码、拓扑码及相关族系。从几何角度来看，精确泡利检测由这些泡利算子的联合高阶数值范围所支配，而秩≥2的此类算子结构在很大程度上尚属未知。基于此视角，我们证明这类码通常形成连通的连续族系，而非互不相交的解区域集合。这些族系可通过从Knill-Laflamme条件导出的单一标量来表征：记为λ^*，它是最大混态码状态上泡利期望值签名向量的欧几里得范数，提供了码联合泡利方差轮廓的单参数概述。在这些连续景观中，稳定子码仅占据可达λ^*谱中离散的零测度子集，揭示了尚未充分探索的、真正非可加性精确码的连续统。通过分析高阶算子压缩的几何结构，我们建立了这一图景，并将其扩展到对称性受限场景——即对误差模型和码投影子同时施加循环对称性和置换对称性。小系统案例通过特征值交错和对称扇区分解揭示了区间、单点及空集三种状态；较大系统则通过Stiefel流形优化和对称自适应参数化进行数值处理。在所有分析的无约束和对称兼容案例中，可达λ^*谱在非空时均构成单一闭区间——尽管通用证明尚待解决。这些结果将稳定子码、对称码及非可加性码族纳入统一的高阶方差框架，为精确量子码的景观提供了连续的几何视角。