全纯量子纠错码
Holomorphic Quantum Error Correction Codes
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中文总结 AI 辅助
本文在Segal-Bargmann空间中提出全纯量子纠错码表示，用微分算子刻画稳定子与恢复，统一了代数、几何与拓扑视角，并建立路径积分表述以量化码距离。
中文摘要 AI 辅助
我们在Segal-Bargmann空间中发展了量子纠错码（QECCs）的全纯表示。通过将量子比特编码为受一次齐次性约束的Schwinger玻色子模式$(z_{a_j}, z_{b_j})$，我们为基本码（三量子比特、五量子比特、七量子比特和九量子比特码）推导了稳定子、综合征提取和恢复的闭式微分算子表示。量子误差被表征为违反该约束的全纯扰动，而综合征测量投影到对易微分算子的本征空间上。限制到单位模变量（$|z|=1$）揭示了环面空间$\T^{2n}$，其中误差综合征表现为绕数空间$\Z^{2n}$中的离散平移，恢复则作为恢复绕数构型的哈密顿流。在完整的Segal-Bargmann空间中，码空间是$\CP^{2^n-1}$的全纯子流形，可纠正误差作为横向法向方向。因此，Knill-Laflamme条件成为码子流形与其误差平移像之间的Fubini-Study正交性条件。拓扑保护源于$U(1)^n$纤维丛结构：沿纤维的全局相位噪声不可观测，而基空间误差需要主动纠正。最后，我们建立了半经典纠错动力学的路径积分表述，并表明通过Segre嵌入的几何纠缠自然量化了码距离。该框架统一了容错量子协议的代数、几何和拓扑视角。
英文摘要
We develop a holomorphic representation of quantum error correction codes (QECCs) within the Segal--Bargmann space. By encoding qubits into Schwinger boson modes $(z_{a_j}, z_{b_j})$ subject to a degree-one homogeneity constraint, we derive closed-form differential operator representations for stabilizers, syndrome extraction, and recovery for fundamental codes (three-, five-, seven-, and nine-qubit codes). Quantum errors are characterized as holomorphic perturbations violating this constraint, while syndrome measurement projects onto eigenspaces of commuting differential operators. Restricting to unit-magnitude variables ($|z|=1$) reveals a toroidal space $\T^{2n}$ where error syndromes manifest as discrete translations in winding number space $\Z^{2n}$, and recovery acts as Hamiltonian flows restoring the winding configuration. In the full Segal--Bargmann space, the code space is a holomorphic submanifold of $\CP^{2^n-1}$, with correctable errors as transverse normal directions. Consequently, the Knill--Laflamme condition becomes a Fubini--Study orthogonality condition between the code submanifold and its error-translated images. Topological protection emerges from the $U(1)^n$ fiber bundle structure: global phase noise along fibers is unobservable, while base-space errors require active correction. Finally, we establish a path-integral formulation for semiclassical error correction dynamics and show that geometric entanglement via the Segre embedding naturally quantifies code distance. This framework unifies algebraic, geometric, and topological perspectives on fault-tolerant quantum protocols.
发表机构
- Wilczek Quantum Center, School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院斐兹克量子中心）
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