一般相空间上的稳定子码
Stabilizer codes over general phase spaces
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中文总结 AI 辅助
本文提出一般相空间上稳定子码的统一理论，涵盖qudit、振子与转子系统，构造不可分解码并推导逻辑算符、距离与解码，贡献在于统一并推广了多种已知码型。
中文摘要 AI 辅助
我们发展了一套稳定子码理论，其稳定子群由对易的qudit泡利算符、振子位移算符和/或平面转子位移算符组成。我们考虑稳定子群在量子相空间中形成广义格子的码，这一条件保证了有限的逻辑维度。我们构造了振子-转子码、转子-qudit码以及振子-转子-qubit码，这些码无法通过广义Clifford变换分解为独立的子系统。我们还引入了一种由部分提升的Golay码产生的振子-qubit码，其中Construction A应用于Golay码坐标的一半。我们从辛对偶格推导出逻辑算符的解析形式，从格对称性确定Clifford-Gaussian门，证明码维度等于稳定子格的协体积，并将校正子子空间组织成向量丛。我们的技术基于Rieffel早期的非交换几何结果，但通常可以简单地解释为将给定的稳定子码与Gottesman-Kitaev-Preskill（GKP）码级联，应用常规格论结果，然后解除级联。我们公式化了距离、最优解码和量子权重枚举器，恢复了Pauli、qudit和GKP版本以及已知的MacWilliams恒等式。我们构造了有限能量码字，其编码是一个等距映射，误差在逆阻尼参数的指数小范围内。
英文摘要
We develop a theory of stabilizer codes whose stabilizer groups consist of commuting qudit Paulis, oscillator displacements, and/or planar-rotor displacements. We consider codes whose stabilizer groups form generalized lattices in quantum phase space, a condition that guarantees a finite logical dimension. We construct oscillator-rotor, rotor-qudit, and oscillator-rotor-qubit codes that cannot be decomposed into separate subsystems by generalized Clifford transformations. We also introduce an oscillator-qubit code arising from a partially lifted Golay code, where Construction A is applied to half of the Golay code coordinates. We derive analytical forms of logical operators from the symplectic dual lattice, determine Clifford-Gaussian gates from lattice symmetries, show that the code dimension equals the covolume of the stabilizer lattice, and organize the syndrome subspaces into a vector bundle. Our technique builds on earlier non-commutative geometric results by Rieffel, but it can often be interpreted simply as concatenating a given stabilizer code with a Gottesman-Kitaev-Preskill (GKP) code, applying conventional lattice-theoretic results, and unconcatenating. We formulate distances, optimal decoding, and quantum weight enumerators, recovering the Pauli, qudit, and GKP versions together with known MacWilliams identities. We construct finite-energy codewords whose encoding is an isometry up to an error exponentially small in the inverse damping parameter.
发表机构
- Institute for Advancing Intelligence, TCG CREST（TCG CREST 先进智能研究所）
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（量子信息与计算联合中心，美国国家标准与技术研究院/马里兰大学）
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