Lee度量中的量子码

中文摘要 AI 辅助

我们针对离散小位移噪声引入了一个量子编码框架，在该框架中，qudit上的错误被建模为低权重的$X$型和$Z$型Pauli位移，类似于连续变量系统中的小相空间位移。这种结构近似地被核自旋噪声所满足，并由Lee度量所刻画，从而推动了经典Lee度量编码理论的量子扩展。我们为任意$q$的$\nmathbb{Z}_q$上的稳定子码发展了这样的形式体系，使用联合和分离的Lee度量来处理错误的$X$和$Z$分量。对于量子比特，联合度量将$Y$错误计数两次，并且可以产生比针对传统Hamming度量设计的码使用更少的物理量子比特来检测和纠正$X$和$Z$错误的码。我们讨论了Clifford门下的Lee权重扩散，并通过Gray映射将$\nmathbb{Z}_4$上的码量子化，发现了具有两倍横向非量子比特Clifford门的码。对于$n$个qudit上的量子CSS Lee-LDPC码，我们证明了Lee距离不能超过$O(n)$，且对$q$一致成立，这展示了利用大$q$ qudit的大内部希尔伯特空间来构造高Lee距离码的意外障碍。在局部Metropolis动力学下，某些经典的$q$元“螺旋重复码”在固定温度下对于足够大的$q$（随系统长度增长）具有指数长的记忆时间，这可以理解为即使在局部一维模型中，有限温度下的自发对称性破缺。这些螺旋重复码的超图积提供了局部二维量子码，它们继承了$Z$错误的自我纠正，但$X$错误的自我纠正仍然是一个开放问题。

英文摘要

We introduce a quantum coding framework for discrete small-shift noise, in which errors on qudits are modeled as low-weight $X$- and $Z$-type Pauli shifts, analogous to small phase-space displacements in continuous-variable systems. This structure is approximately respected by nuclear-spin noise and captured by the Lee metric, motivating a quantum extension of classical Lee-metric coding theory. We develop such a formalism for stabilizer codes over $\mathbb{Z}_q$ for arbitrary $q$, using joint and separate Lee metrics for the $X$- and $Z$-components of errors. For qubits, the joint metric counts $Y$ errors twice and can yield codes that detect and correct $X$ and $Z$ errors with fewer physical qubits than codes designed for the conventional Hamming metric. We discuss Lee-weight spreading under Clifford gates and qubitize codes over $\mathbb{Z}_4$ via the Gray map, finding codes with two-fold transversal non-qubit-Clifford gates. For quantum CSS Lee-LDPC codes on $n$ qudits, we prove that the Lee distance cannot exceed $O(n)$, uniformly in $q$, demonstrating an unexpected obstruction to using the large internal Hilbert space of a large-$q$ qudit to make high-Lee-distance codes. Under local Metropolis dynamics, certain classical $q$-ary ``helical repetition codes'' have exponentially long memory times at fixed temperature for sufficiently large $q$ (that grows with system length), which can be understood as spontaneous symmetry breaking at finite temperature, even in local one-dimensional models. Hypergraph products of these helical repetition codes provide local two-dimensional quantum codes that inherit self-correction for $Z$ errors, but self-correction for $X$ errors remains an open question.

发表机构

• University of Colorado, Boulder（科罗拉多大学博尔德分校）
• Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park（量子信息与计算机科学联合中心，美国国家标准与技术研究院/马里兰大学帕克分校）

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