发表机构
Joint Quantum Institute, NIST/University of Maryland; Joint Center for Quantum Information and Computer Science, NIST/University of Maryland; School of Physics, The University of Sydney; Institute for Physical Sciences and Technology, University of Maryland; Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland（联合量子研究所，美国国家标准与技术研究院/马里兰大学; 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; 悉尼大学物理学院; 马里兰大学物理科学与工程技术研究所; 马里兰大学物理系凝聚态理论中心与联合量子研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文提出 $X\mathcal C^{(k)}_d$ 稳定子形式体系，推广 Pauli 稳定子以涵盖扭曲量子双重模型，证明其可规避 Bravyi-Koenig 界，并展示通过测量该稳定子可确定性制备逻辑魔法态。
AI 中文摘要
近期研究表明，量子双重模型可用于规避 Bravyi-Koenig 界。特别是，这通过码切换协议实现，其中非 Pauli 稳定子相位作为中间码出现。我们引入 $X\mathcal C^{(k)}_d$ 稳定子形式体系，它通过将 Pauli-$X$ 算子与取自素数维 qudit 上 Clifford 层级固定第 $k$ 层的对角算子相结合，推广了 Pauli 稳定子形式体系。这产生了一类自然的超越 Pauli 的稳定子模型，包括许多扭曲量子双重相位。我们对一大类 $X\mathcal C^{(k)}_d$ 稳定子态进行分类，表明它们是具有对角相位的仿射线性子空间上的均匀叠加。特别地，当态所需的稳定子算子来自 Clifford 层级第 $k$ 层时，我们证明产生波函数中相位所需的对角门位于高一层。我们进一步证明，每个二维扭曲量子双重 ${\cal D}^\omega(\mathbb{Z}_p^n)$，其中 $\omega$ 是 III 型上循环的乘积且 $p$ 是奇素数，都仅使用 Clifford（$k=2$）稳定子即可实现，并猜想当 $\omega$ 包含 I 型或 II 型上循环时，对于 $p$ 维 qudit 需要第 $p$ 层 Clifford 层级稳定子。最后，我们证明一对 qudit 环面码可以通过测量 $X\mathcal C^{(k)}_d$ 稳定子来确定性地制备在逻辑魔法态中。这些结果共同为超越 Pauli 稳定子模型的拓扑码及其逻辑门提供了一个具体的稳定子框架。
英文摘要
Recent work has shown that quantum double models can be used to circumvent the Bravyi-Koenig bound. In particular, this is achieved through code-switching protocols in which non-Pauli stabilizer phases appear as intermediate codes. We introduce the $X\mathcal C^{(k)}_d$ stabilizer formalism, which generalizes the Pauli stabilizer formalism by combining Pauli-$X$ operators with diagonal operators drawn from a fixed level $k$ of the Clifford hierarchy on prime-dimensional qudits. This yields a natural class of beyond-Pauli stabilizer models, including many twisted quantum double phases. We classify a large class of these $X\mathcal C^{(k)}_d$ stabilizer states, showing that they are uniform superpositions over affine-linear subspaces with diagonal phases. In particular, when the stabilizers of a state require operators from level $k$ of the Clifford hierarchy, we show that the diagonal gates required to produce the phases in the wavefunction sit one level higher. We further show that every 2D twisted quantum double ${\cal D}^ω(\mathbb{Z}_p^n)$ with $ω$ a product of type-III cocycles and $p$ an odd prime admits a realization using only Clifford ($k=2$) stabilizers, and we conjecture that phases where $ω$ contains a type-I or type-II cocycle require level-$p$ Clifford-hierarchy stabilizers for $p$-dimensional qudits. Finally, we demonstrate that a pair of qudit toric codes may be deterministically prepared in a logical magic state by measuring $X\mathcal C^{(k)}_d$ stabilizers. Together, these results provide a concrete stabilizer framework for topological codes beyond Pauli stabilizer models and their logical gates.
Comments19+14 pages, 4+3 figures
