发表机构
Graduate School of Informatics, Kyoto University; Graduate School of Engineering Science, Osaka University; Center for Quantum Information and Quantum Biology, Osaka University; RIKEN Center for Quantum Computing (RQC)（京都大学大学院情报学府; 大阪大学工学研究科; 大阪大学量子信息与量子生物学中心; 理化学研究所量子计算中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究分析了Clifford模板编译的计算复杂性，发现其从经典可解跨越到NP完全，并存在量子多项式时间算法，揭示了该问题的丰富复杂性景观。
AI 中文摘要
Clifford模板是一个有限有序的可重复Clifford操作族，实例化指定了每个操作被应用的次数。Clifford模板编译问题询问如何选择这些重复次数，使得模板实现Pauli算子的目标变换。例如，在量子纠错中搜索仅使用物理或容错约束允许的Clifford操作的逻辑操作时，会出现此问题。尽管正向Clifford动力学可经典高效模拟，但此逆问题具有尖锐的复杂性转变。对于具有无限制整数指数的交换模板，可行性属于$\mathrm{NP}\cap\mathrm{BQP}$，且构造性量子算法返回特定解以及完整的指数关系格；即使在$k=1$时，恢复重复次数包含有限域$\mathbb{F}_{2^r}^{\times}$上的离散对数。一般而言，将每个指数限制为$\{0,1\}$会移除阿贝尔群闭包，并使可行性对于可变$k$成为NP完全问题，即使对于完全交换的仅CNOT操作和X型Pauli算子也是如此。对于交换的自逆Clifford作用，二元可行性和恢复一个解均可经典多项式时间求解，但对总重复次数施加界限是NP完全的，即使仅对CNOT操作也是如此。这些结果揭示了Clifford模板编译中丰富的复杂性景观，涵盖经典可处理情况、承认量子多项式时间算法的问题以及NP完全变体。
英文摘要
A Clifford template is a finite ordered family of repeatable Clifford operations, and an instantiation specifies how many times each operation is applied. The Clifford template compilation problem asks how to choose these repetition numbers so that the template realizes a target transformation of Pauli operators. This problem arises, for example, when searching for logical operations in quantum error correction using only Clifford operations permitted by physical or fault-tolerance constraints. Although forward Clifford dynamics is efficiently classically simulable, this inverse problem has sharp complexity transitions. For commuting templates with unrestricted integer exponents, feasibility lies in $\mathrm{NP}\cap\mathrm{BQP}$ and a constructive quantum algorithm returns a particular solution together with the full exponent-relation lattice; already at $k=1$, recovering the repetition number contains finite-field discrete logarithm over $\mathbb{F}_{2^r}^{\times}$. In general, restricting every exponent to $\{0,1\}$ removes the Abelian-group closure and makes feasibility NP-complete for variable $k$, even for exactly commuting CNOT-only operations and X-type Paulis. For commuting self-inverse Clifford actions, both binary feasibility and recovery of one solution are classically polynomial-time solvable, but imposing a bound on the total repetition count is NP-complete, even for CNOT-only operations. These results reveal a rich complexity landscape within Clifford template compilation, spanning classically tractable cases, problems admitting quantum polynomial-time algorithms, and NP-complete variants.
Comments12 pages, 2 figures, 1 table
