发表机构
Barcelona Supercomputing Center; Los Alamos National Laboratory; Johannes Kepler University; École Polytechnique Fédérale de Lausanne (EPFL); Princeton University（巴塞罗那超级计算中心; 洛斯阿拉莫斯国家实验室; 约翰内斯·开普勒大学; 洛桑联邦理工学院; 普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究matchgate-Clifford群中最优精确合成问题的复杂性，证明其依赖于门集连通图：路径和完全图可多项式时间求解，树图则NP完全，并给出优于现有编译器的门最优方案。
AI 中文摘要
Clifford电路和matchgate电路是经典可模拟量子电路的两个典型家族。它们的交集，即matchgate-Clifford群，在随机费米子协议和matchgate合成中扮演重要角色。其伴随作用与$2n$个Majorana模式的单位行列式符号置换群同构，我们研究该群中的最优精确合成问题。即，给定目标酉算子和门集，输出一个使用最少操作实现目标的$n$量子比特电路。我们证明该问题的复杂性强烈依赖于门集。特别地，我们研究由不同连通图构成的Majorana辫门集。对于路径图和完全图，我们证明该问题可在$\mathcal{O}\left(n^2\right)$时间内经典求解，并提供显式的门最优编译器。此外，我们证明当连通图为树时，最优合成问题的判定版本变为NP完全问题。最后，我们将我们的最优编译器在多达$n=80$量子比特的链上与\texttt{Qiskit}和\texttt{Tket}的编译器进行基准测试，在总门数上分别获得$\times2.57$和$\times2.28$的常数因子改进。
英文摘要
Clifford and matchgate circuits are canonical families of classically simulable quantum circuits. Their intersection, the matchgate-Clifford group, plays an important role in randomized fermionic protocols and in matchgate synthesis. Its adjoint action is isomorphic to the group of unit-determinant signed permutations of $2n$ Majorana modes, and we study optimal exact synthesis in this group. That is, given a target unitary and a gate set, output an $n$-qubit circuit implementing the target using the fewest operations. We show that the complexity of this problem strongly depends on the gate set. In particular, we study gate sets consisting of Majorana braids with different connectivity graphs. For path and complete graphs, we prove that the problem is classically solvable in $\mathcal{O}\left(n^2\right)$ time, and we provide explicit gate-optimal compilers. In addition, we prove that when the connectivity graph is a tree, the decision version of the optimal synthesis problem becomes NP-complete. Finally, we benchmark our optimal compiler on chains of up to $n=80$ qubits against those of \texttt{Qiskit} and \texttt{Tket}, obtaining circuits with constant factor improvements $\times2.57$ and $\times2.28$ in the total number of gates, respectively.
Comments20 + 21 pages, 4 + 1 figures
