The Clifford group is a finite subgroup of the unitary group generated by the Hadamard, the CNOT, and the Phase gates. This group plays a prominent role in quantum error correction, randomized benchmarking protocols, and the study of entanglement. Here we consider the problem of finding a short quantum circuit implementing a given Clifford group element. Our methods aim to minimize the entangling gate count assuming all-to-all qubit connectivity. First, we consider circuit optimization based on template matching and design Clifford-specific templates that leverage the ability to factor out Pauli and SWAP gates. Second, we introduce a symbolic peephole optimization method. It works by projecting the full circuit onto a small subset of qubits and optimally recompiling the projected subcircuit via dynamic programming. CNOT gates coupling the chosen subset of qubits with the remaining qubits are expressed using symbolic Pauli gates. Software implementation of these methods finds circuits that are only 0.2% away from optimal for 6 qubits and reduces the two-qubit gate count in circuits with up to 64 qubits by 64.7% on average, compared with the Aaronson-Gottesman canonical form.

中文翻译：

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使用模板和符号泡利门的 Clifford 电路优化

Clifford 群是由 Hadamard、CNOT 和相位门生成的酉群的有限子群。该小组在量子纠错、随机基准协议和纠缠研究中发挥着重要作用。在这里，我们考虑找到实现给定 Clifford 群元素的短量子电路的问题。我们的方法旨在假设全对全量子位连接，最大限度地减少纠缠门数。首先，我们考虑基于模板匹配和设计 Clifford 特定模板的电路优化，这些模板利用了分离泡利和 SWAP 门的能力。其次，我们介绍了一种符号窥视孔优化方法。它的工作原理是将整个电路投影到一个小的量子位子集上，并通过动态编程以最佳方式重新编译投影的子电路。将所选量子位子集与剩余量子位耦合的 CNOT 门使用符号泡利门表示。与 Aaronson-Gottesman 规范形式相比，这些方法的软件实现发现与 6 个量子位的最佳电路仅相差 0.2%，并且将最多 64 个量子位的电路中的双量子位门数平均减少了 64.7%。