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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带有模板和符号Pauli Gates的Clifford电路优化

Clifford组是由Hadamard，CNOT和Phase Gates生成的unit组的有限子组。该小组在量子误差校正，随机基准测试协议和纠缠研究中发挥着重要作用。在这里，我们考虑找到一个实现给定Clifford群元素的短量子电路的问题。我们的方法旨在假设所有量子位连通性，以最小化纠缠门数。首先，我们考虑基于模板匹配的电路优化，并设计Clifford专用模板，这些模板利用了分解Pauli和SWAP门的能力。其次，我们介绍了一种象征性的窥孔优化方法。它通过将整个电路投影到一小部分量子位上并通过动态编程来最佳地重新编译投影的子电路来工作。使用符号泡利门表示将量子位的选定子集与其余量子位耦合的CNOT门。与Aaronson-Gottesman规范形式相比，这些方法的软件实现可发现与6个量子位的最优电路仅相距0.2％的电路，并且将多达64个量子位的电路中的两个量子位门数平均减少了64.7％。