The Clifford group plays a central role in quantum randomized benchmarking, quantum tomography, and error correction protocols. Here we study the structural properties of this group. We show that any Clifford operator can be uniquely written in the canonical form $F_{1}HSF_{2}$ , where $H$ is a layer of Hadamard gates, $S$ is a permutation of qubits, and $F_{i}$ are parameterized Hadamard-free circuits chosen from suitable subgroups of the Clifford group. Our canonical form provides a one-to-one correspondence between Clifford operators and layered quantum circuits. We report a polynomial-time algorithm for computing the canonical form. We employ this canonical form to generate a random uniformly distributed $n$ -qubit Clifford operator in runtime $O(n^{2})$ . The number of random bits consumed by the algorithm matches the information-theoretic lower bound. A surprising connection is highlighted between random uniform Clifford operators and the Mallows distribution on the symmetric group. The variants of the canonical form, one with a short Hadamard-free part and one allowing a circuit depth $9n$ implementation of arbitrary Clifford unitaries in the Linear Nearest Neighbor architecture are also discussed. Finally, we study computational quantum advantage where a classical reversible linear circuit can be implemented more efficiently using Clifford gates, and show an explicit example where such an advantage takes place.

中文翻译：

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Hadamard-Free 电路揭示了 Clifford Group 的结构

Clifford 小组在量子随机基准测试、量子断层扫描和纠错协议中发挥着核心作用。在这里，我们研究该组的结构特性。我们证明任何 Clifford 算子都可以唯一地写成规范形式 $F_{1}HSF_{2}$ ， 在哪里 $H$ 是一层哈达玛门， $S$ 是量子位的排列，并且 $F_{i}$ 是从 Clifford 群的合适子群中选择的参数化 Hadamard-free 电路。我们的规范形式提供了 Clifford 算子和分层量子电路之间的一一对应关系。我们报告了一种用于计算规范形式的多项式时间算法。我们采用这种规范形式来生成随机均匀分布的 $n$ -qubit Clifford 运算符在运行时 $O(n^{2})$ . 算法消耗的随机比特数与信息论下限相匹配。随机均匀 Clifford 算子与对称群上的 Mallows 分布之间突出显示了令人惊讶的联系。规范形式的变体，一种具有短的无哈达玛部分，另一种允许电路深度 $9n$ 还讨论了线性最近邻体系结构中任意 Clifford 幺正的实现。最后，我们研究了计算量子优势，其中使用克利福德门可以更有效地实现经典的可逆线性电路，并展示了一个明确的例子，其中出现了这种优势。