论文标题
无哈达马电路暴露了克利福德集团的结构
Hadamard-free circuits expose the structure of the Clifford group
论文作者
论文摘要
克利福德组在量子随机基准测试，量子断层扫描和误差校正方案中起着核心作用。在这里，我们研究该组的结构特性。我们表明，任何Clifford操作员都可以用规范的形式$ f_1hsf_2 $唯一地写成，其中$ h $是哈达姆大门的一层，$ s $是Qubits的排列，$ f_i $是从Clifford Group的合适子组中选择的参数化Hadamard的无用电路。我们的规范形式在Clifford操作员和分层量子电路之间提供一对一的对应关系。我们报告了用于计算规范形式的多项式时间算法。我们采用此规范表格来生成一个随机分布的$ n $ qubit Clifford运算符，以运行时$ O（n^2）$。该算法消耗的随机位数与信息理论下限匹配。在随机统一的克利福德操作员和对称组上的绿色摩洛龙分布之间突出了一个令人惊讶的连接。还讨论了在线性最接近的邻居体系结构中实现任意的Clifford Unitaries的典型形式的变体，其中一个零件短，允许电路深度$ 9N $实现。最后，我们研究了计算量子优势，其中可以使用Clifford门更有效地实现经典的可逆线性电路，并显示出发生这种优势的明确示例。
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 $F_1HSF_2$, 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.
