We present a quantum circuit implementation of the quantum hashing algorithm (quantum fingerprinting) for a quantum device with restrictions on the application of two-qubit gates by a qubit connectivity graph. We present an optimization technique for the shallow circuit for quantum hashing in the case of a cactus as a qubit connectivity graph. The algorithm has $O(n^3)$ complexity to build the circuit, where $n$ is the number of qubits and $m$ is the number of connections (edges) in the graph. It is improvement compared to the existing exponential-time algorithm in the case of arbitrary graphs. The algorithm uses solution for the shortest non-simple 1-covering path problem as a subroutine. We present an $O(n^3)$-time solution for this graph-theory problem in the case of a cactus. This result can be interesting independently. The algorithm also used for improving of the quantum circuit for Quantum Fourier Transform.

翻译：我们提出了一种量子哈希算法（量子指纹识别）的量子电路实现方案，该方案适用于受量子比特连接图限制双量子比特门应用的量子设备。针对仙人掌图作为量子比特连接图的情形，我们提出了一种量子哈希浅层电路的优化技术。该算法构建电路的计算复杂度为 $O(n^3)$，其中 $n$ 表示量子比特数，$m$ 表示图中的连接边数。与现有针对任意图结构的指数时间复杂度算法相比，这一结果实现了改进。该算法将最短非简单1-覆盖路径问题的求解作为子程序。针对仙人掌图情形，我们为该图论问题提出了一个 $O(n^3)$ 时间复杂度的解决方案，该结果本身可能具有独立研究价值。该算法还可用于优化量子傅里叶变换的量子电路实现。