Circuits of Quantum Hashing and Quantum Fourier Transform for a Cactus as a Qubit Connectivity Graph
This work addresses the practical challenge of implementing quantum algorithms on devices with restricted qubit connectivity, offering a polynomial-time solution for cactus graphs.
The authors present a quantum circuit implementation of quantum hashing for a cactus qubit connectivity graph, achieving O(n^3) complexity to build the circuit, an improvement over the exponential-time algorithm for arbitrary graphs. They also apply this to improve the Quantum Fourier Transform circuit.
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.