Circuits of Quantum Hashing and Quantum Fourier Transform for a Cactus as a Qubit Connectivity Graph

TL;DR

This paper introduces optimized quantum circuits for hashing and Fourier transform tailored to cactus connectivity graphs, improving efficiency for restricted qubit interactions.

Contribution

It presents an $O(n^3)$-time algorithm for quantum hashing and Fourier transform circuits on cactus graphs, surpassing previous exponential-time methods for arbitrary graphs.

Findings

Quantum hashing circuit complexity reduced to $O(n^3)$

Optimized shallow circuits for cactus connectivity graphs

Improved quantum Fourier Transform circuits for specific graph structures

Abstract

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.