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

2026-05-20

AI Summary

This work addresses the challenge of limited two-qubit gate connectivity in cactus-structured quantum hardware by proposing an efficient circuit optimization method that integrates graph theory with quantum compilation. The key innovation lies in the novel introduction of cactus graphs into quantum circuit design and the formulation of the shortest non-simple 1-covering path problem, which enables a polynomial-time algorithm with O(n³) complexity—significantly improving upon conventional exponential-complexity approaches. Leveraging this framework, the method successfully constructs shallow quantum hash (fingerprinting) circuits and efficient quantum Fourier transforms, achieving substantial reductions in circuit depth and resource overhead while preserving functional correctness.

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.

Problem

• quantum hashing
• qubit connectivity graph
• cactus graph
• Quantum Fourier Transform
• quantum circuit

Innovation

• quantum hashing
• cactus graph
• qubit connectivity
• shallow quantum circuit
• quantum Fourier transform