Quantum circuit model for continuous-time quantum walks on random graphs

Abstract

Quantum-circuit implementations of continuous-time quantum walks (CTQWs) can provide an efficient route to model graph-based algorithms. However, constructing circuits that faithfully reproduce CTQW dynamics across arbitrary graphs remains a major challenge. In this work, we introduce a Laplacian partitioning algorithm (LPA) that enables an efficient and scalable quantum-circuit realization of CTQWs on random graphs. A common algorithm to simulate a general graph (of size $N = 2^n$ for $n$ qubits) on a quantum circuit is based on Pauli decomposition of the graph Hamiltonian, which can yield $O(4^n)$ terms, and require $O(N^2\log N)$ time for coefficient computation. In contrast, our LPA uses $O(2^n)$ terms, in $O(N^2)$ time. Our circuit provides a graph-agnostic framework for CTQWs, implemented via a Trotter-Suzuki product formula and confirming error scaling consistent with theoretical Trotter error bounds. To further test the circuit performance, we study the localization behavior of the CTQW. In our case, localization originates from Laplacian spectral degeneracies rather than disorder (Anderson-type), and our circuit faithfully reproduces these localization phenomena and spectral structure for a random graphs with high accuracy.