Continuous-time quantum walk on a random graph using quantum circuits

Abstract: Quantum walks, particularly continuous-time quantum walks (CTQW), have emerged as powerful tools for modeling quantum transport, simulating complex dynamics, and developing quantum algorithms with potential speedups over classical counterparts. In this work, we present a scalable quantum circuit formalism to simulate CTQW on random graph structures, especially focusing on Erd\H{o}s-R\'enyi random graphs. Our quantum circuit construction efficiently implements the time evolution of the graph Laplacian, using the Trotterization scheme. We investigate key dynamical properties, \emph{i.e.,} the localization behavior of the CTQW. Our quantum circuit implementation over random graph ensures that the circuit design can work on any graph structure, thereby laying the foundation for realizing CTQW-based quantum simulations efficiently.