Quantum simulation of perfect state transfer on weighted cubelike graphs

A continuous-time quantum walk on a graph evolves according to the unitary operator $e^{-iAt}$, where $A$ is the adjacency matrix of the graph. Perfect state transfer (PST) in a quantum walk is the transfer of a quantum state from one node of a graph to another node with $100\%$ fidelity. It can be shown that the adjacency matrix of a cubelike graph is a finite sum of tensor products of Pauli $X$ operators. We use this fact to construct an efficient quantum circuit for the quantum walk on cubelike graphs. In \cite{Cao2021, rishi2021(2)}, a characterization of integer weighted cubelike graphs is given that exhibit periodicity or PST at time $t=\pi/2$. We use our circuits to demonstrate PST or periodicity in these graphs on IBM's quantum computing platform~\cite{Qiskit, IBM2021}.