Simulation of Anderson Localization with One-dimensional Quantum Random Walks in Disordered Conditions

Quantum walks with disorder lead to various interesting phenomena, such as Anderson localization. A major challenge in simulating quantum walks with disorder is that the phase of the walker can be disturbed. In this work, we investigated the effect of disorder on the behavior of quantum random walks and the relationship between phase shifts and localized states in the system. Using the quantum computing platform Qiskit, we simulated a quantum circuit by constructing a walkeroperator and adding a Trotterized circuit to create disorder at different levels in a one-dimensional structure. The eigenvalues of the system were determined from the phase shifts. The simulation showed that the eigenvalues obtained from the quantum circuit provided insights into particle localization, which became more pronounced as disorder increased, supporting theoretical predictions.