Experimentally implementing the step-dependent discrete-time quantum walk on quantum computers

Abstract

The discrete-time quantum walk (DTQW) with step-dependent (SD) scattering operator is implemented on quantum computer. The probabilities of different states, with their respective fidelities, are calculated. This is done by generalizing the coin with a rotation gate using the quantum gate model. The CNOT gates in the shift operator are replaced with the alternative to CNOT gates Rx( π). They are applied on a quantum device and a quantum simulator (QS). The fidelities varied around 50% and the probability distribution of SD-DTQW for the angle π/4 spread symmetrically, while the step-independent DTQW (SI-DTQW) tended to peak at one side. The symmetric probability distribution of SD-DTQW can help in better control of the walk on QS. In the case of angle π/2, the SI-DTQW spread equally across the states with four peaks, while the SD-DTQW spread with two peaks to one side.