Experimentally Implementing the Step-Dependent Discrete Time Quantum Walk on Quantum Computers

The Discrete-Time Quantum Walk (DTQW) with step-dependent scattering operator (SDS) is implemented on Quantum Computer (QC). 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 (QGM). The CNOT gates in the shift operator are replaced with the alternative to CNOT gates Rx(π). They are applied on a Quantum Device (QD) and a Quantum Simulator (QS). The fidelities vary around $50\%$ and the probability distribution of step-dependent DTQW (SD-DTQW) for the angle π/4 spreads symmetrically, while the step-independent DTQW (SI-DTQW) tends to peak at the one side. The symmetric distribution of probability of SD-DTQW can help in better control of the walk on QS. In the case of angle π/2, the SI-DTQW spreads equally across the states with four peaks, while the SD-DTQW spreads with two peaks to one side. Some other angles are simulated on QS for about 30 steps revealing some interesting features of SD-DTQW.