Discrete-time quantum walk search on the cycle graph with weighted self-loop

The evolution operator of the lackadaisical quantum walk on a weighted cycle graph with self-loop weight \(w>0\) is examined. With Grover oracle, the effect of different values of \(w\) on the success probability of the search under various hyperparameter combinations is analyzed. The success probability function of the first few steps is provided algebraically, which in turn highlights the role of \(w\) in governing the evolutionary behavior of the quantum walk search. These studies subsequently allow the numerical results to be obtained for various hyperparameter combinations, showing that the highest success probability for the weighted cycle graph with \(N\) vertices is achieved using a flip-flop shift operator, a weighted coin superposition initial state, and \(w=\frac{1.26}{N}\). A comparison is also made with other known oracle, demonstrating that the proposed configuration provides a better trade-off between success probability and runtime. An extension of the study to the SKW scheme is also included, and it demonstrates that the scheme provides more variability and potential for quantum walk search.