Search for Multiple Adjacent Marked Vertices on the Hypercube by a Quantum Walk with Partial Phase Inversion

Abstract: In this paper, we analyze the application of the Multi-self-loop Lackadaisical Quantum Walk on the hypercube that uses partial phase inversion to search for multiple adjacent marked vertices. We evaluate the influence of the relative position of non-adjacent marked vertices. The use of self-loops and the composition of their weights are an essential part of the construction process of new quantum search algorithms based on lackadaisical quantum walks, however, other aspects have been considered, such as, for example, the type of marked vertices. Part of the energy of a quantum system is retained in states adjacent to the target state. This behavior causes the probability amplitudes of these states to be amplified to values equivalent to those of the target state, reducing their chances of being observed. Here we show experimentally that with the use of partial phase inversion, it is possible to amplify their probability amplitudes to values close to 1 even in scenarios with adjacent marked vertices. We also show that the relative position of the non-adjacent marked vertices did not significantly influence the results. The lackadaisical quantum walk generalization to only a single self-loop and the ideal composition of a weight value was sufficient to obtain advances to quantum search algorithms based on quantum walks. However, the results show that many other aspects need to be taken into account for the construction of new quantum algorithms. It was possible to add gains in the maximum probabilities of success compared to other results found in the literature. In one of the most significant cases, the probability of success increased from $p \approx 0.38$ to $p > 0.99$. Therefore, the use of partial phase inversion brings new contributions to the development of new quantum search algorithms based on quantum walks and the use of multiple self-loops.