  Quantum walks are the quantum counterpart of classical random walks and
provide an intuitive framework for building new quantum algorithms. The
lackadaisical quantum walk, which is a quantum analog of the lazy random walk,
is obtained by adding a self-loop transition to each state allowing the walker
to stay stuck in the same state, being able to improve the performance of the
quantum walks as search algorithms. However, the high dependence of a weight
$l$ makes it a key parameter to reach the maximum probability of success in the
search process. Although many advances have been achieved with search
algorithms based on quantum walks, the number of self-loops can also be
critical for search tasks. Believing that the multiple self-loops have not yet
been properly explored, this article proposes the quantum search algorithm
Multiself-loop Lackadaisical Quantum Walk with Partial Phase Inversion, which
is based on a lackadaisical quantum walk with multiple self-loops where the
target state phase is partially inverted. Each vertex has $m$ self-loops, with
weights $l' = l/m$, where $l$ is a real parameter. The phase inversion is based
on Grover's algorithm and acts partiality, modifying the phase of a given
quantity $s \leqslant m$ of self-loops. On a hypercube structure, we analyzed
the situation where $s=1$ and $1 \leqslant m \leqslant 30$ and investigated its
effects in the search for 1 to 12 marked vertices. Based on two ideal weights
$l$ used in the literature, we propose two new weight values. As a result, with
the proposal of the Multiself-loop Lackadaisical Quantum Walk with partial
phase inversion of target states and the new weight values for the self-loop,
this proposal improved the maximum success probabilities to values close to 1.
This article contributes with a new perspective on the use of quantum
interferences in the construction of new quantum search algorithms.

Quantum Physics