Multi-self-loop Lackadaisical Quantum Walk with Partial Phase Inversion

• URL: http://arxiv.org/abs/2305.01121v3
• Date: Mon, 18 Nov 2024 11:34:01 GMT
• Title: Multi-self-loop Lackadaisical Quantum Walk with Partial Phase Inversion
• Authors: Luciano S. de Souza, Jonathan H. A. de Carvalho, Henrique C. T. Santos, Tiago A. E. Ferreira,
• Abstract summary: This article proposes the quantum search algorithm Multi-self-loop Lackadaisical Quantum Walk with Partial Phase Inversion. The proposed algorithm is based on Grover's algorithm and acts partially, modifying the phase of a given quantity $s m$ of self-loops. It improves the maximum success probabilities to values close to $1$ in $O (sqrt(n+m)cdot N)$, where $n$ is the hypercube degree.
• Score: 3.8436076642278754
• License:
• Abstract: The lackadaisical quantum walk, a quantum analog of the lazy random walk, is obtained by adding a weighted self-loop transition to each state. Impacts of the self-loop weight $l$ on the final success probability in finding a solution make it a key parameter for the search process. The number of self-loops can also be critical for search tasks. This article proposes the quantum search algorithm Multi-self-loop Lackadaisical Quantum Walk with Partial Phase Inversion, which can be defined as a lackadaisical quantum walk with multiple self-loops, where the target state phase is partially inverted. In the proposed algorithm, 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 partially, modifying the phase of a given quantity $s < m$ of self-loops. On a hypercube structure, we analyzed the situation where $1 \leqslant m \leqslant 30$. We also propose two new weight values based on two ideal weights $l$ used in the literature. We investigated the effects of partial phase inversion in the search for $1$ to $12$ marked vertices. As a result, this proposal improved the maximum success probabilities to values close to $1$ in $O (\sqrt{(n+m)\cdot N})$, where $n$ is the hypercube degree. This article contributes with a new perspective on the use of quantum interferences in constructing new quantum search algorithms.

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