23 research outputs found
Multiself-loop Lackadaisical Quantum Walk with Partial Phase Inversion
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
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 self-loops, with
weights , where is a real parameter. The phase inversion is based
on Grover's algorithm and acts partiality, modifying the phase of a given
quantity of self-loops. On a hypercube structure, we analyzed
the situation where and and investigated its
effects in the search for 1 to 12 marked vertices. Based on two ideal weights
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.Comment: 16 pages, 4 figures, 3 table
On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids
Quantum computing holds the promise of improving the information processing
power to levels unreachable by classical computation. Quantum walks are heading
the development of quantum algorithms for searching information on graphs more
efficiently than their classical counterparts. A quantum-walk-based algorithm
that is standing out in the literature is the lackadaisical quantum walk. The
lackadaisical quantum walk is an algorithm developed to search two-dimensional
grids whose vertices have a self-loop of weight . In this paper, we address
several issues related to the application of the lackadaisical quantum walk to
successfully search for multiple solutions on grids. Firstly, we show that only
one of the two stopping conditions found in the literature is suitable for
simulations. We also demonstrate that the final success probability depends on
the space density of solutions and the relative distance between solutions.
Furthermore, this work generalizes the lackadaisical quantum walk to search for
multiple solutions on grids of arbitrary dimensions. In addition, we propose an
optimal adjustment of the self-loop weight for such scenarios of arbitrary
dimensions. It turns out the other fits of found in the literature are
particular cases. Finally, we observe a two-to-one relation between the steps
of the lackadaisical quantum walk and the ones of Grover's algorithm, which
requires modifications in the stopping condition. In conclusion, this work
deals with practical issues one should consider when applying the lackadaisical
quantum walk, besides expanding the technique to a wider range of search
problems.Comment: Extended version of the conference paper available at
https://doi.org/10.1007/978-3-030-61377-8_9 . 21 pages, 6 figure