On applying the lackadaisical quantum walk algorithm to search for multiple solutions on grids

Abstract

Quantum computing promises to improve 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 standing out in the literature is the lackadaisical quantum walk. The lackadaisical quantum walk is an algorithm developed to search graph structures whose vertices have a self-loop of weight l. This paper addresses several issues related to applying the lackadaisical quantum walk to search for multiple solutions on grids successfully. Firstly, we show that only one of the two stopping conditions found in the literature is suitable for simulations. In the most discrepant case shown here, a stopping condition is prematurely satisfied at the step T=288 with a success probability Pr=0.593276, while the suitable condition captures the actual amplification that occurred until T=409 with Pr=0.878178. We also demonstrate that the final success probability depends on both the space density of solutions and the relative distance between solutions. For instance, we show here that decreases in the density of solutions can even take a success probability of 0.849178 to 0.961896. In contrast, increases in the relative distances can even take a success probability of 0.871665 to 0.940301. 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 l for such d-dimensional grids. It turns out other fits of l found in the literature are particular cases. Our experiments demonstrate that successful searches for multiple solutions with higher than two dimensions are possible by achieving success probabilities such as 0.999979, with the value of l proposed here, where it would be 0.637346, with the value of l proposed in previous works. Finally, we observe a two-to-one relation between the steps of the lackadaisical quantum walk and Grover’s algorithm, which requires modifications in the stopping condition. That modified stopping condition can escape intermediary fluctuations that would produce premature stops at T=6 with Pr=0.000878 where the system can evolve until T=354 with Pr=0.99999, as an example that we show here. In conclusion, this work deals with practical issues one should consider when applying the lackadaisical quantum walk, besides expanding the technique to a broader range of search problems.