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arXiv:2210.13920 · Entropy

Quantum Spatial Search with Electric Potential: Long-Time Dynamics and Robustness to Noise

Thibault Fredon, Julien Zylberman, Pablo Arnault, Fabrice Debbasch

Published Dec 5, 2022• 4 citations
arXivHTMLPDF

Abstract

We present various results on the scheme introduced in a previous work, which is a quantum spatial-search algorithm on a two-dimensional (2D) square spatial grid, realized with a 2D Dirac discrete-time quantum walk (DQW) coupled to a Coulomb electric field centered on the the node to be found. In such a walk, the electric term acts as the oracle of the algorithm, and the free walk (i.e., without electric term) acts as the “diffusion” part, as it is called in Grover’s algorithm. The results are the following. First, we run long time simulations of this electric Dirac DQW, and observe that there is a second localization peak around the node marked by the oracle, reached in a time O(N), where N is the number of nodes of the 2D grid, with a localization probability scaling as O(1/lnN). This matches the state-of-the-art 2D-DQW search algorithms before amplitude amplification We then study the effect of adding noise on the Coulomb potential, and observe that the walk, especially the second localization peak, is highly robust to spatial noise, more modestly robust to spatiotemporal noise, and that the first localization peak is even highly robust to spatiotemporal noise.

DOI
10.3390/e24121778

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We present various results on the scheme introduced in a previous work, which is a quantum spatial-search algorithm on a two-dimensional (2D) square spatial grid, realized with a 2D Dirac discrete-time quantum walk (DQW) coupled to a Coulomb electric field centered on the the node to be found. In such a walk, the electric term acts as the oracle of the algorithm, and the free walk (i.e., without electric term) acts as the “diffusion” part, as it is called in Grover’s algorithm. The results are the following. First, we run long time simulations of this electric Dirac DQW, and observe that there is a second localization peak around the node marked by the oracle, reached in a time O(N), where N is the number of nodes of the 2D grid, with a localization probability scaling as O(1/lnN). This matches the state-of-the-art 2D-DQW search algorithms before amplitude amplification We then study the effect of adding noise on the Coulomb potential, and observe that the walk, especially the second localization peak, is highly robust to spatial noise, more modestly robust to spatiotemporal noise, and that the first localization peak is even highly robust to spatiotemporal noise.
Electric Dirac quantum walks, which are a discretisation of the Dirac equation for a spinor coupled to an electric field, are revisited in order to perform spatial searches. The Coulomb electric field of a point charge is used as a non local oracle to perform a spatial search on a 2D grid of N points. As other quantum walks proposed for spatial search, these walks localise partially on the charge after a finite period of time. However, contrary to other walks, this localisation time scales as N for small values of N and tends asymptotically to a constant for larger Ns, thus offering a speed-up over conventional methods.
Quantum-walk-based algorithms that search a marked location among NNN locations on a ddd-dimensional lattice succeeds in time O(N)O(\sqrt{N})O(N​) for d>2d>2d>2, while this is not found to be possible when d=2d=2d=2. In this paper, we consider a spatial search algorithm using continuous-time quantum walk on a two-dimensional square lattice with the existence of additional long-range edges. We examined such a search on a probabilistic graph model where an edge connecting non-nearest-neighbor lattice points iii and jjj apart by a distance $|i\ensuremath{-}j|$ is added by probability ${p}_{ij}={|i\ensuremath{-}j|}^{\ensuremath{-}\ensuremath{\alpha}}\phantom{\rule{1em}{0ex}}(\ensuremath{\alpha}\ensuremath{\ge}0)$. Through numerical analysis, we found that the search succeeds in time O(N)O(\sqrt{N})O(N​) when $\ensuremath{\alpha}\ensuremath{\le}{\ensuremath{\alpha}}_{c}=2.4\ifmmode\pm\else\textpm\fi{}0.1$. For $\ensuremath{\alpha}>2$, the expectation value of the additional long-range edges on each node scales as a constant when $N\ensuremath{\rightarrow}\ensuremath{\infty}$, which means that search time of O(N)O(\sqrt{N})O(N​) is achieved on a graph with average degree scaling as a constant.
The task of finding an entry in an unsorted list of NNN elements famously takes O(N)O(N)O(N) queries to an oracle for a classical computer and O(N)O(\sqrt{N})O(N​) queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large NNN, the random walks converge to the same evolution, both taking Nln⁡(1/ϵ)N \ln(1/\epsilon)Nln(1/ϵ) time to reach a success probability of 1−ϵ1-\epsilon1−ϵ. In contrast, the discrete-time quantum walk asymptotically takes πN/22\pi\sqrt{N}/2\sqrt{2}πN​/22​ timesteps to reach a success probability of 1/21/21/2, while the continuous-time quantum walk takes πN/2\pi\sqrt{N}/2πN​/2 time to reach a success probability of 111.
Abstract Quantum search algorithms are crucial for exploring large solution spaces, but their robustness to environmental perturbations, such as noise or disorder, remains a critical challenge. We examine the impact of biased disorder potentials modeled by a bivariate Gaussian distribution function on the dynamics of quantum walks in spatial search problems. Building on the Ambainis-Kempe-Rivosh (AKR) model for searching on a two-dimensional grid, we incorporate potential fields to investigate how changes in standard deviation and normalization of the bivariate Gaussian function impact the performance of the search algorithm. Our results show that the quantum walk closely mirrors the AKR algorithm when the standard deviation is small but exhibits a rapid decay in success probability as the standard deviation increases. This behavior demonstrates how the bivariate Gaussian can effectively model a noisy oracle within the AKR algorithm. Additionally, we compare the AKR-based model with an alternative quantum walk model using a Hadamard coin and standard shift. These findings contribute to understanding the robustness of quantum walk search algorithms, and provide insights into how quantum walks can be applied to optimization algorithms.
We address quantum spatial search on graphs and its implementation by continuous-time quantum walks in the presence of dynamical noise. In particular, we focus on search on the complete graph and on the star graph of order NNN, also proving that noiseless spatial search shows optimal quantum speedup in the latter, in the computational limit $N\ensuremath{\gg}1$. The noise is modeled by independent sources of random telegraph noise (RTN), dynamically perturbing the links of the graph. We observe two different behaviors depending on the switching rate of RTN: fast noise only slightly degrades performance, whereas slow noise is more detrimental and, in general, lowers the success probability. In particular, we still find a quadratic speedup for the average running time of the algorithm, while for the star graph with external target node, we observe a transition to classical scaling. We also address how the effects of noise depend on the order of the graphs and discuss the role of the graph topology. Overall, our results suggest that realizations of quantum spatial search are possible with current technology and indicate the star graph as the perfect candidate for the implementation by noisy quantum walks, owing to its simple topology and nearly optimal performance for just a few nodes also.
Abstract Quantum walks are a powerful framework for the development of quantum algorithms, with lackadaisical quantum walks (LQWs) standing out as an efficient model for spatial search. In this work, we investigate how broken-link decoherence affects the performance of LQW-based search on a two-dimensional toroidal grid. We show through numerical simulations that while decoherence drives the loopless walk toward a uniform distribution and eliminates its search capability, the inclusion of self-loops significantly mitigates this effect. In particular, even under noise, the marked vertex remains identifiable with probability well above uniform, demonstrating that self-loops enhance the robustness of LQWs in realistic scenarios. These findings extend the known advantages of LQWs from the noiseless setting to noisy environments, consolidating self-loops as a valuable resource for designing resilient quantum search algorithms.