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arXiv:1812.06676 · Europhysics Letters (EPL)

Continuous-time quantum walks on dynamical percolation graphs

Claudia Benedetti, Matteo A. C. Rossi, Matteo G. A. Paris

Published Jan 4, 2019• 30 citations
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Abstract

We address continuous-time quantum walks on graphs in the presence of time- and space-dependent noise. Noise is modeled as generalized dynamical percolation, i.e. , classical time-dependent fluctuations affecting the tunneling amplitudes of the walker. In order to illustrate the general features of the model, we review recent results on two paradigmatic examples: the dynamics of quantum walks on the line and the effects of noise on the performances of quantum spatial search on the complete and the star graph. We also discuss future perspectives, including extension to many-particle quantum walk, to noise model for on-site energies and to the analysis of different noise spectra. Finally, we address the use of quantum walks as a quantum probe to characterize defects and perturbations occurring in complex, classical and quantum, networks.

DOI
10.1209/0295-5075/124/60001

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We address continuous-time quantum walks on graphs in the presence of time- and space-dependent noise. Noise is modeled as generalized dynamical percolation, i.e. , classical time-dependent fluctuations affecting the tunneling amplitudes of the walker. In order to illustrate the general features of the model, we review recent results on two paradigmatic examples: the dynamics of quantum walks on the line and the effects of noise on the performances of quantum spatial search on the complete and the star graph. We also discuss future perspectives, including extension to many-particle quantum walk, to noise model for on-site energies and to the analysis of different noise spectra. Finally, we address the use of quantum walks as a quantum probe to characterize defects and perturbations occurring in complex, classical and quantum, networks.
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.
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Abstract Continuous-time quantum walks may be exploited to enhance spatial search, i.e., for finding a marked element in a database structured as a complex network. However, in practical implementations, the environmental noise has detrimental effects, and a question arises on whether noise engineering may be helpful in mitigating those effects on the performance of the quantum algorithm. Here we study whether time-correlated noise inducing non-Markovianity may represent a resource for quantum search. In particular, we consider quantum search on a star graph, which has been proven to be optimal in the noiseless case, and analyze the effects of independent random telegraph noise (RTN) disturbing each link of the graph. Upon exploiting an exact code for the noisy dynamics, we evaluate the quantum non-Markovianity of the evolution, and show that it cannot be considered as a resource for this algorithm, since its presence is correlated with lower probabilities of success of the search.
We address the dynamics of continuous-time quantum walks on one-dimensional disordered lattices inducing dynamical noise in the system. Noise is described as time-dependent fluctuations of the tunneling amplitudes between adjacent sites, and attention is focused on non-Gaussian telegraph noise, going beyond the usual assumption of fast Gaussian noise. We observe the emergence of two different dynamical behaviors for the walker, corresponding to two opposite noise regimes: slow noise (i.e., strong coupling with the environment) confines the walker into few lattice nodes, while fast noise (weak coupling) induces a transition between quantum and classical diffusion over the lattice. A phase transition between the two dynamical regimes may be observed by tuning the ratio between the autocorrelation time of the noise and the coupling between the walker and the external environment generating the noise. We also address the non-Markovianity of the quantum map by assessing its memory effects, as well as evaluating the information backflow to the system. Our results suggest that the non-Markovian character of the evolution is linked to the dynamical behavior in the slow noise regime, and that fast noise induces a Markovian dynamics for the walker.
Quantum walks, both discrete (coined) and continuous time, form the basis of several quantum algorithms and have been used to model processes such as transport in spin chains and quantum chemistry. The enhanced spreading and mixing properties of quantum walks compared with their classical counterparts have been well studied on regular structures and also shown to be sensitive to defects and imperfections in the lattice. As a simple example of a disordered system, we consider percolation lattices, in which edges or sites are randomly missing, interrupting the progress of the quantum walk. We use numerical simulation to study the properties of coined quantum walks on these percolation lattices in one and two dimensions. In one dimension (the line), we introduce a simple notion of quantum tunnelling and determine how this affects the properties of the quantum walk as it spreads. On two-dimensional percolation lattices, we show how the spreading rate varies from linear in the number of steps down to zero as the percolation probability decreases towards the critical point. This provides an example of fractional scaling in quantum-walk dynamics.
We address the properties of continuous-time quantum walks with Hamiltonians of the form $\mathcal{H}=L+\ensuremath{\lambda}{L}^{2}$, with LLL the Laplacian matrix of the underlying graph and the perturbation $\ensuremath{\lambda}{L}^{2}$ motivated by its potential use to introduce next-nearest-neighbor hopping. We consider cycle, complete, and star graphs as paradigmatic models with low and high connectivity and/or symmetry. First, we investigate the dynamics of an initially localized walker. Then we devote attention to estimating the perturbation parameter $\ensuremath{\lambda}$ using only a snapshot of the walker dynamics. Our analysis shows that a walker on a cycle graph spreads ballistically independently of the perturbation, whereas on complete and star graphs one observes perturbation-dependent revivals and strong localization phenomena. Concerning the estimation of the perturbation, we determine the walker preparations and the simple graphs that maximize the quantum Fisher information. We also assess the performance of position measurement, which turns out to be optimal, or nearly optimal, in several situations of interest. Besides fundamental interest, our study may find applications in designing enhanced algorithms on graphs.