Quantum spatial search on graphs subject to dynamical noise

Abstract

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 N, also proving that noiseless spatial search shows optimal quantum speedup in the latter, in the computational limit N 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 speed-up 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 also indicate the star graph as the perfect candidate for the implementation by noisy quantum walks, owing to its simple topology and nearly optimal performance also for just few nodes.

Continuous-time quantum walks on dynamical percolation graphs

“…with (H s C N ) 0 = I, the probability of finding the target node is found to be p w (t) = | w|e −iHt |s | 2 = 1 N cos 2 t N . Recently, the same quadratic speedup has been proved also for the star graph [33], i.e. a graph where only a central node is connected to all the other (N − 1) nodes.…”

“…By properly manipulating the expression of the Hamiltonian and after using perturbation theory [33], one obtains that the initial state |s evolves into the state |w + O(N −1/2 ) after a time t opt = T . This indicates that for very large values of N the algorithm is optimal even for external target nodes.…”

Multiple-scale perturbation method on integro-differential equations: Application to continuous-time quantum walks on regular networks in non-Markovian reservoirs

“…Grover's algorithm has proven to be equivalent to a continuous-time quantum walk on a complete network [26]. In addition, a speed-up by continuous-time quantum walks on a star network has also been found [27], and this has encouraged further study of continuous-time quantum walks on regular networks. Here, a network is said to be regular if it has a repeating pattern in its network topology.…”

“…Unlike a classical random walker, the propagation of the quantum walker is coherent, i.e., besides the randomness inherited from quantum-mechanical probability amplitudes the coherence between different sites also governs the system dynamics [21]. The study of continuoustime quantum walks is mathematically based on network theory [22][23][24] and is closely related to other fields in quantum information theory, e.g., universal quantum computation [25], quantum algorithms [26][27][28], and perfect state transfer [29]. A continuous-time quantum walk can be experimentally implemented, usually by such quantum optical systems as waveguides [30] or Rydberg atoms [31].…”

Continuous-time quantum walks in the presence of a quadratic perturbation

“…In these systems, the graph Laplacian L (also referred to as the Kirchhoff matrix of the graph) plays the role of the free Hamiltonian, i.e., it corresponds to the kinetic energy of the particle. Perturbations to ideal CTQWs have been investigated earlier [4][5][6][7][8][9][10][11][12], however with the main focus being on the decoherence effects of stochastic noise rather than the quantum effects induced by a perturbing Hamiltonian. A notable exception exists, though, given by the quantum spatial search, where the perturbation induced by the so-called oracle Hamiltonian has been largely investigated as a tool to induce localization on a desired site [13][14][15][16][17][18].…”