Continuous-time quantum walks on dynamic graphs

Abstract

Continuous-time quantum walks (CTQWs) on static graphs provide efficient methods for search and sampling as well as a model for universal quantum computation. We consider an extension of CTQWs to the case of dynamic graphs, in which an ordered sequence of graphs governs free evolution of the quantum walk. We then consider how perfect state transfer during the quantum walk can be used to design dynamic graphs that implement a universal set of quantum logic gates. We give explicit examples for a complete logical basis, and we validate implementations using numerical simulations for quantum teleportation and addition circuits. Finally, we discuss the potential for realizing CTQWs on dynamic graphs using actively controlled quantum optical waveguides.

High fidelity state transfer via continuous-time quantum walks from distributed states

“…We have outlined an underlying idea based on the dynamic control of these graphs [23] suggesting potential applications as quantum memories [25] and sequential transportation of quantum cargo [26]. We hope our findings might be helpful to the development of new protocols of quantum communication.…”

“…Figure 6 illustrates how we can use these features to confine and transport a quantum state by means of a dynamic control of circular graphs [23]. First, we timeevolve the distributed state inside the small left circular graph (t < t 1 ).…”

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

“…We define E λ := ε max + λε 2 max , where ε max is the highest-energy eigenvalue of L [see Eqs. (10) and (12) for even and odd N, respectively].…”

“…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].…”

Isolated vertices in continuous-time quantum walks on dynamic graphs

“…In this paper, we focus on the quantum walk effected by the adjacency matrix, such as a single excitation in a spin network with XY interactions [6]. For convenience and in alignment with several prior works [8,9], we choose the jumping rate to be −1, so the Hamiltonian is equal to the adjacency matrix. This corresponds to walking backward in time, but the results can easily be adapted * thomaswong@creighton.edu to forward-time evolution.…”