Quantum search with a continuous-time quantum walk in momentum space

Abstract

The atom-optics kicked rotor can be used to prepare specific momentum distributions on a discrete basis set. We implement a continuous-time quantum walk and a quantum search protocol in this momentum basis. In particular we propose ways to identify a specific marked state from the final momentum distribution after the walker's evolution. Our protocol is guided by current experimental possibilities making it accessible to experimentally implemented quantum walks with Bose-Einstein condensates.

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

“…The QFI of an initially localized state in the cycle graph is F q (t ) = 136t 2 (48) and it is independent of N. We numerically evaluate the FI (45) for the probability distribution in Eq. (18). The results are shown in Fig.…”

“…Numerical results suggest that revivals in the starting vertex are most likely not exact. Indeed, to be exact, the periods of the cosine functions entering the definition of the probability (18) have to be commensurable and such periods strongly depend on the choice of N and λ. Results are for N = 5 and λ = 0.2.…”

“…Whether or not the (de)localization is achievable is most likely related to the choice of N and λ. This choice in turn might result in the commensurability or incommensurability of the periods of the cosine functions entering the definition of the probability (18). For large N the IPR approaches 1/N, since the probability distribution approaches the uniform one.…”

“…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]. In fact, quantum walks have found several applications ranging from universal quantum computation [19] to quantum algorithms [20][21][22][23][24][25] and to the study of excitation transport on networks [26][27][28] and biological systems [29,30].…”

“…Map of the probability distribution(18) as a function of the position (vertex) and λ at t = 4. The walker is initially localized in the center of the cycle graph (N = 100).…”