Scalable networks for discrete quantum random walks

Abstract

Recently, quantum random walks ͑QRWs͒ have been thoroughly studied in order to develop new quantum algorithms. In this paper we propose scalable quantum networks for discrete QRWs on circles, lines, and also in higher dimensions. In our method the information about the position of the walker is stored in a quantum register and the network consists of only one-qubit rotation and ͑controlled͒ n -NOT gates, therefore it is purely computational and independent of the physical implementation. As an example, we describe the experimental realization in an ion-trap system.

Limit distributions of two-dimensional quantum walks

“…At the end of the present paper, we refer to the fact that recent papers propose implementations of not only one-dimensional but also two-dimensional quantum walks using optical equipments [32,33], ion-trap systems [34], and ultra-cold Rydberg atoms in optical lattices [35,36]. We hope that combinations of experiments and theoretical works of quantum physics will make significant contribution to development of quantum informatics.…”

Quantum walks, quantum gates, and quantum computers

“…A similar construction for the simulation of discrete-time quantum walks on a quantum computer was conducted by Fujiwara et al [17]. Results in this section can be viewed as analogous to this work, extended to the construction of quantum circuits for simulating continuous-time quantum walks.…”

“…Several recent papers have also proposed experimental implementations of quantum walks for quantum information processing [16,17], in various systems such as ion traps, optical lattices and optical cavities. Some of these proposals involve walks in real space, whereas others are purely computational walks (eg., a walk in the Hilbert space of a quantum register [17]).…”

“…Some of these proposals involve walks in real space, whereas others are purely computational walks (eg., a walk in the Hilbert space of a quantum register [17]). To our knowledge, two quantum walk experiments have been carried out: a quantum walk on the line, using photons [18], and a walk on a N = 4 length cycle, using a 3 qubit NMR quantum computer [19].…”

Scattering quantum random-walk search with errors

“…In general, it is possible to design scalable networks made of controlled NOT gates and one qubit rotations to realize QRWs, thereby utilizing any implementation of such gates [6]. The schemes proposed to directly implement QRWs include ion traps [7], nuclear magnetic resonance [8] which was also experimentally verified [9], cavity quantum electrodynamics [10,11], optical lattices [12], optical traps [13], optical cavity [14], and classical optics [15].…”