Trojan Quantum Walks

Abstract

We investigate the transport properties and entanglement between spin and position of one-dimensional quantum walks starting from a qubit over position states following a delta-like (local state) and Gaussian (delocalized state) distributions. We find out that if the initial state is delocalized enough and a NOT gate reflects this state backwards, then the interference pattern extinguishes the position dispersion without preventing the propagation of the state. This effect allows the creation of a Trojan wave packet, a non-spreading and non-stationary double-peak quantum state.Keywords Spreading · Entanglement · Gaussian states · Quantum walks 1 IntroductionFrom the epic poem Iliad, the term "Trojan" was used to name a group of asteroids which share the Jupiter's orbit around the Sun. The center of mass of the Trojan asteroids is steady relative to Jupiter, once they are trapped on stable Lagrange regions (L 4 and L 5 ) of this celestial mechanical system [1]. In the quantum-mechanical context, Trojan wave packets have non-spreading and nonstationary behavior and they have been observed as a localized Rydberg electron over a circular orbit with dispersion suppressed by external fields [2,3].Quantum walks are the quantum analogue of the classical random walks [4]. The quantum walker is a spin-1/2 particle positioned over discrete positions in a one-dimensional lattice. The quantum walk state is composed by a tensor product between the internal (spin) and external (position) degrees of freedom. The time

Connecting velocity and entanglement in quantum walks

“…This generates a probability distribution with relative velocities ± √ 2 between the two peaks. On one hand, for q = 0 the reflection is maximal, and this means that the spin state which comes from left (right) is reflected to right (left) trapping the state [29,32]. On the other hand, for q = 1, the two spin states are split generating two peaks with relative velocities ±2.…”

“…Therefore, it is reasonable to state that these two sorts of walks represent borderline spreading and entanglement behaviors. We carry out extensive numerical calculations of average entanglement and variance over time of Hadamard and Fourier walks starting from a Gaussian state with σ 0 = 10, as performed in earlier works [10,11,18,29]. The average variance and entanglement over time depicted in Fig.…”

Quantum Corralling

“…Quantum walks are a promising framework to implement a variety of quantum tasks, such as quantum search algorithms [7,8] and universal quantum computation [9,10]. A rich dynamical behavior can be engineered in a quantum walk, ranging from diffusive to ballistic transport [11][12][13][14][15][16][17][18], and many physical systems can be used to experimentally build a quantum walk [19,20].…”

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

“…Here we consider an initial distributed state, i.e., a superposition of vertex (position) states weighted by a distribution function as employed in some experimental platforms [16,17] and theoretical works [18][19][20][21][22]. Once a walk starts, the state spreads over a cycle graph, and the periodic boundary allows it to superpose itself.…”