Tracing Quantum Walks in Energy Space

Ting-Hsu Chang, Tein-Sheng Lin, Chia-Hung Chien +2 more

Quantum walks can be implemented in the hypercube and general graphs, where hypercube is a regular graph. According to quantum walk algorithm, unitary property of quantum walks can be preserved. In the regular graph, to develop quantum walks algorithm is focused on quantum search algorithm. Quantum switching is the reversible and parallel computation circuits, where parallel computation circuit can achieve the better performance in time complexity and space complexity. Especially, this switching can trace the behavior of quantum walks from input sequence to output sequence, looking as quantum search algorithm. Furthermore, application of unicasting and multicasting can be implemented in this circuit.

Michael S. Underwood, David L. Feder

A. Romanelli, A. Auyuanet, R. Siri +2 more

Luis Alberto Bru, Germán J. de Valcárcel, Giuseppe Di Molfetta +3 more

We consider the two-dimensional alternate quantum walk on a cylinder. We concentrate on the study of the motion along the open dimension, in the spirit of looking at the closed coordinate as a small or ``hidden'' extra dimension. If one starts from localized initial conditions on the lattice, the dynamics of the quantum walk that is obtained after tracing out the small dimension shows the contribution of several components which can be understood from the study of the dispersion relations for this problem. In fact, these components originate from the contribution of the possible values of the quasimomentum in the closed dimension. In the continuous space-time limit, the different components manifest as a set of Dirac equations, with each quasimomentum providing the value of the corresponding mass. We briefly discuss the possible link of these ideas to the simulation of high-energy physical theories that include extra dimensions. Finally, entanglement between the coin and spatial degrees of freedom is studied, showing that the entanglement entropy clearly overcomes the value reached with only one spatial dimension.

Chris Jones

We examine a discrete-time quantum walk with two-step memory for a particle on a onedimensional infinite space. The walk is defined with a four-state memory space analogous to the two-state coin space commonly used in discrete time quantum walks, and a method is presented for calculating the time evolution by using the Fourier transform. An integral expression for the probability is calculated, and this is used to produce numerical solutions for the probability distribution as a function of the time step and position. The results show two peaks in the probability distribution. One peak propagates ballistically with time, which is a common feature of quantum walks. The other peak is stationary with time and located at the initial site of the particle. This feature is not common in quantum walks and suggests that tracing the immediate history of the particle using two-step memory may represent the beginning of a transition to a classical system.

Sanjeev Naguleswaran, Ian Fuss, Lang White