Implementations of Quantum Random Walks

Abstract

In this research, the implementations of quantum random walks in superconducting circuit-QED are studied. In particular, a walk that moves across the Fock states of a quantum harmonic oscillator by a Jaynes-Cummings model is investigated, which is difficult to implement because of different Rabi frequencies for different Fock states. Theoretically, the lower boundary vacuum state of the harmonic oscillator causes a reflection of the probability amplitudes in the distribution. A walk that moves across a grid of coherent states |nα+imα〉 in phase space is then investigated. A setup for a 1D and 2D quantum random walk is suggested, using controlled displacements of a resonator dispersively coupled to one or two superconducting transmon qubits in circuit-QED, followed by Hadamard gates. From numerical simulations it was observed that the 2D walk commutes for α·β = 0 mod π/2 for which the variance is proportional to the number of steps t squared. For other values of α·β the horizontal and vertical displacements do not commute, resulting in extra phase factors. The numerical simulations showed that for most values of α· β with a larger distance to 0 mod π/2 than 0.01, the probability distribution of the
walk collapses to a distribution centered around origin within t = 100 steps, similar to a classical random walk. Exceptions are the 2D walks for α·β = ±π/6 mod π/2 or ±
π/4 mod π/2, for which the variance is still proportional to the number of steps squared.