Optimizing the discrete time quantum walk using a SU(2) coin

Abstract

We present a generalized version of the discrete time quantum walk, using the SU(2) operation as the quantum coin. By varying the coin parameters, the quantum walk can be optimized for maximum variance subject to the functional form ${\ensuremath{\sigma}}^{2}\ensuremath{\approx}{N}^{2}$ and the probability distribution in the position space can be biased. We also discuss the variation in measurement entropy with the variation of the parameters in the SU(2) coin. Exploiting this we show how the quantum walk can be optimized for improving the mixing time in an $n$-cycle and for a quantum walk search.