Uniform Measure for Continuous-Time Quantum Walk on Hypercube

In this paper, we introduce a generalized continuous-time quantum walk (CTQW) on a hypercube, which can be interpreted as a CTQW with a “coin”. We examine its properties from a perspective of probability distribution and spectral theory, and obtain formulae for calculating its probability distributions and spectra, respectively. In particular, we prove that, for a wide range of initial states, the walk produces the uniform measure on the hypercube as its a stationary measure. Finally, we show that the time-reversal symmetry of the walk can be described directly by a unitary operator. Some other results are also proven.