In this expository note, we study several families of periodic graphs which satisfy a sufficient condition for the ergodicity of the associated continuous-time quantum walk. For these graphs, we compute the limiting distribution of the walk explicitly. We uncover interesting behavior where in some families, the walk is ergodic in both horizontal and sectional directions, while in others, ergodicity only holds in the horizontal (large N) direction. We compare this to the limiting distribution of classical random walks on the same graphs.
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional ``coin space'' must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of a DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on d-regular, d-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a ``coined'' continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at a spatial search by a DTQW over the two-dimensional (2D) torus (a grid with periodic boundary conditions) of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$, where it was shown that a coined DTQW can search in time O(NlogN), but a standard CTQW takes $\mathrm{\ensuremath{\Omega}}(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2D torus exhibits the O(NlogN) scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in Krovi and Brun, Phys. Rev. A 75, 062332 (2007).
In this work, the application of continuous time quantum walks (CTQW) to the Maximum Clique (MC) problem was studied. Performing CTQW on graphs can generate distinct periodic probability amplitudes for different vertices. We found that the intensities of the probability amplitudes at some frequencies imply the clique structure of special kinds of graphs. Recursive algorithms with time complexity O(N^6) in classical computers were proposed to determine the maximum clique. We have experimented on random graphs where each edge exists with different probabilities. Although counter examples were not found for random graphs, whether these algorithms are universal is beyond the scope of this work.