Stochastic Qwak on a Maze#

Models of Stochastic Quantum Walks on different kinds of graphs have been used in the literature to model the remarkably efficient energy transport phenomena in light-harvesting complexes, in which the external noisy environment plays a fundamental role. Here, we will show how to run a Stochastic Quantum Walk on a particular kind of graph, the perfect maze, by using the QWAK package.

A perfect maze is defined to be a maze with one and only one path connecting the entrance with the exit. For example, one can observe, in the random maze in the following figure that there is only one path connecting the maze entrance (blue node) to the exit (red node). We can generate a random perfect maze, by looking at earlier works in the literature that worked with quantum walks on mazes and then just extract their adjacency matrix in order to define our Stochastic Quantum Walker on QWAK. To properly define the walker we need to specify some additional parameters with respect to the CTQW case. First, the noise parameter \(p \in [0,1]\), that specifies the amount of noise that our SQW has. If a sink node (or exit node) is present we also need to specify its index \(n\) and the sink rate \(\Gamma\) of the corresponding Linblad operator. We can thus simply instantiate our SQW object as follows:

1p = 0.1
2n = 99
3Gamma = 0.99
4
5graph = nx.from_numpy_array(maze_graph.adjacency)
6qwak = StochasticQWAK(graph, noiseParam=p,
7                      sinkNode=n, sinkRate=Gamma)

stochasticMazeWalker

Now that we have instantiated our StochasticQWAK object, we can make it evolve by the usual runWalk command. For example, we can see that if we run SQWs with different noise parameters \(p\) and plot the exit probability from the sink node as a function of time as reported in the previous figure, we obtain that the \(p=0.1\) walker has a much higher exit rate than both the purely quantum and purely classical walker (\(p=0\) and \(p=1\) respectively). This is a result analogous to well-known noise-assisted transport phenomena on networks that we were able to reproduce in just a few lines of code with the QWAK package.