Thermodynamic formalism for dissipative quantum walks

We consider the dynamical properties of dissipative continuous-time quantum walks on directed graphs. Using a large-deviation approach we construct a thermodynamic formalism allowing us to define a dynamical order parameter, and to identify transitions between dynamical regimes. For a particular class of dissipative quantum walks we propose a quantum generalization of the the classical PageRank vector, used to rank the importance of nodes in a directed graph. We also provide an example where one can characterize the dynamical transition from an effective classical random walk to a dissipative quantum walk as a thermodynamic crossover between distinct dynamical regimes.

Figure 1

The directed graph that we consider as an example for the application of the thermodynamic formalism to QSWs.

Figure 2

The dynamical free energy $\theta \left(s\right)$ for the QSW associated with the graph in Fig. 1.

Figure 3

The normalized dynamical activity of the six nodes of the graph in Fig. 1. For this particular example the ranking provided by the activity order parameter does not change as a function of $s$ (see the text for more details).

Figure 4

A comparison between the classical pagerank (left bars), the dynamical activity at $s=0$ (middle bars), and the populations in the steady state (right bars). These values are obtained with respect to the graph shown in Fig. 1. Note that the extreme left values in the plot correspond to the classical pagerank.

Figure 5

The stability parameter quantifying fluctuations in the trajectories for different values of $s$. Each line corresponds to a node in the graph of Fig. 1. The inset shows the sum with respect to all nodes of the same quantities, and it shows a maximum close to $s=-1$.

Figure 6

The activity order parameter for the fourth node in the graph of Fig. 1 (triangles), and the total fluctuations of the dynamical trajectories on the same graph (circles). The presence of a maximum in the fluctuations corresponding to the region where the dynamical order parameter is changing is an indication of a crossover phenomenon.

Figure 7

A directed two-node graph.

Figure 8

Simulation results for the directed two-node graph in Fig. 7. See the main text for comments.