Topology identification of autonomous quantum dynamical networks

Topology identification comprises reconstructing the Hamiltonian, and thus the corresponding interaction terms, by properly processing measurements of its density operator within a fixed time interval. It finds application in several quantum technology contexts, ranging from quantum communication to quantum computing or sensing. In this paper we provide analytical conditions for the solvability of the topology identification problem for autonomous quantum dynamical networks (i.e., as in our case, not explicitly depending on time via the use of an external drive). The solvability condition is then converted in an algorithm for quantum network reconstruction that is easily implementable on standard computer facilities. The obtained algorithm is tested for Hamiltonian reconstruction on numerical examples based on the quantum walks formalism.

Figure 1

Plot of the mean solvability rate $\overline{\mathfrak{s}}$ as a function of $d\in [2,30]$ (number of nodes) and of the pairs $(\tau ,{\tilde{n}}_s)$, with $\tau \in \{1,2,3\}$ and ${\tilde{n}}_s\in \{n_s/20,n_s/10,n_s/5\}$. Specifically, for the considered values of $d$ and $p_{\mathrm{link}}=0.5$, the topology identification problem is solved for 100 random Erdős-Rényi quantum networks. The black, orange, and blue curves refer, respectively, to $\tau =1,2,3$.

Figure 2

Plot of the relative reconstruction error $\epsilon$ in identifying single random Erdős-Rényi quantum networks with $p_{\mathrm{link}}=0.5$. The relative error $\epsilon$ is plotted as a function of $d\in [2,12]$ for different pairs $(\tau ,{\tilde{n}}_s)$, with $\tau \in \{1,2\}$ and ${\tilde{n}}_s\in \{n_s/20,n_s/10,n_s/5,n_s\}$. In the figure, the red and blue curves respectively refer to $\tau =2,3$.