Operator delocalization in quantum networks

We investigate the delocalization of operators in nonchaotic quantum systems whose interactions are encoded in an underlying graph or network. In particular, we study how fast operators of different sizes delocalize as the network connectivity is varied. We argue that these delocalization properties are well captured by Krylov complexity and show, numerically, that efficient delocalization of large operators can only happen within sufficiently connected network topologies. Finally, we demonstrate how this can be used to furnish a deeper understanding of the quantum charging advantage of a class of Sachdev-Ye-Kitaev (SYK)-like quantum batteries.

Figure 1

The Watts-Strogatz algorithm can be visualized best in terms of network diagrams and their associated adjacency matrices, here represented by an $N\times N$ array coded black where a connection exists and white otherwise. In the above, for example, we take $N=11$. The left set of networks all have $p=0$ with $k=2$, 4, and 10, reading from top to bottom. The regularity of these graphs are clear in the adjacency matrices.The set of graphs on the right implement the Watts-Strogatz algorithm on the $k=2,N=11$ lattice with $p$ increasing from top to bottom. The increasing randomness seen in the graph is reflected in the increasingly crossword-puzzle-like resemblance of the corresponding adjacency matrices.

Figure 2

$C_K\left(t\right)$ for systems having $L=24$. $C_K\left(t\right)$ is computed for small (size 1) and large (size $L/2$) operators, for the full ${\mathrm{SYK}}_2$ model, compared against the Watts-Strogatz Hamiltonians having $k=1,2$ and both low and large rewiring probability ($p=0.1$ and $p=0.9$, respectively). The results are averaged over 1000 realizations of disorder and graph.

Figure 3

The quantity $R\left(L\right)$, computed at different system sizes and for the full ${\mathrm{SYK}}_2$ model, compared against the Watts-Strogatz Hamiltonians having $k=1,2$ and both low and large rewiring probability ($p=0.1$ and $p=0.9$, respectively). All the results are averaged over 1000 different realizations of disorder and underlying graph.

Figure 4

The maximum charging power, $P_{\max }\left(L\right)$ for the same models considered in Fig. 3.