Operator Delocalization in Quantum Networks

Abstract

We investigate the delocalization of operators in non-chaotic 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 SYK-like quantum batteries.

Figures (6)

• Figure 1: $C_K(t)$ for systems having $L = 24$. $C_K(t)$ is computed for small (size $1$) and large (size $L/2$) operators, for the full 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 2: The quantity $R(L)$, computed at different system sizes and for the full 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 3: The maximum charging power, $P_{\mathrm{max}}(L)$ for the same models considered in figure \ref{['fig:complexity_ratios']}.
• Figure S1: The maximum charging power, $P_{\mathrm{max}}(L)$ for the $x$-battery and for the $z$-battery, both based on a quench Hamiltonian given by the full SYK$_2$ model.
• Figure S2: The comparison between the average power, $P_{\mathrm{av}}(t)$ as computed via the perturbative formula, equation \ref{['eq:P_inst_SYK2_pert']} (solid lines) and via exact diagonalization (dashed lines).