Memory preservation and cooperative shielding in complex quantum networks

TL;DR

The paper investigates transport in complex quantum networks governed by the XXZ Hamiltonian, revealing a memory-preserving mechanism driven by cooperative shielding in highly connected systems. By exploiting symmetry, it derives exact energy-band structures for all-to-all networks and shows that the number of observable frequencies in local magnetization spectra equals the number of initial excitations, with degeneracy lifting governed by angular-momentum selection rules. Extending to small-world and power-law networks, it demonstrates that cooperative shielding can slow excitation spreading and preserve memory, with the effect depending on topology and initial conditions, as quantified by the Inverse Participation Ratio and spectral analysis. These results illuminate universal features of memory and localization in complex quantum networks and point to potential applications in neuroscience-inspired models and biomimetic quantum technologies. They also suggest practical guidelines for engineering transport properties in quantum simulators by tuning connectivity, range, and initial-state preparation.

Abstract

Complex quantum networks are powerful tools in the modeling of transport phenomena, particularly for biological systems, and enable the study of emergent phenomena in many-body quantum systems. High connectivity and long-range interactions induce strong constraints on the system dynamics. Here, we study the transport properties of a quantum network described by the paradigmatic XXZ Hamiltonian, with non-trivial graph connectivity and topology, and long-range interactions. We show how long-range interactions induce memory preserving effects and strongly affect the spreading of the excitations due to cooperative shielding. We describe the memory-preserving effect in all-to-all connected regular networks with distance-independent couplings. Indeed, the memory of the number of initially injected excitations is preserved over long times, encoded in the number of frequencies present in the dynamics. Interestingly, we find that memory-preserving effects occur also in less regular graphs, such as quantum networks with either power-law node connectivity or complex, small-world type, architectures. We discuss the implications of these properties in biology-related problems, such as an application to Weber's law in neuroscience, and their implementation in specific quantum technologies via biomimicry. We also show how the presence of long-range interaction strongly affects the dynamics of the excitations in small-world networks and power law all-to-all coupled networks. Indeed, because of cooperative shielding blue, as the connectivity or the range of interaction increases, the initial excitation spreads more slowly among the network and becomes strongly dependent on the initial conditions.

Figures (10)

• Figure 1: The role of symmetry on the quantum network dynamics. $(a)$ Quantum spin-1/2 XXZ network. Excitations along the $z$-axis are depicted by red circles (spin up) in a background of non-excited blue circles (spin downs). The system presents two-body interactions by the spin-exchange coupling $J_{ij}$ and the anisotropy $\Delta_{ij}$. $(b)-(e)$ Temporal evolution of the local $z$-magnetization for the all-to-all connected XXZ spin network, revealing how the network preserves time and space memory of the excitations introduced over time. $(b)$ refers to a network with $L=7$ sites and one excitation quench positioned at site 4 along $x$, i.e. with initial state $|\phi(0)\rangle=(|\downarrow\downarrow\downarrow\uparrow\downarrow\downarrow\downarrow\rangle+|\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\downarrow\rangle)/\sqrt{2}$; the arrows represent the spin direction along the $z$-axis. Note that the signal at site 4 starts from a value of 0, instead of +0.5, because $\langle\phi(0)|\hat{\mathcal{S}}_4^z|\phi(0)\rangle$=0. $(d)$ refers to a network with $L=9$ sites and four excitations quenches along $z$ positioned at $i=2, 3, 6, 8$, thus the initial state is $|\downarrow\uparrow\uparrow\downarrow\downarrow\uparrow\downarrow\uparrow\downarrow\rangle$. Example parameters chosen are $\Delta_{ij}$=0 and $J_{ij}/2=1$, d$t$=0.06 (units of $J/2$) and Periodic Boundary Conditions (PBC). Plots $(c)$ and $(e)$ depict the normalized power spectra of $\langle \hat{\mathcal{S}}_i^z\rangle$ in site $i=4$ for cases $(b)$ and $(d)$ respectively, revealing how the number of peaks precisely reflects the number of excitations in the system. The vertical dashed red line corresponds to the possible frequency of the dynamics as computed in Section \ref{['sec:xx-ham']}. The data has been detrended before the Fourier transform by removing the zero-frequency component.
• Figure 2: Symmetries and structure of the energy levels for the all-to-all connected XX Hamiltonian. $(a)$ Diagram for a system with $L=7$ sites and constant exchange coupling $J/2=1$ (unperturbed study). The vertical axis represents the exact values of the eigenvalues of $\hat{\mathcal{H}}_0$, while the numbers on the left of the horizontal lines account for the degeneracy of the corresponding level; different colors and symbols represent different subspaces corresponding to different number of excitations along the $z$-axis. Red triangles and lines: levels corresponding to 1 excitation. Blue lozenges and lines: levels referring to 2 excitations. Double-headed arrows : the three possible energy gaps described by Eq.\ref{['2exc_freq']}: the dotted arrow represents the frequency that is not manifested in the $z$-magnetization spectrum. Green bars: remaining energy levels. $(b)$ The same diagram for the case $L=9$, with the red bars contoured by stars representing the levels with 4 excitations denoted by the quantum number $l$ on the right.
• Figure 3: Multiple excitations with all-to-all connected XX network: frequency analysis. $(a)$ Spectrum's peaks for the local $z$-magnetization signal in the case of two excitations: the corresponding positions are 5 (higher prominence) and 7 (lower prominence). The system parameters are the same as in Fig. \ref{['fig:state-art']}, with the signal being picked up from site 4. $(b)-(d)$ Bar plot representing the distribution of the coefficients $|d_{mn}|^2$ in Eq.\ref{['d_coeff']}, showing that the number of frequencies perfectly matches with the simulated spectra in Fig. \ref{['fig:state-art']}. As in the case of the spectra, the zero-frequency is not shown to highlight the oscillating components of the signals. The coefficients are summed over the levels' degeneracies and normalized to the total sum of the coefficients. $(b)$$L=7$ and two excitations. $(c)$$L=7$ and three excitations. $(d)$$L=9$ and four excitations.
• Figure 4: Construction of small-world graphs and their memorization properties. $(a)$ Small-world network construction (see text). Top to bottom: a regular graph with $L=7$ sites and $k=2$ nearest-neighbor connections, and different rewired links (green and orange) with respect to the original graph, according to the value of the probability $\beta$. $(b)-(e)$ Examples of intensity spectra for average $z$-magnetization at the initial site for one quenched excitation along the $z$-axis in four different networks with $L=33$ sites. $(b)$ All-to-all network, with delta-like peaks at frequencies $\omega^*=\pm JL/2$=33, see Eq.\ref{['freq_spliten']}. $(c)$ Small-world, highly-connected graph with $(k,\beta)$=(15,0.5), possessing additional low-frequency components. $(d)$ Small-world, highly-connected graph with $(k,\beta)$=(13,0.5), having small high-frequency contributions. $(e)$ Sparsely connected small-world with $(k,\beta)$=(1,0.2), having no high-frequency peaks. The data has been detrended before the Fourier transform by removing the zero-frequency component.
• Figure 5: Understanding transport in small-world graphs at early times. $(a)$ Concept of the cascade model (see text). Top: an initial excitation (red circle) in a generic network (black edges) is located at site 1, as denoted by the state $|s=1\rangle=|\uparrow\downarrow\dots\downarrow\rangle$ and the red circle opposed to the blue ones. Bottom: after a short temporal evolution (black arrow) the initial excitation has spread (gradient-colored circles) across site 1's first neighbors (orange edges), such that the state is a superposition of $|s=1\rangle$ and states $|f_2\rangle=|\downarrow\uparrow\downarrow\dots\downarrow\rangle$, $|f_4\rangle$, $|f_5\rangle$ and $|f_7\rangle$. $(b)-(c)$ Comparison of the IPR's time evolution as computed from the cascade model (orange curves) and from a numerical simulation (blue curves) for the following small-world networks: $(b)$$k=1$, initial site with degree 2; $(c)$$k=3$, initial site with degree 4.