Information Dynamics in Quantum Harmonic Systems: Insights from Toy Models

TL;DR

This paper develops a Gaussian-based framework to study information dynamics and computational resources in quantum harmonic systems, focusing on a two-body coupled oscillator model with an external magnetic field and a one-body ion transport setup. By deriving time-dependent Gaussian states via Ermakov equations and applying the Nielsen geometric approach, the authors quantify how coupling, detuning, and fields shape synchronization, mutual information, and circuit depth, revealing a divergence between informational and dynamical correlations in nonlinear regimes. The work introduces explicit analytical expressions for circuit depth, demonstrates a fidelity-complexity trade-off in quantum control, and highlights how smooth, adiabatic protocols minimize excitations and resource costs while preserving high fidelity. These results provide concrete guidelines for optimizing control strategies in quantum technologies such as trapped ions and superconducting qubits, and establish a platform for exploring more complex non-Gaussian dynamics and experimental validations.

Abstract

This study investigates the dynamics of quantum information and computational resources using a tractable model of coupled harmonic oscillators. We precisely characterize the interplay between mutual information, synchronization, and circuit complexity, demonstrating that they serve as complementary yet distinct measures of quantum correlations. Our analysis reveals how coupling strength, detuning, and external magnetic fields modulate these quantities, with synchronization and mutual information exhibiting marked divergence in nonlinear regimes. By employing exact Gaussian methods, we compute the circuit depth required to prepare target states and connect increased fidelity to more regular dynamical behavior. Furthermore, we analyze single-ion transport in a harmonic trap, comparing sudden and adiabatic protocols. We introduce a nonadiabaticity metric to quantify the fidelity-complexity trade-off, showing that smooth control sequences significantly minimize operational errors by suppressing excitations. These results provide a refined understanding of quantum correlations and offer concrete principles for optimizing control strategies in quantum technologies.

Figures (6)

• Figure 1: The plots show the variation of synchronization (top) and mutual information (bottom) as a function of the frequency detuning $\omega_2 - \omega_1$ for different values of the coupling strength $g$ and cutoff frequency $\omega_c$. In the top panel, for $\omega_c = 1$, synchronization is shown for $g = 0.5$ (blue) and $g = 1.5$ (red). In the bottom panel, for $g = 1$, mutual information is plotted for $\omega_c = 1$ (blue) and $\omega_c = 3$ (red).
• Figure 2: Time evolution of synchronization (top) and mutual information (bottom) for the quench model under various conditions. In the top panel, for $g = 1$ (blue) and $g = 1.5$ (red), synchronization is shown with frequency $\omega_c$ varying between 1 (blue) and 3 (red). In the bottom panel, mutual information is depicted for $g = 1$ (blue) and $g = 1.5$ (red), with frequency $\omega_c$ set to 1 (blue) and 3 (red). The plots also highlight the average values of both synchronization and mutual information over time.
• Figure 3: Left panel: Schematic diagram of the circuit depth as a function of the coupling strength $g$ at early times following the quench, where we fix $\omega_R = 1.0$, $\omega_1 = 2.0$, and $\omega_2 = 2.01$. Right panel: In the steady state approximation circuit depth versus coupling strength $g$ is shown for $\omega_1 = 1.0$, $\omega_2 = 1.2$, $\omega_c = 1.5$, and $\omega_R = 1.0$. The curve exhibits an initial growth for small $g$.
• Figure 4: Left panel: Circuit depth versus external field $\omega_c$ for fixed $g = 0.5$, showing the suppression of complexity at large $\omega_c$. The curve follows the predicted $\log(\omega_c/\omega_R)$ scaling when $\omega_c$ exceeds the other energy scales. Right panel: Circuit depth versus detuning $\Delta = \omega_1^2 - \omega_2^2$ on log-log scales, for $g = 0.5$, $\omega_c = 1.0$.
• Figure 5: Left: Schematic diagram of complexity and fidelity as a function of time for two corresponding amplitudes for two kinds of displacements. We have set $m=d_0=L=1$, $T=2$, $\beta=1$ and $\omega=2$. Right: Nonadiabaticity parameter $\mathcal{Q}(t)$ for two transport protocols: sudden (blue) and smooth sinusoidal (red). The sudden displacement leads to sustained excitation, while the smooth protocol suppresses transient energy buildup, indicating more adiabatic and efficient transport.