Growth and spreading of quantum resources under random circuit dynamics

TL;DR

The paper investigates how three quantum resources—nonstabilizerness, coherence, and fermionic non-Gaussianity—propagate and decay in one-dimensional random brick-wall circuits. Using two setups (resource-generating gates on a free initial state and free gates with a localized resource cluster) they uncover a universal rise–peak–fall in local resource content with peak times scaling as log L_A, followed by an exponential approach to maximally mixed states. In the second setup, resources spread ballistically with light-cone fronts, and specific features such as left/right moving modes for non-Gaussianity emerge, indicating a robust, dimension-independent transport phenomenology. Across qubits and qutrits, the results establish a unified baseline for local resource dynamics in ergodic quantum many-body systems and offer a framework to study information scrambling and thermalization through the lens of quantum resources.

Abstract

Quantum many-body dynamics generate nonclassical correlations naturally described by quantum resource theories. Quantum magic resources (or nonstabilizerness) capture deviation from classically simulable stabilizer states, while coherence and fermionic non-Gaussianity measure departure from the computational basis and from fermionic Gaussian states, respectively. We track these resources in a subsystem of a one-dimensional qubit chain evolved by random brickwall circuits. For resource-generating gates, evolution from low-resource states exhibits a universal rise-peak-fall behavior, with the peak time scaling logarithmically with subsystem size and the resource eventually decaying as the subsystem approaches a maximally mixed state. Circuits whose gates do not create the resource but entangle neighboring qubits, give rise to a ballistic spreading of quantum resource initially confined to a region of the initial state. Our results give a unified picture of spatiotemporal resource dynamics in local circuits and a baseline for more structured quantum many-body systems.

Figures (7)

• Figure 1: Our setup. (a) Local resource dynamics: We track the resource content of a subsystem $A$ under brick-wall circuit dynamics with resource-generating gates starting from a resource-free initial state, uncovering a universal rise–peak–fall profile. (b) Resource spreading: Starting from an initial state where the resource is confined to a small region and evolving with circuits whose gates do not generate the resource, we scan the spatiotemporal position of $A$ and expose ballistic spreading of the resource.
• Figure 2: Local dynamics of (a) LRoM $\mathcal{L}(\rho_A)$ ($L=24$), (b) relative entropy of coherence $C_d(\rho_A)$ ($L=128$), and (c) relative entropy of non-Gaussianity $\mathcal{NG}(\rho_A)$ ($L=128$) for subsystems of size $L_A$ under resource-generating circuits initialized in the free state $\ket{\Psi_1}$. Results are averaged over at least $10^4$ circuit realizations; the dilution parameter $\epsilon$ in each panel is chosen for clarity. Main panels: the rise-peak-fall structure of resource content of $\rho_A(t)$. Top insets: exponential relaxation of the monotones at large $t$. Bottom insets: peak times scale logarithmically with subsystem size, $\tau^{m} \sim \log L_A$.
• Figure 3: Spreading of (1) nonstabilizerness ($L=24$), (2) coherence ($L=256$), and (3) non-Gaussianity ($L=128$) from a localized resource cluster under circuits comprising resource-free gates. The results are averaged over $10^4$ circuit realizations. Panels (a) display the emergence of ballistic light cone structure of the spreading; panels (b) show the evolution of the resource content of $\rho_A$ at fixed $x_r$ ($-x_r=(5,4,3), (47.5,41.5,35.5)$ and $(19,17,15)$ for $R=\mathcal{L},c, N$, respectively.); panels (c) highlight the exponential attenuation of the peak local resource with distance along the light cone $x = v_R t + \mathrm{const}$ (with $R=\mathcal{L}, c, N$).
• Figure 4: Dynamics of mana $\mathcal{M}(\rho_A)$ in qutrit chains. Panel (1): evolution from a resource-free initial state under resourceful two-qutrit Haar gates for $L=16$ and various $L_A$, showing a universal rise-peak-fall profile (main), late-time exponential decay (top inset), and peak times scaling logarithmically with $L_A$ (bottom inset). Panel (2): $\mathcal{M}(\rho_A)$ for $L=14$ with an initial magical cluster of size $L_M=5$ and $L_A=3$ under resource-free gates, exhibiting a light-cone spreading pattern and late-time exponential decay to a free state. Results averaged over at least $10^4$ circuit realizations and overall dilution $\epsilon$ in each panel are chosen for clarity.
• Figure 5: Coherence spreading for a chain of $L = 64$ qubits from initially localized resource cluster, evolved under random coherence preserving gates and computed via sparse-vector simulations, with $L_c=0$ and $L_A = 8$. Panel (a) reveals a clear two-front structure in the dynamics, with a sharply defined inner light cone. Panel (b-c) illustrates the late-time relaxation of the local resource content back to a resource-free steady state and exponential attenuation of the peak local resource with distance from the initial cluster.