Anticoncentration and nonstabilizerness spreading under ergodic quantum dynamics

TL;DR

The paper investigates how anticoncentration and nonstabilizerness (magic) spread under ergodic quantum dynamics, testing whether random-circuit predictions extend to Floquet and Hamiltonian 1D systems. It finds a sharp dichotomy: Floquet dynamics saturate PE and SE at Haar-like values on times $t_{\mathrm{sat}} \propto \log N$, consistent with random-circuit expectations, while energy-conserving Hamiltonian dynamics (MFIM) exhibit slower, power-law relaxation with $t_{\mathrm{sat}} \propto N$ and saturate below Haar due to conservation laws and atypical eigenstates. The study highlights the central role of conserved quantities in constraining Hilbert-space delocalization and magic spreading, and it discusses implications for simulation and computational tasks, as well as extensions to other symmetries and disorder. These results refine our understanding of how ergodic dynamics approach typicality and guide future exploration of symmetry-protected dynamics in many-body quantum systems.

Abstract

Quantum state complexity metrics, such as anticoncentration and nonstabilizerness, or ``magic'', offer key insights into many-body physics, information scrambling, and quantum computing. Anticoncentration and equilibration of magic resources under dynamics of random quantum circuits occur at times scaling logarithmically with system size, a prediction that is believed to extend to more general ergodic dynamics. This work challenges this idea by examining the anticoncentration and magic spreading in one-dimensional ergodic Floquet models and Hamiltonian systems. Using participation and stabilizer entropies to probe these resources, we reveal significant differences between the two settings. Floquet systems align with random circuit predictions, exhibiting anticoncentration and magic saturation at time scales logarithmic in system size. In contrast, Hamiltonian dynamics deviate from the random circuit predictions and require times scaling approximately linearly with system size to achieve saturation of participation and stabilizer entropies, which remain smaller than that of the typical quantum states even in the long-time limit. Our findings establish the phenomenology of participation and stabilizer entropy growth in ergodic many-body systems and emphasize the role of conservation laws in constraining anticoncentration and magic dynamics.

Figures (12)

• Figure 1: Anticoncentration and magic resources under the dynamics of a many-body system with time-evolution operator $U(\Delta t)$. The distribution of probabilities $p_{\pmb{x}}$ changes from being concentrated at small evolution time $t$, (a), to being anticoncentrated at large $t$, (b), and following the Porter-Thomas distribution Porter56mullane2020sampling (blue line). (c) Entanglement, PE, $\mathcal{S}_2$, and magic resources (quantified by SE, $\mathcal{M}_2$) increase and reach, up to a fixed accuracy $\epsilon$ their saturation values at times $t^{(\mathrm{ent})}_{\mathrm{sat}}$, $t^{(\mathcal{S}_2)}_{\mathrm{sat}}$, $t^{(\mathcal{M}_2)}_{\mathrm{sat}}$.
• Figure 2: Anticoncentration and magic spreading in KIM for $N$ qubits. (a) PE, $\mathcal{S}^{(2)}$, saturates to $S^{\mathrm{Haar}}_2$ (dashed line). (b) An exponential decay of $\Delta \mathcal{S}_2 = A_s e^{-\alpha_S t}$ with time $t$, where $\alpha_S=0.28(2)$, and $A_s \propto N$, see the inset. (c) the SE, $\mathcal{M}_2$, abruptly saturates to $\mathcal{M}^{\mathrm{Haar}}_2$ (dashed line). (d) The difference $\Delta \mathcal{M}_2$ is well fitted by an exponential decay, $\Delta \mathcal{M}_2 = A_m e^{-\alpha_M t}$, with $\alpha_M=0.59(3)$, and $A_m \propto N$, see the inset. (e) $\Delta \mathcal{S}_2$ and $\Delta \mathcal{M}_2$ decay to a given $\epsilon \lesssim O(1)$ at times $t_{\mathrm{sat}} \propto \log_2(N)$ scaling logarithmic with system size $N$.The results are averaged over more than $200$ initial states $\ket{\Psi_0}$\ref{['eq:initial_state']}.
• Figure 3: Anticoncentration and magic spreading in MFIM for $N$ qubits. (a) The PE, $\mathcal{S}^{(2)}$, increases towards the long-time saturation value $\mathcal{S}^{\infty}_2$ whose difference with $\mathcal{S}_2^{\mathrm{Haar}}$ remains non-zero, see the inset in (b). The difference $\Delta \mathcal{S}_2$, see panel (b), follows a power-law decay, $\Delta \mathcal{S}_2 \propto t^{- \beta_S}$ with $\beta_S \approx 1$. (c) The SE, $\mathcal{M}_2$, saturates to $\mathcal{M}^{\infty}_2$ smaller than the Haar value $\mathcal{M}^{\mathrm{Haar}}_2$, as shown in the inset in (d). The difference $\Delta \mathcal{M}_2$, shown in (d), is well fitted by a power-law decay $\Delta \mathcal{M}_2 \propto t^{-\beta_M}$ with $\beta_M \approx -1.5$. (e) $\Delta \mathcal{S}_2$ and $\Delta \mathcal{M}_2(t)$ decay to a given $\epsilon$ at times $t_{\mathrm{sat}} \propto N$ scaling linearly with system size $N$.
• Figure 4: Anticoncentration and magic dynamics in random unitary circuits with $U(1)$ symmetry. (a), (c) Algebraic decay of the differences $\Delta \mathcal{S}_2 \propto t^{-\beta_S}$ and $\Delta \mathcal{M}_2 \propto t^{-\beta_M}$, with $\beta_S \approx 1.25$ and $\beta_M \approx 3$. (b), (d) The saturation time scales extensively, $t_{\mathrm{sat}} \propto N$, for $U(1)$-symmetric circuits—in stark contrast to the $t_{\mathrm{sat}} \propto \ln(N)$ behavior observed in Haar-random brick-wall circuits without symmetry (red lines).
• Figure 5: Anticoncentration and magic dynamics in Floquet model \ref{['eq:UFdef']}. (a), (c) The exponential decay of the differences $\Delta \mathcal{S}_2 \propto e^{-\beta_S t}$ and $\Delta \mathcal{M}_2 \propto e^{-\beta_M t}$ for system size $N=20$ and $0<\theta<1/2$. The decay slows down as $\theta$ approaches $1/2$. (b), (d) The saturation times follow $t_{\mathrm{sat}} \propto \ln(N)$ for any $0<\theta<1/2$ ($\epsilon=0.2$ for PE and $\epsilon=0.1$ for SE) and scale extensively only for $\theta=1/2$, i.e., when the energy is conserved.