Trotter transition in BCS pairing dynamics

TL;DR

The paper analyzes how Trotterization induces chaotic thermalization in a strongly interacting quantum many-body system by studying the reduced-BCS model in its mean-field limit. Using a split-step, symplectic $\mathrm{SABA}_2$ integrator, it identifies a finite-Trotter transition at $\tau_c \approx \sqrt{N}$ between a weakly chaotic, long-range-network regime and a memoryless, fully ergodic regime, with distinct scaling laws for the maximum Lyapunov exponent: $\Lambda_1 \propto \tau^\eta$ in the LRN regime and $\tau\Lambda_1 \approx 2\ln(\tau/\sqrt{N}) + C_N$ in the memoryless regime. The large-$\tau$ universal behavior is connected to the kicked-top map, enabling analytical insight into the transition and its scaling. These results have implications for quantum simulation on gate-based hardware, suggesting new observables (e.g., Loschmidt echo) to probe beyond mean-field theory and offering a universal framework for chaos and thermalization in digital quantum dynamics.

Abstract

We study universal aspects of thermalization induced by Trotterization, a procedure routinely used in gate-based quantum computation. We use the reduced-BCS model -- quantum integrable with a classically integrable mean-field limit -- where the effects of Trotter chaos are expected to be particularly stark. The resulting Trotterized chaotic dynamics is characterized by its Lyapunov spectrum and rescaled Kolmogorov-Sinai entropy. The chaos quantifiers depend on the Trotterization time step $τ$. We observe a Trotter transition at a finite step value $τ_c \approx \sqrt{N}$. While the dynamics is weakly chaotic for time steps $τ\ll τ_c$, the regime of large Trotterization steps is characterized by short temporal correlations. We derive two different scaling laws for the two different regimes by numerically fitting the maximum Lyapunov exponent data. The scaling law of the large \(τ\) limit agrees well with the one derived from the kicked top map. Beyond its relevance to current quantum computers, our work opens new directions -- such as probing observables like the Loschmidt echo, which lie beyond standard mean-field description -- across the Trotter transition we uncover.

Figures (13)

• Figure 1: We show $\log_{10} \Lambda_1$ as a function of $\log_{10} \tau$ for $N = 32$ and $64$. We have included the error bars. For a fixed $N$, we choose a configuration where all the spins point in random directions. The linear fits for $N = 32$ and $N=64$ to the first few points in the small $\tau$ regime are given by $y = 1.40x - 3.71$ and $y = 1.29x - 4.22$ respectively. On the other hand, the linear fit to the last few points in the large $\tau$ regime for both $N = 32$ and $N=64$ is given by $y = -0.85x + 0.39$. In the memoryless regime, the $N$ dependence of $\Lambda_1$ is indeed quite weak, see Fig. \ref{['Fig:Large_tau_MLE']}.
• Figure 2: Initiating from a fully random spin configuration where each spin points in a random direction, we show rescaled Lyapunov spectra for various step sizes $\tau$ for $N = 32$ in panels (a) and (c) and for $N = 64$ in panels (b) and (d). The small $\tau$ regime spectra in panels (a) and (b) $\mkern 1.5mu\overline{\mkern-1.5mu\Lambda\mkern-1.5mu}\mkern 1.5mu(\rho) \equiv \Lambda_i/\Lambda_1$ are obtained for end time $T_\mathrm{end} = 10^7$ and for step sizes $\tau = 0.359 \text{ (limegreen)}, 0.464, 0.599, 0.774$ and $1.0$ (dodgerblue). They show an approximate power law dependence on the normalized index $\rho \equiv i/N$ -- e.g., compare with the LRN spectra shown in Ref. Gabriel. The large $\tau$ regime spectra in panels (c) and (d) $\mkern 1.5mu\overline{\mkern-1.5mu\Lambda\mkern-1.5mu}\mkern 1.5mu(\rho)$ are obtained for step sizes $\tau = 10^{0} \text{ (limegreen)}, 10^1, \ldots$ and $10^7$ (dodgerblue) and for number of time steps $N_\mathrm{steps} = 10^7$ and $10^6$ respectively. They show a faster than an exponential decay as a function of the normalized index $\rho$. The transition from one regime to another takes place at $\tau_{c} \approx \sqrt{N}$. As seen in panels (c) and (d), the rescaled spectra remain similar to those of the small $\tau$ regime up to $\tau \approx 10$ but beyond this point their behavior changes abruptly.
• Figure 3: We show the rescaled Kolmogorov-Sinai entropy $\kappa$ as a function of $\log_{10} \tau$ for $N = 32$ and $64$. In the inset, we show a magnified $\kappa$ versus $\tau$ plot for the small $\tau$ LRN regime.
• Figure 4: We show the $\log$-$\log$ plot of $\Lambda_1$ versus $\tau$ for $N = 2, 3, 4, 5, 6, 7, 8, 16, 32$ and $64$ with $50 \leq \tau \leq 5 \times 10^4$. The numerical values of $\Lambda_1$ (dark blue circles) for $N = 2$ at different $\tau$ values coincide with the data (black cross) obtained from our semi-analytic method requiring only ensemble averaging and no time averaging for $N = 2$ with particle-hole symmetric initial condition. In the inset, we show the $\log$-$\log$ plot of $\Lambda_1$ versus $N$ for $\tau = 5\times 10^4$. Here lime-green circles represent values obtained from our numerical calculation (linear fit: $y = -0.076x - 3.45$), whereas black crosses indicate results from our semi-analytic method (linear fit: $y = -0.067x - 3.45$).
• Figure 5: We show $\tau \Lambda_1$ (obtained from numerics) as a function of $\ln (\tau/\sqrt{N})$ for $N = 16, \; 32,\; 64$ and $128$ with completely random initial spin configurations. The slope of the linear fit to the $N = 16$ data is $1.96 \pm 0.02$, whereas the slope for the $N = 32,\; 64$ and $128$ data is $1.99 \pm 0.01$.