Prethermalization, shadowing breakdown, and the absence of Trotterization transition in quantum circuits

TL;DR

This work develops a framework based on Ruelle-Pollicott resonances of the truncated propagator to quantify how long Trotterized quantum circuits faithfully reproduce Hamiltonian dynamics. It shows that the shadowing time is finite in chaotic many-body systems, with an almost-conserved energy $H_0$ exhibiting a prethermal plateau of duration $T=1/\Delta$ where $\Delta=1-|\lambda_1|$, while other observables decay and the plateau vanishes in the thermodynamic limit, implying no Trotterization transition. The diffusion constant $D$ is extracted from the $k$-dependence of the leading RP resonance via $1-\lambda_1(k)\approx D k^2$ and corroborated by nonequilibrium steady-state and Green-Kubo calculations. Demonstrations on 1D and 2D kicked Ising and 1D kicked XX models establish the method's generality and its capacity to connect prethermalization, discrete time crystals, and transport to exact TD-limit behavior.

Abstract

One of premier utilities of present day noisy quantum computers is simulation of many-body quantum systems. We study how long in time is such a discrete-time simulation representative of a continuous time Hamiltonian evolution, namely, a finite time-step introduces so-called Trotterization errors. We show that the truncated operator propagator (Ruelle-Pollicott resonances) is a powerful tool to that end, as well as to study prethermalization and discrete time crystals, including finding those phenomena at large gate duration. We show that the effective energy is more stable than suggested by Trotter errors -- a manifestation of prethermalization -- while all other observables are not. Even the most stable observable though deteriorates in the thermodynamic limit. Different than in classical systems with the strongest chaos, where the faithfulness time (the shadowing time) can be infinite, in quantum many-body chaotic systems it is finite. A corollary of our results is also that, opposite to previous claims, there is no Trotterization transition in non-integrable many-body quantum systems. We demonstrate our results on a one-dimensional (1d) kicked Ising model, as well as on 1d kicked XX model and 2d kicked Ising model. The truncated propagator is also used to calculate the energy diffusion constant in the tilted-field Ising model with high accuracy.

Figures (18)

• Figure 1: Shadowing in classical chaotic systems. A true shadow trajectory (blue) from a slightly perturbed initial condition closely follows (shadows) the noisy one (red), that otherwise exponentially separates from the original exact trajectory (dotted blue).
• Figure 2: Truncated propagator and Ruelle-Pollicott resonances. ( A) Momentum-resolved operator propagator $M(k)$ is truncated to infinite system size operators with their density supported on at most $r$ sites (sine-shaped curves suggest quasimomentum modulation of local densities, e.g., of $a^{(\boldsymbol{{\alpha}})}_j=\sigma^{\rm z}_{j}$ having $r=1$). ( B) Spectrum of non-unitary $M(k=0)$ lies inside a dashed unit circle. Shown are spectra at truncation $r=8$ and different gate duration $\tau$ in the kicked Ising model (\ref{['eq:KIU']}). Spectrum can be used to identify prethermalization (e.g. $\tau=0.8$) as well as a discrete time crystal ($\tau=3.0$). The largest eigenvalue (e.g., zoom-in inset with red arrow in the top middle frame) is shown in (C). ( C) Dependence of the spectral gap $\Delta$ on $\tau$. Circles and crosses mark $\tau$ for which we show convergence with $r$, and correlation functions, respectively. ( D) Autocorrelation function of energy $H_0$ (\ref{['eq:H0']}) and magnetization $Z=\sum_j \sigma^{\rm z}_j$ (right). Dotted curve is exponential decay with the rate $\Delta$ from (C), showing that the inverse gap $T=1/\Delta$ gives the duration of the prethermalization plateau. In the thermodynamic limit $L \to \infty$ the plateau at large time disappears for any $\tau$ -- there is no Trotterization transition with $\tau$.
• Figure 3: Kicked Ising quantum circuit. Sequence of gates for one Floquet step of the kicked Ising model (\ref{['eq:KIU']}).
• Figure 4: Prethermalization vs. Heisenberg time. Finite-size asymptotic value of magnetization correlation function (e.g., from Fig. \ref{['fig2']}C we can read $C_Z(\infty) \approx 0.1$ for $L=24$ and $\tau=1.0$, reached at $t>10^5$) is shown for various sizes $L=14-28$ and $\tau$. To observe the correct thermodynamic limit $\lim_{t \to \infty} \lim_{L \to \infty}$, in which $C_Z(\infty)$ decays as $\propto 1/2^L$ (green line), one must have $T \ll t_{\rm H}$ -- the prethermalization time $T=1/\Delta$ has to be smaller than the Heisenberg time $t_{\rm H} \sim 2^L$.
• Figure 5: Single almost-conserved operator. ( A) Largest eigenvalues of $M(k)$ for different quasimomenta $k$: even and odd sector in $k=0$, and $k=\pi$. Only one gap is very small (dashed red, also Fig. \ref{['fig2']}C), all other scale as $\sim \tau^2$ for small $\tau$. ( B) Correlation function of staggered magnetization $S=\sum_j (-1)^j \sigma^{\rm z}_j$ deteriorates fast already for small $\tau$ ($L=32$, finite-size effects are negligible on shown times; black curve is Hamiltonian evolution by $H_0$). Two insets zoom-in on specific time windows.