Quantum Chaos and Universal Trotterisation Behaviours in Digital Quantum Simulations

TL;DR

This work analyzes digital quantum simulation (DQS) using Trotterisation across three paradigmatic models (Ising, Heisenberg, and Dicke) to uncover universal, pre-threshold, and post-threshold behaviours. By combining full-system dynamics, operational error metrics, and a novel X^2_RMT statistic for eigenvector distributions, the authors demonstrate a universal Trotterisation threshold signaling a transition from regular (quasiperiodic) to quantum-chaotic dynamics driven by digitisation errors. They reveal stable regions—digital aliasing windows where chaos signatures partially recede—and show that, beyond the threshold, eigenvector statistics align with Random Matrix Theory (RMT) predictions across models and system sizes. The findings offer a unified framework for understanding Trotter errors, offer perturbative extrapolation and error-correction opportunities, and provide rigorous tools (X^2_RMT) to diagnose chaos with potential experimental applicability. Overall, the work advances a universal picture of Trotterisation in DQS that informs resource optimization and the design of near-term quantum simulations.

Abstract

Digital quantum simulation (DQS) is one of the most promising paths for achieving first useful real-world applications for quantum processors. Yet even assuming rapid progress in device engineering and development of fault-tolerant quantum processors, algorithmic resource optimisation will long remain crucial to exploit their full power. Currently, Trotterisation provides state-of-the-art resource scaling. And recent theoretical studies of Trotterised Ising models suggest that even better performance than expected may be possible up to a distinct breakdown threshold in empirical performance. Here, we study multiple paradigmatic DQS models with experimentally realisable Trotterisations, and evidence the universality of a range of Trotterisation performance behaviours, including not only the threshold, but also new features in the pre-threshold regime that is most important for practical applications. In each model, we observe a distinct Trotterisation threshold shared across widely varying performance signatures; we further show that an onset of quantum chaotic dynamics causes the performance breakdown and is directly induced by digitisation errors. In the important pre-threshold regime, we are able to identify new distinct regimes displaying qualitatively different quasiperiodic performance behaviours, and show analytic behaviour for properly defined operational Trotter errors. Our results rely crucially on diverse new analytical tools, and provide a previously missing unified picture of Trotterisation behaviour across local observables, the global quantum state, and the full Trotterised unitary. This work provides new insights and tools for addressing important questions about the algorithm performance and underlying theoretical principles of sufficiently complex Trotterisation-based DQS, that will help in extracting maximum simulation power from future quantum processors.

Figures (22)

• Figure 1: Trotterisation performance for the A2A-Ising model (with $j=256$; see Sec. \ref{['Sec:Models']} for definition), quantified in this figure with the average point-to-point absolute error in expectation values over $t = 200~(2\pi g^{-1})$ (see Sec. \ref{['Sec:ErrorAnalyses']} for definitions and more detailed error analyses; data from Fig. \ref{['Fig:7_TrotterErrors']} (g)), breaks down rapidly after reaching a certain Trotter step size, observed as the large increase in the errors.
• Figure 2: Pictorial schematics of the physical models analysed in this paper.
• Figure 3: Full dynamics (with the corresponding Fourier transforms (FTs)) for each model system, showing (a--c) local observables (FTs in (g--i)) and (d--f) simulation fidelity (FTs in (j--l)). Left to right, columns show results for A2A-Ising (with $j = 64$), Heisenberg (chain of $N = 8$ qubits), and Dicke (with $j=6$) models, respectively, with x-axes common to each column showing step sizes. The y-axes of the colour plots (left axes) are the simulation times in (a--f) and the FT frequencies in (g--l). The black lines (right y-axes) in (a--f) are the time averages of the colour plots. The colour plots (a--f) are shown using a linear colour scale ranging from -1 to 1, and the FTs (normalised to have a unit maximum amplitude for each $\tau$) in (g--l) are shown using a logarithmically scaled colour map ranging from $10^{-6}$ to 1. In each model, dynamical quantities share a common Trotterisation threshold at a critical step size separating quasiperiodic from quantum chaotic regimes and two pre-threshold regimes of different step size dependencies. (a--c) Quasiperiodicity in expectation values of example local observables is clearly observed prior to the threshold, and destroyed after it: shown for (a) magnetisation $\braket{\bar{J}_{z}} = \frac{1}{j}\braket{J_{z}}$, (b) polarization of qubit one $\braket{\sigma_{z}^{1}}$, and (c) normalised photon number $\langle n\rangle/n_{j}$ (with $n_{j} = 7\times\dim_j$). (d--f) Simulation fidelity $\mathcal{F}(\psi_{\text{dig}}(t), \psi_{\text{ide}}(t))$ decays rapidly with time beyond the same threshold, with clear quasiperiodic oscillations in the near-threshold region showing why the time-averaged simulation fidelity starts decreasing strongly already before the threshold. These markedly universal behaviours across the different models are better demonstrated with the FT in (g--l), which show: (i) distinct stable frequencies at small step sizes, (ii) "branching" of frequencies at larger step sizes, and (iii) effectively flat spectra beyond the threshold. In each model, the dynamical signatures of quantum chaos (a--c) disappear in certain regimes beyond the threshold, referred to as stable regions, showing revival of regular dynamics, without a corresponding return of accurate target system simulation (d--f).
• Figure 4: Time-averages of dynamical signatures for each model compared across system size: (a--c) local observables and (d--f) simulation fidelity. Left to right, columns show results for A2A-Ising, Heisenberg, and Rabi-Dicke models, respectively, with line colours corresponding to system sizes given in legends in (a--c), and x-axes common to each column showing step sizes. $\langle . \rangle_{t}$ represents time averaging over periods $t = 200~(2\pi g^{-1})$ (A2A-Ising), $t = 50~(2\pi g^{-1})$ (Heisenberg), and $t = 200~(2\pi g^{-1})$ (Rabi-Dicke). (a--c) Expectation values of example local observables: (a) magnetisation $\langle \braket{\bar{J}_{z}}\rangle_{t}$, (b) polarization of qubit one $\langle \braket{\sigma_{z}^{1}}\rangle_{t}$, and (c) normalised photon number $\langle\langle n\rangle\rangle_{t}/n_{j}$. Dotted silver lines are time averages of sampled ideal dynamics, sampled at intervals of $\tau$, to illustrate the emergence of sampling errors at large step sizes, even in the absence of digitisation errors. (d--f) While time-averaged simulation fidelities $\langle\mathcal{F}(\psi_{\text{dig}}(t), \psi_{\text{ide}}(t))\rangle_{t}$ do not show the same sudden threshold because of pre-threshold quasiperiodic oscillations, they show a secondary drop at the threshold. Where they occur (e.g., for large enough $j$ values in the A2A-Ising and Rabi-Dicke models), stable regions are readily observed in time-averaged signatures, and are generally observed at the same position independent of size.
• Figure 5: Histograms showing eigenvector statistics for the $j=256$ A2A-Ising Trotter step unitary (TSU) for representative step sizes in different regimes: (a) $\tau = 0.002~(2\pi g^{-1})$ (before the threshold), (b) $\tau = 0.4~(2\pi g^{-1})$ (on the threshold), (c) $\tau = 0.5~(2\pi g^{-1})$ (quantum chaotic regime beyond the threshold), and (d) $\tau = 0.7~(2\pi g^{-1})$ (stable region in the quantum chaotic regime). Agreement with random matrix theory (RMT) distributions, shown for different time-reversal symmetries by black dashed and dotted lines in each plot, provides strong evidence for quantum chaotic dynamics. However, while the visual agreement and disagreement in the quantum-chaotic and regular regimes, respectively, in (a) and (c) seem clear, apparently similar visual agreement in (d) would then be misleading, as it corresponds to a stable region where the behaviours of dynamical signatures clearly deviate from quantum chaotic, and even (b) shows reasonable visual agreement at a step size in the transition where the dynamics still exhibits clearly quasiperiodic behaviour. By contrast, the reduced chi-squared goodness-of-fit test statistic $X^{2}_{\rm RMT}$, compared here against the COE distribution and labelled in coloured text on each figure, provides quantitative and objective evidence for agreement or disagreement with RMT and successfully predicts the observation of quantum chaotic dynamics in each case: for example, $X^{2}_{\rm RMT}$ shows clear disagreement with quantum chaotic dynamics in both transition regions (b) and even stable regions (d).