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Quantum localization bounds Trotter errors in digital quantum simulation
Markus Heyl, Philipp Hauke, Peter Zoller
TL;DR
Digital quantum simulation must control errors introduced when continuous many-body evolution is discretized into gate sequences. This paper interprets Trotterized evolution through quantum localization and finds a sharp step-size threshold: local-observable errors remain controllable in the localized regime but become associated with quantum chaos at larger steps. The results indicate that relatively large Trotter steps can reduce gate counts while preserving controlled local-observable dynamics, subject to finite-size and long-time scope limitations.
Problem
Trotterization errors are conventionally bounded through the global unitary, while the behavior and physical origin of errors on local observables remain to be explained.
Method
The paper interprets Trotterized evolution as a periodically driven Floquet system and analyzes localization, conserved quantities, and perturbative corrections in the Trotter step.
Results
A sharp threshold separates a localized small-step regime with controllable local-observable Trotter errors from a delocalized quantum-chaotic regime at larger steps.
Takeaways & Limitations
Relatively large Trotter steps can retain controlled errors for local observables, reducing the number of gates needed and diminishing the influence of extrinsic gate errors.
Takeaways & Limitations
The numerical studies reach up to N = 20 qubits, and quantitative extrapolation to N →∞ requires larger-system studies; generic periodically driven systems may eventually heat indefinitely.
Abstract
from arXiv · showhide
A fundamental challenge in digital quantum simulation (DQS) is the control of inherent errors. These appear when discretizing the time evolution generated by the Hamiltonian of a quantum many-body system as a sequence of quantum gates, called Trotterization. Here, we show that quantum localization-by constraining the time evolution through quantum interference-strongly bounds these errors for local observables. Consequently, for generic quantum many-body Hamiltonians, Trotter errors can become independent of system size and total simulation time. For local observables, DQS is thus intrinsically much more robust than what one might expect from known error bounds on the global many-body wave function. This robustness is characterized by a sharp threshold as a function of the Trotter step size. The threshold separates a regular region with controllable Trotter errors, where the system exhibits localization in the space of eigenstates of the time-evolution operator, from a quantum chaotic regime where the trajectory is quickly scrambled throughout the entire Hilbert space. Our findings show that DQS with comparatively large Trotter steps can retain controlled Trotter errors for local observables. It is thus possible to reduce the number of quantum gate operations required to represent the desired time evolution faithfully, thereby mitigating the effects of imperfect individual gate operations
Introduction
Digital quantum simulation represents many-body time evolution with elementary gates, but Trotterization introduces errors whose conventional global bounds grow with simulation time and system size. The paper explains why local-observable errors can instead remain small through quantum localization at sufficiently small Trotter steps.
- DQS and Trotterization: Digital quantum simulators approximate target many-body evolution by repeatedly applying gate sequences implementing decomposed Hamiltonian terms.The target Hamiltonian is split into implementable terms, whose evolutions are repeated using the Suzuki–Trotter formula.
- Error challenge: Global-unitary Trotter-error bounds grow quadratically with total simulation time and linearly with system size for generic many-body systems.These scalings make practical simulations challenging even though they remain polynomial in computational-complexity terms.
- Local-observable accuracy: For a quantum Ising chain, magnetization deviations from ideal evolution can remain bounded at long times and be substantially smaller than global-unitary bounds suggest.Figure 1b compares magnetization dynamics for different Trotter steps and shows error collapse after normalization for sufficiently small steps.
- Physical interpretation: The paper links these weak local-observable errors to quantum localization in Hilbert space at small Trotter steps.This localization constrains time evolution through quantum interference and provides a physical interpretation of Trotter errors.
Results
The Trotterized evolution can be viewed as a Floquet system whose eigenstate localization separates controlled local-observable errors from quantum-chaotic behavior. In the localized regime, long-time errors remain bounded and scale quadratically with the Trotter step, with little dependence on system size.
- Floquet interpretation: The Trotter sequence defines a periodically driven Floquet system with period τ=t/n and effective Hamiltonian HF.This reframes digital simulation as stroboscopic dynamics governed by a Floquet Hamiltonian.
- Benchmark model: The study uses the quantum Ising chain as an experimentally relevant benchmark and reports that the findings also apply to other model systems.The benchmark is implemented through Trotterized evolution of a local-interaction many-body Hamiltonian.
- Quantum chaos threshold: A sharp threshold in τ separates a regular localized regime from a quantum-chaotic regime where Floquet eigenstates become broadly delocalized.The inverse participation ratio and out-of-time-ordered correlator both identify this transition.
- Local-observable robustness: In the localized regime, magnetization errors remain bounded at long times and scale quadratically with τ.Different τ-dependent error trajectories collapse after normalization by (hτ)^2, indicating regular rather than rapidly diverging dynamics.
- Simulation accuracy: Simulation accuracy QE also shows a sharp localized-to-chaotic crossover and only weak quadratic dependence on τ for small steps.The long-time accuracy measure approaches full delocalization in the quantum-chaotic regime and remains weakly perturbed for small τ.
- Perturbative regime: The small-τ behavior of local-observable errors is captured by perturbation theory, with analytical results matching numerical data.The analysis treats the target Hamiltonian as an approximately conserved quantity and finds no notable dependence on N in the asymptotic regime.
Discussion
The authors find controllable intrinsic Trotter errors for local observables, while identifying finite-size extrapolation and extrinsic experimental errors as important boundaries. Larger Trotter steps can reduce gate counts and thereby lessen the influence of extrinsic errors.
- Scope and robustness: Controllable intrinsic Trotter errors for local observables persist across generic short-range systems and some long-range systems, including the lattice Schwinger model.The reported numerical evidence includes systems beyond the specific model studied in the main text.
- Scope and robustness: N = 20 qubits is the largest system size used in the numerical studies.The authors state that this size lies within realized and expected digital quantum simulator ranges.
- Experimental procedure: Estimating the threshold τ* experimentally can begin with small-N calculations of QE, followed by decreasing τ at larger N until sufficient convergence is reached.In the perturbative regime, nonzero-τ data can be used to extrapolate to ideal dynamics.
- Thermodynamic-limit considerations: A worst-case indefinitely heating scenario still permits a Trotter step with at most logarithmic dependence on total time t and no dependence on system size N for fixed accuracy.The stated bound uses τ = τ0/log(ct/∆) when τ ≪ τ0.
- Experimental implications: Reducing the number of gates is important because timing errors, decoherence, and faulty pulses can strongly affect DQS accuracy, whereas slow gate-coupling drifts are relatively benign.The paper therefore connects controlled intrinsic errors at relatively large Trotter steps with reduced exposure to extrinsic errors.
Materials and Methods
The methods interpret Trotterized evolution as Floquet dynamics and combine numerical diagnostics with perturbative calculations. Simulations use specific gate sequences, long-time averages, and expansions of the Floquet Hamiltonian to evaluate errors in energy and observables.
- Numerical model and gates: The main-text simulations use a quantum Ising chain whose Trotterized dynamics is implemented with a sequence of two elementary gates.The method notes that many terms commute, allowing a small gate set.
- Long-time numerics: Real-time observables are evolved for 2 · 10^4 periods and averaged over the final 10^4 periods to estimate their asymptotic values.The averaging removes remaining finite-size temporal fluctuations.
- Localization diagnostics: The inverse participation ratio is obtained dynamically through a stroboscopic mean, providing an experimentally accessible route that can reach larger systems than exact diagonalization.The resulting quantity is identified with the Loschmidt echo.
- Chaos diagnostics: The OTO correlator is computed using forward and backward state evolution with operator insertions, but its runtime scales as n^2 and limits accessible simulation times.The main-text simulations use n = 10^3 and average over the final 300 periods.
- Perturbative analysis: For the considered parameters, full diagonalization gives qE = 0.18, consistent with the dynamical calculation in the small-Trotter-step limit.The derivation uses the diagonal ensemble and model-specific properties of the initial state.
Supplementary Materials to
The supplementary materials extend the analysis to the lattice Schwinger model and examine timing and ensemble errors in digital quantum simulation. They find a sharp localization-to-chaos threshold and controllable, quadratic Trotter errors for local observables at small steps.
- Lattice Schwinger model: The lattice Schwinger model provides a second benchmark for DQS and represents an interacting lattice gauge theory without general exact real-time solution methods.Its gauge fields are integrated out using Gauss law, producing asymmetric long-range fermionic interactions.
- Simulation setup: The Schwinger-model simulations use exact diagonalization, a specified gate sequence, a bare-vacuum Néel-state initialization, and long-time stroboscopic averaging.The simulations use 2 · 10^4 periods and average over the final 10^4 periods.
- Localization threshold: A sharp IPR threshold separates localized dynamics at small τ from full delocalization over accessible Hilbert-space states at large τ.The accessible-state count accounts for the model’s conserved total-spin sector.
- Trotter errors: For small τ, the particle-production deviation ∆ν and simulation-accuracy errors scale quadratically with τ, whereas large-step chaotic dynamics produces uncontrolled Trotter errors.The long-time particle number ν and simulation accuracy QE both signal the quantum many-body chaos threshold.
- Extrinsic imperfections: Timing errors arise from inaccurate gate lengths, while ensemble errors arise from slowly drifting gate couplings.The supplementary analysis treats these as distinct extrinsic imperfections beyond Trotterization.
A. Timing error
Timing fluctuations are modeled as random gate-strength variations and, in the fast-driven regime, as effective Markovian noise. Their influence becomes significant on a time scale proportional to η^-2 rather than η^-1.
- Noise model: Timing errors replace the intended gate duration τ with pulse-dependent fluctuations whose independent random variables have vanishing mean and variance proportional to η.The fluctuating gates are described by a noise process after extending the pulse variables to continuous time.
- Effective dynamics: In the fast-driven regime, the timing-noise spectrum is flat to leading order, so averaged dynamics follows a Markovian Master equation with Lindblad operators H_l.The relevant frequencies must be small compared with τ^-1.
- Heating mechanism: The controlled evolution contains the time-averaged Hamiltonian plus a Trotterization perturbation, while timing fluctuations generate heating at a rate proportional to η^2τ times the commutator scale.The heating contribution is suppressed by an additional factor η^2 relative to the commutator scale.
- Relevance time: The timing error is insignificant for t ≪ 1/(τη^2) but becomes severe for t ≳ 1/(τη^2).Numerical data collapse after rescaling time with η^2, supporting a relevance time proportional to η^-2.
B. Ensemble error
Ensemble errors model slowly drifting gate couplings that remain fixed during each experimental run but vary between runs. Unlike timing errors, they do not heat the system; they average observables across nearby Hamiltonians.
- Error model: Slow parameter drifts are represented by independently fluctuating gate couplings that are constant within one run and randomly different between runs.Each run therefore evolves under a slightly modified Hamiltonian and gate sequence.
- Physical effect: In contrast to timing errors, ensemble errors do not cause heating but average expectation values over a family of slightly different Hamiltonians.In the perturbative regime this averaging is generally benign, except near hypersensitive regimes such as quantum phase transitions.