[Paper Review] Adaptive Trotterization for time-dependent Hamiltonian quantum dynamics using piecewise conservation laws

Hongzheng Zhao, Marin Bukov+2 more|arXiv (Cornell University)|Jul 19, 2023

Quantum Computing Algorithms and Architecture59 references4 citations

TL;DR

This paper introduces tADA-Trotter, an adaptive Trotterization algorithm for time-dependent quantum Hamiltonians that uses instantaneous conservation laws of effective time-independent Hamiltonians to bound local and global errors. By dynamically adjusting the time step based on measured deviations in energy and energy variance, the method suppresses Trotter-induced heating and achieves higher simulation accuracy than fixed-step Trotterization with reduced circuit depth.

ABSTRACT

Digital quantum simulation relies on Trotterization to discretize time evolution into elementary quantum gates. On current quantum processors with notable gate imperfections, there is a critical tradeoff between improved accuracy for finer timesteps, and increased error rate on account of the larger circuit depth. We present an adaptive Trotterization algorithm to cope with time-dependent Hamiltonians, where we propose a concept of piecewise "conserved" quantities to estimate errors in the time evolution between two (nearby) points in time; these allow us to bound the errors accumulated over the full simulation period. They reduce to standard conservation laws in the case of time-independent Hamiltonians, for which we first developed an adaptive Trotterization scheme [PRX Quantum 4, 030319]. We validate the algorithm for a time-dependent quantum spin chain, demonstrating that it can outperform the conventional Trotter algorithm with a fixed step size at a controlled error.

Motivation & Objective

• To address the challenge of balancing simulation accuracy and circuit depth in noisy intermediate-scale quantum (NISQ) devices during digital quantum simulation of time-dependent Hamiltonians.
• To extend adaptive Trotterization—previously limited to time-independent systems—by introducing a criterion for step-size adaptation in time-dependent settings.
• To suppress Trotter-induced heating effects through global error control, enabling reliable long-time simulations.
• To provide a feedback-based algorithm that uses measurable quantities (energy and energy variance) to guide step-size adaptation in real time.
• To validate the method on a driven quantum spin chain, demonstrating superior performance over fixed-step Trotterization.

Proposed method

• The method discretizes time evolution into intervals [t, t+δt], within which an effective time-independent Hamiltonian H_{[∞]}^{t,δt} is defined via a perturbative Magnus expansion in δt.
• Within each interval, the algorithm enforces instantaneous conservation laws: the expectation values of H_{[∞]}^{t,δt} and its variance should remain unchanged after evolution.
• The adaptive step size δt is determined by measuring the deviation of these expectation values before and after Trotterized evolution, with δt maximized as long as errors remain bounded.
• A global error is defined as the cumulative sum of local errors in the instantaneous conservation laws, which is constrained via feedback to prevent error accumulation.
• The algorithm uses a truncated Magnus expansion H_{[k]}^{t,δt} to approximate H_{[∞]}^{t,δt} and applies a Trotter decomposition U_{[λ]}(t+δt,t) for time evolution.
• The feedback loop adjusts δt dynamically, reducing it when local errors exceed thresholds and increasing it when errors are small, especially in regions of slow dynamics.

Figure 1: Schematics of tADA-Trotter for a time-dependent Hamiltonian. (a) The expectation values of the instantaneous conserved Hamiltonian $H_{[\infty]}^{t,\delta t}$ coincide at times $t$ and $t+\delta t$ . We maximize $\delta t$ as long as errors in this conservation law are bounded, i.e., devia

Experimental results

Research questions

• RQ1 Can instantaneous conservation laws of effective time-independent Hamiltonians be used to define a reliable error metric for adaptive Trotterization in time-dependent systems?
• RQ2 How can global error accumulation from local Trotter errors be controlled to suppress Trotter-induced heating in long-time simulations?
• RQ3 Can adaptive step sizing based on measured energy and energy variance outperform fixed-step Trotterization in terms of accuracy and circuit depth?
• RQ4 Does the algorithm maintain accuracy under strong time-dependent driving and complex many-body dynamics?
• RQ5 Can the framework be generalized to open quantum systems or higher-order Trotter schemes?

Key findings

• tADA-Trotter successfully suppresses Trotter-induced heating by constraining the accumulation of errors in instantaneous conservation laws, enabling reliable long-time simulations.
• The algorithm achieves significantly better accuracy than fixed-step Trotterization, with simulation errors only becoming visible at t > 11, while fixed-step errors appear much earlier (t > 5).
• The adaptive step size varies dynamically over a range of [0.1, 0.7], decreasing to 0.1 during rapidly changing dynamics and increasing to 0.7 in quiescent regions, demonstrating flexibility.
• Accumulated errors in the expectation value of H_{[k]} and its variance remain bounded below specified thresholds for most of the simulation, with occasional overruns due to tight early-time tolerances.
• The method reduces circuit depth while maintaining accuracy, as evidenced by the red (globally bounded) data closely following the exact solution, unlike the blue (fixed-step) data which diverges early.
• For a driven spin chain with ω = 0.8 and τ = 30, tADA-Trotter achieves a total simulation time of t ~ 20 with N = 100 Trotter steps, demonstrating practical feasibility.

Figure 2: Comparison between tADA-Trotter and fixed-step Trotter algorithms. Inset depicts the stepsize that is varying in time. It takes larger values when external driving fields are weak. We use the following Hamiltonian parameters for numerical simulation, $J_{z}=1,h_{x}=3,h_{z}=0.5,\tau=30,\ome