Making Trotterization Adaptive and Energy-Self-Correcting for NISQ Devices and Beyond

Simulation of continuous-time evolution requires time discretization on both classical and quantum computers. A finer time step improves simulation precision but it inevitably leads to increased computational efforts. This is particularly costly for today’s noisy intermediate-scale quantum computers, where notable gate imperfections limit the circuit depth that can be executed at a given accuracy. Classical adaptive solvers are well developed to save numerical computation times. However, it remains an outstanding challenge to make optimal usage of the available quantum resources by means of adaptive time steps. Here, we introduce a quantum algorithm to solve this problem, providing a controlled solution of the quantum many-body dynamics of local observables. The key conceptual element of our algorithm is a feedback loop that self-corrects the simulation errors by adapting time steps, thereby significantly outperforming conventional Trotter schemes on a fundamental level and reducing the circuit depth. It even allows for a controlled asymptotic long-time error, where the usual Trotterized dynamics faces difficulties. Another key advantage of our quantum algorithm is that any desired conservation law can be included in the self-correcting feedback loop, which has a potentially wide range of applicability. We demonstrate the capabilities by enforcing gauge invariance, which is crucial for a faithful and long-sought-after quantum simulation of lattice gauge theories. Our algorithm can potentially be useful on a more general level whenever time discretization is involved also concerning, e.g., numerical approaches based on time-evolving block-decimation methods.

Popular Summary

Quantum computers have the potential to outperform classical computers in certain tasks. One area of interest is simulating the time evolution of many-body systems when they are far from equilibrium. To simulate the continuous evolution of these systems, we need to discretize time into smaller pieces, both on regular computers and on quantum computers. Using smaller time steps improves the accuracy of the simulation, but it requires more computational power. This is particularly costly for today's noisy intermediate-scale quantum computers, where notable gate imperfections limit the simulation time that can be achieved at a given accuracy. Classical adaptive solvers are well developed to save numerical computation times. However, it remains an outstanding challenge to make optimal usage of the available quantum resources by means of adaptive time steps.

Here, we introduce a quantum algorithm to solve this problem, providing a controlled solution of the quantum many-body dynamics of local observables. The key conceptual element of our algorithm is a feedback loop that self-corrects the simulation errors by adapting the time-step size, thereby significantly outperforming conventional fixed-step schemes on a fundamental level and reducing the circuit depth. It even allows for a controlled long-time error, where the usual fixed-step scheme faces difficulties. Another key advantage of our algorithm is that any desired conservation law can be included in the self-correcting feedback loop, which has a potentially wide range of applicability.

Our algorithm can be applied in more general situations that involve time discretization, for instance, quantum imaginary time evolution, time-dependent Hamiltonian simulation, and classical numerical approaches based on time-evolving block decimation methods.

Figure 1

(a) A schematic of the ADA Trotter protocol. Given a quantum state at time $t_m$ and a Trotter expansion that approximates a target Hamiltonian, one tries to use the largest possible Trotter step size as long as errors in conservation laws are bounded. (b) ADA Trotter generally outperforms conventional Trotter: states propagated via ADA Trotter (orange) are constrained in a energy shell where energy and its variance are approximately conserved. In contrast, a fixed step size is chosen for conventional Trotter (blue) and larger errors may occur. (c) An illustration of the local dynamics obtained by ADA Trotter, which approximates the exact dynamics (black) closely at both short and long times. The inset shows the selected adaptive step size, which fluctuates in time. Smaller step sizes are chosen at early times to capture the rapid local oscillations, whereas an overall growing trend can be observed at longer times. We use $J_z=-1$, $h_x=-2$, $h_z=0.2$, $L=24$, and an initial state polarized in the negative $y$ direction for our numerical simulations.

Figure 2

A comparison between ADA Trotter and fixed-step Trotterization. (a) and (b) depict the time evolution of the magnetization in the x and z directions, respectively. The variance constraint is crucial for ADA Trotter: with variance constraint $d_{\delta {\mathcal{E}}^2}=1$ (orange circle), ADA Trotter reproduces the exact dynamics, whereas notable deviation appears for $d_{\delta {\mathcal{E}}^2}=\mathrm{\infty }$ (dashed red) without variance control. ADA Trotter outperforms fixed-step Trotter: for a large fixed Trotter step ($\delta t=0.36$, green triangle), notable local errors appear at early times; a smaller $\delta t$ can be employed (blue diamond) to suppress errors but the total simulation time is limited. (c) and (d) depict the expectation value of target Hamiltonian H and its variance, deviations of which are constrained by $d_{\mathcal{E}}$ and $d_{\delta {\mathcal{E}}^2}$ for ADA Trotter. Fixed-step Trotter can lead to a sufficiently large error in the energy at early times. (e) Higher moments of the target Hamiltonian are better preserved with the variance constraint. We use $J_z=-1$, $h_x=-1.7$, $h_z=0.5$, $d_{\mathcal{E}}=0.03$, and $L=24$ for our numerical simulations.

Figure 3

The deviation in the expectation value for the $z$ magnetization between the ADA Trotter and exact quench results in the long-time limit, as a function of (a) the energy-density deviation $d_{\mathcal{E}}$ and (b) the variance deviation $d_{\delta {\mathcal{E}}^2}$. (a) The local error decreases for a smaller energy deviation and can be captured by the microcanonical prediction (gray). (b) The variance deviation leads to local errors as finite-size effects, which vanish linearly in $d_{\delta {\mathcal{E}}^2}/L$. Parameters $J_z=1$, $h_x=1$, and $h_z=0.3$ and initial states $\exp (-i{\theta }_y\sum _j{\sigma }_j^y)|\downarrow \cdots \downarrow ⟩$ are used for our numerical simulations.

Figure 4

The dynamics of the expectation values of the gauge symmetry generator in (a) and its variance in (b). Additional constraints on gauge violation preserve the gauge symmetry (red); otherwise errors grow quickly in a short time (blue). A soft constraint is used to prevent ADA Trotter from freezing. We use $J=0.5$, $\mu =0.5$, $k=0.5$, $d_{\mathcal{E}}=0.1$, $d_{\delta {\mathcal{E}}^2}=0.2$, $\lambda =0.3$, $L=6$, and $V=\sum _j[s_{j,j+1}^{+}/\sqrt{S(S+1)}+{\sigma }_j^{+}{\sigma }_{j+1}^{-}+\text{h.c.}]$ for our numerical simulations.