Optimal Control Design Guided by Adam Algorithm and LSTM-Predicted Open Quantum System Dynamics

TL;DR

The paper addresses the challenge of achieving high-fidelity quantum control in noisy, non-Markovian environments by predicting open-system dynamics with LSTM-NNs and performing gradient-based pulse optimization with the Adam algorithm. It introduces a two-step control design for adiabatic speedup in a two-level system: first optimizing the driving trajectory s(t) and then a global zero-area pulse c(t), both guided by LSTM-predicted dynamics. The approach yields fidelity improvements and orders-of-magnitude speedups compared with direct RK4 simulations, while maintaining accuracy in extrapolated regimes. The framework offers a scalable path for rapid control design applicable to quantum computing, communication, and sensing, with prospects for extension to higher-dimensional systems and real experimental data.

Abstract

The realization of high-fidelity quantum control is crucial for quantum information processing, particularly in noisy environments where control strategies must simultaneously achieve precise manipulation and effective noise suppression. Conventional optimal control designs typically requires numerical calculations of the system dynamics. Recent studies have demonstrated that long short-term memory neural networks (LSTM-NNs) can accurately predict the time evolution of open quantum systems. Based on LSTM-NN predicted dynamics, we propose an optimal control framework for rapid and efficient optimal control design in open quantum systems. As an exemplary example, we apply our scheme to design an optimal control for the adiabatic speedup in a two-level system under a non-Markovian environment. Our optimization procedure entails two steps: driving trajectory optimization and zero-area pulse optimization. Fidelity improvement for both steps have been obtained, showing the effectiveness of the scheme. Our optimal control design scheme utilizes predicted dynamics to generate optimized controls, offering broad application potential in quantum computing, communication, and sensing.

Figures (7)

• Figure 1: Schematic illustration of the machine learning algorithm for open quantum system dynamics prediction and control optimization. Top left: The predicted evolution of a two-level system (e.g., a spin) under driving $s(t)$ and control $c(t)$. Top right: The expectation values of the initial states are encoded into the initial hidden states of the LSTM-NNs via an encoder. The initial state is first encoded into the LSTM hidden state via a multilayer perceptron (MLP). This encoded state is then used by the network to autoregressively generate the system’s evolution trajectory across the entire time interval, given the control pulses and environmental parameters. Middle: Detailed architecture of the LSTM-NNs, highlighting the input–output relations between the quantum system and the network. Bottom: The process of pulse optimization via the Adam algorithm. The dynamics of the system are predicted through the LSTM-NNs. The fidelity of the final state is taken as the loss function.
• Figure 2: Verification of the high accuracy prediction ability of the LSTM-NNs for the system dynamics under arbitrary control $s(t), c(t)$, and arbitrary environmental parameter $\Gamma \in [0,0.05]$, $\gamma \in [1,30]$, $T\in [5,15]$. We train the NN on random driving generated via multi-frequency random sampling method based on Fourier transform for a fixed time interval $T_{tot}=2.5$ and predict the dynamics for arbitrary driving field in a large time interval $T_{tot}=5$: (a) Random Fourier Synthesis (RFS) driving and control, (b) linear ($s(t)=t/T_{tot}$) and sine ($c(t)=I\sin(2\pi t)$ ($I\approx54.4\ \text{to satisfy the third zero of } J_0(I\tau/\pi))$), (c) sine ($s(t)=\frac{1}{2}\sin(\pi t/T_{tot}-\frac{\pi}{2})+\frac{1}{2}$) and quench ($\tau=2,I\approx37.7 \ \text{to satisfy}\ \ I\tau = 2k\pi\ \text{where}\ \ k=6$ ). The initial states for three cases are the ground states of the time-dependent Hamiltonian $H(t)$ at $t=0$. These three drivings are not seen by the NN during training. The time window $[2.5, 5]$ highlighted in light purple corresponds to the region where the NN are not trained. The first line is the driving $c(t)$ and $s(t)$. The 2-4th line are the dynamics of $\langle \sigma_x \rangle$,$\langle \sigma_y \rangle$, and $\langle \sigma_z \rangle$, respectively. The last line shows the evolution of the error of the NN.
• Figure 3: The fidelity $F$ versus the rescaled time $t/T_{tot}$ for different total evolution time $T_{tot}$ without $(\Gamma=0)$ and with $(\Gamma=0.03)$ environment.
• Figure 4: The fidelity versus the rescaled time $t/T_{tot}$ for linear ($s(t)=t/T_{tot}$), sine ($s(t)=\frac{1}{2}\sin(\pi t/T_{tot}-\frac{\pi}{2})+\frac{1}{2}$)
• Figure 5: The fidelity improvement $F_{im}^{s(t)}$ versus environmental parameters $\Gamma, \gamma$ and $T$. $F_{im}^{s(t)}=F_{opt}^{s(t)}-F_{lin}^{s(t)}$, with $F_{opt}^{s(t)}$ and $F_{lin}^{s(t)}$ represent the final fidelities corresponding to the optimized and the linear $s(t)$, respectively. $\Gamma=0.03, \gamma=4, T=5$, and the other two are fixed when one of them varies.