AI Predicts Quantum System Behaviour for Faster, More Reliable Control

Scientists are increasingly focused on achieving high-fidelity control in quantum systems, a necessity for robust information processing, especially given the challenges posed by environmental noise. JunDong Zhong and ZhaoMing Wang, from their respective institutions, alongside et al., present a novel optimal control framework that leverages the predictive power of long short-term memory neural networks (LSTM-NNs) to streamline control design for open quantum systems. This research significantly advances the field by replacing computationally intensive numerical simulations of system dynamics with LSTM-NN predictions, enabling rapid, efficient optimisation. They demonstrate the effectiveness of their approach through the design of an optimal control scheme for adiabatic speedup in a two-level system, achieving fidelity improvements across both trajectory and pulse optimisation stages and offering potential applications in diverse areas such as computing and communication.

Scientists recognise quantum control as 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 require 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 process.

Optimisation of pulse sequences for enhanced adiabatic speedup is crucial for practical quantum computation

Scientists are investigating adiabatic speedup in a two-level system within a non-Markovian environment. Their optimization procedure comprises two steps: driving trajectory optimization and zero-area pulse optimization. Fidelity improvement has been observed for both steps, demonstrating the effectiveness of the scheme.

This optimal control design scheme utilizes predicted dynamics to generate optimized controls, offering broad application potential in quantum computing, communication, and sensing. High-precision and robust quantum control is an essential prerequisite for high-quality quantum information processing tasks.

Dynamical decoupling has been widely utilized in various experimental platforms, such as superconducting quantum circuits, Rydberg atoms, and trapped ions. The realization of high-fidelity control depends on the design of the pulse profile, for example, optimal radio frequency pulse sequences in superconducting quantum circuits.

Both analytical and numerical methods have been developed to design these pulse sequences. Analytical methods include zero-area pulse schemes, which prevent transitions between subspaces. Global optimal pulses for two-level quantum systems against offset or control-field uncertainties have been derived via the Pontryagin maximum principle.

The fastest possible pulses implementing single-qubit phase gates via geometrical optimization have also been investigated. Numerous traditional numerical methods for pulse design exist, including stochastic gradient descent, the Adam algorithm, Gradient Ascent Pulse Engineering, the Krotov method, the Chopped Random Basis method, and the distributed proximal policy optimization algorithm.

Recently, artificial intelligence-driven optimal control design has surged, encompassing techniques like reinforcement learning and supervised learning. In practice, environmental noise degrades control performance, making optimal control design in noisy environments difficult, especially for analytical methods.

The Lindblad quantum master equation is widely used to describe dynamics in the presence of an environment. Non-Markovian cases for bosonic baths have been derived via the quantum state diffusion equation approach for both zero and finite temperatures. The influence of the baths on the system is characterized by environmental parameters such as system-bath coupling strength, bath temperature, and bath characteristic frequency.

A pulse design approach, based on the master equation, uses ideal pulses derived in the absence of an environment as a foundation, followed by fine-tuning to mitigate noise. However, this process requires continuous solving of the master equation through numerical methods, including finite difference, Runge-Kutta, and Low-Storage Runge-Kutta.

These methods are slow and struggle with large Hilbert space dimensions. Recently, machine learning-based neural networks have been applied to simulate open quantum system dynamics, addressing these challenges. Various architectures have been explored, such as physics-informed neural networks, deep quantum neural networks, variational methods, autoregressive neural networks, time series prediction, and recurrent neural networks.

The intuition is that an efficient description of the system can be learnt from data without explicit modelling assumptions. Neural network-based learning techniques can learn complex functional dependencies in time series directly from data, avoiding theoretical assumptions often required when deriving non-Markovian quantum master equations.

LSTM-NN, a recurrent neural network designed for dynamical systems with long-range temporal correlations, is well-suited for predicting non-Markovian dynamics. It can quickly learn underlying correlations to correctly predict the evolution of open quantum systems, with a runtime shorter than direct numerical simulations.

LSTM-NNs can predict dynamics well for trained time intervals and extrapolate to longer, untrained windows. Their memory allows them to build an implicit representation of higher-order correlations using a non-Markovian strategy. Owing to their efficiency, LSTM-NNs are employed to predict system dynamics in optimal pulse design, eliminating the need for direct numerical calculation, effectively acting as a solver for the master equation.

This prediction is integrated with conventional optimization algorithms for optimal pulse design, offering faster performance compared to numerical calculation. Recurrent neural networks have previously been employed for quantum control. In this paper, the researchers consider optimal control design in adiabatic speedup as a demonstration of their scheme.

Adiabatic speedup allows quick evolution into target states of otherwise slow adiabatic dynamics, with schemes including transitionless quantum driving, counteradiabatic control, rapid adiabatic passage, and stimulated Raman adiabatic passage. Recently, schemes for accelerating the adiabatic process via zero-area pulses were investigated, with stochastic learning achieving higher adiabatic fidelities than ideal pulses.

The researchers use LSTM-NNs and the Adam algorithm for optimal design of both driving trajectory and zero-area pulse. The system dynamics are predicted by the LSTM-NNs instead of numerical calculation using Runge-Kutta. This strategy substantially improves adiabatic fidelity, demonstrating rapid and efficient performance in optimal control design.

For the adiabatic evolution, a time-dependent Hamiltonian H(t) can be written as H(t) = [1 − s(t)]Hi + s(t)Hf, where s(t) is the driving trajectory. Hi and Hf denote the initial and final Hamiltonians, respectively, with s(0) = 0 and s(Ttot) = 1. The problem focuses on obtaining a ground state at s(Ttot) by evolving the system from a trivial ground state at s(0).

With the computational problem encoded in the ground state of Hf, adiabatic quantum computation can be realized. For a closed system, adiabatic speedup can be achieved by adding a leakage elimination operator Hamiltonian, suppressing transitions between subspaces. The system Hamiltonian becomes H(t) = [1 + c(t)][(1 − s(t))Hi + s(t)Hf], where c(t) denotes the pulse control function.

Ideal pulse control conditions have been theoretically derived via P-Q partitioning techniques. Various pulse shapes, such as rectangular and sine pulses, have been considered. For rectangular pulses, the ideal pulse condition is Iτ = 2kπ, k = 1, 2, 3, etc.

For sine pulses c(t) = Isin(πt/τ), the ideal control conditions satisfy J0(Iτ/π) = 0, where I is the pulse intensity, τ is the pulse half period, and J0(x) is the zero-order Bessel function of the first kind. If the system is immersed in bosonic baths, a non-Markovian quantum master equation describes the dynamics: ∂ρ/∂t = −i[H, ρ] + ([L, ρO†z] − [L†, ρOz]) + ([L, ρO†w] − [L†, ρOw]), where ρ is the reduced density matrix.

Using this equation, the non-Markovian dynamics can be fully analyzed. The researchers present a schematic illustration of the machine learning algorithm for open quantum system dynamics prediction and control optimization. The predicted evolution of a two-level system under driving s(t) and control c(t) is shown.

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. 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.

The detailed architecture of the LSTM-NNs highlights the input, output relations between the quantum system and the network. The process of pulse optimization via the Adam algorithm is also shown, with the fidelity of the final state taken as the loss function. By combining LSTM-NNs and the Adam algorithm, the researchers first focus on the optimal choice of time sequence s(t), then on the global optimal control design c(t) in the adiabatic speedup.

For simplicity, the researchers consider a two-level system with Hi = σz and Hf = σx, where σz and σx are Pauli matrices. The initial state is the ground state of σz, i.e., |E0(0)⟩ = |0⟩. According to the adiabatic theorem, if Ttot is infinitely long, the system will evolve along an adiabatic path, resulting in the final state |E0(Ttot)⟩ = 1/√2(|0⟩ − |1⟩).

When the system is subject to a non-Markovian, finite-temperature heat bath, the fidelity F = p ⟨E0(t)| ρ(t) |E0(t)⟩ measures the adiabaticity, where ρ(t) is the reduced density matrix and |E0(t)⟩ is the instantaneous ground state of the Hamiltonian. The Lindblad operator is taken as L = σ− throughout.

LSTM-NN is a specialized type of recurrent neural networks designed to model dynamical systems with long-term temporal correlations. Unlike conventional recurrent neural networks, LSTM-NN introduces a gating mechanism for explicit regulation of information flow over time, mitigating the vanishing and exploding gradients common in long sequences.

The introduction of independent memory cells retains key information over long times, selectively forgetting past information and controllably writing new information through gated structures. LSTM-NNs can learn long-term and short-term temporal correlations, efficiently predicting the evolution of open quantum systems, including complex non-Markovian cases.

Adiabatic fidelity variation with evolution time and driving trajectory characteristics reveals key insights into process robustness

Total evolution times of 3, 7, and 10 were examined to identify the approximate adiabatic regime, both without (Γ = 0) and with (Γ = 0.03) an environment, using parameters γ = 2 and T = 10. Adiabatic fidelity increased with increasing total evolution time up to approximately 10 when no environment was present, indicating entry into the adiabatic regime.

However, in the presence of an environment, fidelity was diminished, and did not consistently increase with total evolution time. For a short total evolution time of 3, the environment had minimal impact on fidelity, but this impact grew with longer times of 7 and 10, even in the adiabatic regime without an environment.

Comparisons of fidelity versus rescaled time for linear, sine, and optimized driving trajectories s(t) were conducted both without and with (Γ = 0.03, γ = 2, T = 10) an environment, using a total evolution time of 5. The sine trajectory initially outperformed the linear one, but the linear trajectory became superior in the latter half of the total duration, a trend observed in both scenarios.

The optimal trajectory consistently demonstrated the highest fidelity at the final time, t = Ttot, across all cases. Optimization was constrained to maintain s(t) values between 0 and 1, with initial and final values of 0 and 1, respectively. Fidelity improvement, defined as F s(t) im = F s(t) opt −F s(t) lin, was calculated using parameters Γ = 0.03, γ = 4, and T = 5, while varying one parameter at a time.

Results indicated that F s(t) im gradually decreased with increasing values of Γ, γ, and T, suggesting that enhancing adiabatic fidelity through s(t) optimization becomes more challenging in stronger environmental conditions. Employing a zero-area pulse control scheme further improved adiabatic fidelity, even in a non-adiabatic regime with a short total evolution time of 5, limiting noise accumulation.

Ideal pulse control, using c(t) = I sin(2πt) with I ≈ 54.4, significantly improved fidelity with and without an environment, satisfying the third zero of J0(Iτ/π). Fidelity decreased with increasing environmental parameters Γ, γ, and T, demonstrating that control performance deteriorates as environmental influence intensifies.

LSTM neural networks predict open quantum system dynamics for accelerated control optimisation by learning from trajectory data

Researchers have developed a novel quantum control framework integrating long short-term memory neural networks (LSTM-NNs) to predict the dynamics of open quantum systems. This approach circumvents the need for extensive numerical calculations of system dynamics, traditionally required for optimal control design.

By training the LSTM-NN on data generated from a non-Markovian master equation, the network learns to accurately predict the evolution of key observables within the quantum system. The method was demonstrated using a two-level system undergoing adiabatic speedup in a non-Markovian environment, achieving fidelity improvements through optimized driving trajectories and pulse shapes.

This LSTM-NN-based approach offers a computational advantage of one to two orders of magnitude compared to conventional methods, while maintaining prediction accuracy for target observables. The technique is particularly suited to processing real experimental data, as it requires only partial observation of the quantum system’s state.

Limitations acknowledged by the researchers include the LSTM-NN’s prediction of only expectation values, rather than the full quantum state, and the current focus on relatively simple systems. Future work will focus on extending this method to higher-dimensional systems and more complex tasks, such as high-fidelity state transfer and quantum error correction, potentially benefiting superconducting quantum processors. These findings establish a clear path toward accelerating quantum control design and facilitating practical applications in computing and communication.