⚛️ quantum physics

Physics-informed neural networks for quantum control

This paper introduces a physics-informed neural network (PINN) framework for optimal quantum control that efficiently solves state-to-state transfer problems in open quantum systems with high probability, short evolution times, and low energy consumption, while demonstrating superior flexibility in adapting to varying physical parameters and initial conditions compared to traditional optimization techniques.

Original authors: Ariel Norambuena, Marios Mattheakis, Francisco J. González, Raúl Coto

Published 2026-07-21

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Ariel Norambuena, Marios Mattheakis, Francisco J. González, Raúl Coto

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to steer a tiny, invisible boat through a stormy ocean. This isn't a normal boat, though; it's a quantum system, a realm where particles behave like waves, can be in two places at once, and are incredibly sensitive to the slightest touch. In this world, "quantum control" is the art of steering these particles from one state to another—like moving a quantum bit from "off" to "on"—without crashing into the rocks of noise and energy loss. Scientists have been trying to find the perfect steering wheel (a control pulse) to do this, but the ocean is so complex that traditional math often gets stuck in the mud, taking forever to calculate the best path or failing when the weather changes.

Enter a new kind of navigator: the Physics-Informed Neural Network, or PINN. Think of a standard neural network as a student who learns by memorizing thousands of flashcards of past storms. It's good, but if a new storm hits that it hasn't seen before, it might get confused. A PINN, however, is like a student who has memorized the flashcards and the fundamental laws of physics written in the stars. It doesn't just guess based on data; it knows the rules of the universe (like how energy flows or how particles interact) are non-negotiable. This paper introduces a way to use these "rule-following" AI navigators to steer quantum systems, finding the perfect control signals to move particles quickly, efficiently, and accurately, even when the conditions are tricky.

The authors of this paper, Ariel Norambuena and colleagues, propose a fresh computational method to solve these optimal quantum control problems. Instead of relying on massive datasets of previous experiments, they built a neural network that is "informed" by the physics equations themselves. They set up a game where the network has to minimize a "loss function"—essentially a scorecard that penalizes it for breaking the laws of physics, failing to reach the target, or using too much energy. By playing this game, the network discovers the smooth, optimal control functions needed to guide quantum states.

To test their idea, the team first tackled a simple two-level system, which is like a quantum coin that can be heads or tails. Their goal was to steer this coin into a specific "mixed" state (a balanced blend of heads and tails) known as a Gibbs state, which is useful for simulating high-temperature superconductivity. The PINN successfully found a control signal that drove the system to this target with a fidelity (accuracy) of 0.99, meaning it was almost perfect. Interestingly, the network figured out that the control signal needed to settle at a specific constant value to maintain this state, a result that matched theoretical predictions but was found without the network ever being told the theory beforehand.

Next, the team moved to a more complex three-level system, shaped like the Greek letter Lambda (Λ). This setup is famous for a technique called STIRAP, where scientists use two laser pulses to move a particle from one state to another without ever letting it get stuck in a "lossy" middle state that might leak energy. Traditionally, STIRAP requires a very specific, counterintuitive order: the "Stokes" pulse must start before the "Pump" pulse. However, in their simulations, the PINN discovered a control strategy that worked just as well, but with a twist: it turned on both pulses at roughly the same time. The network didn't know about the "dark state" theory or the rules of STIRAP; it simply found a path that minimized energy loss and reached the target with high fidelity (0.97) and in record time (2.0 arbitrary units), beating several standard methods that took much longer or used significantly more energy.

The paper also highlights the flexibility of this approach. When the researchers changed the initial conditions or introduced "detuning" (a mismatch in the laser frequencies that usually ruins the transfer), the PINN adapted instantly. Unlike standard methods that often fail when the parameters shift, the PINN remained robust, delivering high-quality transfers even when the environment changed. The authors also showed that their method could scale up, successfully simulating control for systems with up to five qubits (quantum bits), maintaining high fidelity even as the complexity grew.

In these simulations, the PINN proved to be a versatile tool that suggests a new way to design quantum controls. It offers a data-free path to finding smooth, energy-efficient control pulses that respect the underlying physics. While the results are currently based on simulations, the authors suggest that this method could be a powerful asset for future quantum technologies, from building better quantum computers to manipulating complex entangled states, all while using less energy and adapting to changes in real-time.

Technical Summary: Physics-Informed Neural Networks for Quantum Control

Problem Statement
Optimal Quantum Control (QC) is essential for manipulating quantum systems, ranging from state preparation to quantum sensing. Traditional QC techniques, such as Lyapunov control, optimal pulse sequences, and differentiable programming, often rely on complex optimization processes that are system-dependent and computationally cumbersome. While machine learning (ML) has been applied to quantum systems, standard Neural Networks (NNs) typically require large datasets to learn dynamics, which are often unavailable or difficult to generate for open quantum systems. Furthermore, integrating time evolution and control optimization within a single deep learning framework without preliminary data remains an unexplored challenge. The specific problem addressed is the inverse design of control functions for open quantum systems to achieve state-to-state transfer or state preservation with high fidelity, short evolution times, and low energy consumption, without relying on pre-existing training data.

Methodology
The authors introduce a novel architecture based on Physics-Informed Neural Networks (PINNs) to solve optimal control problems in open quantum systems. This approach is a data-free inverse modeling technique where the target dynamical behavior serves as the objective rather than a dataset.

System Formulation: The quantum system is modeled as an $n$ -dimensional non-autonomous dynamical system $\dot{x} = A(\lambda, u(t))x(t)$ , where $x(t)$ is the state vector (derived from the density matrix for open systems), $u(t)$ is the control vector, and $\lambda$ represents system parameters.
Neural Architecture: A feed-forward NN is employed as a universal approximator. The input is a discrete time vector $t$ $t$ , and the outputs are the state solution $N_x(t)$ $N_{x} (t)$ and the control solution $N_u(t)$ $N_{u} (t)$ .
- Activation Function: The hidden layers utilize a sine activation function, $\sin(\cdot)$ , chosen for its proven efficacy in solving nonlinear dynamical systems and high-dimensional partial differential equations.
- Hard Constraints: To ensure exact satisfaction of initial conditions ( $x(0)=x_0, u(0)=u_0$ ), the network outputs are parameterized as $x(t) = x(0) + f(t)N_x(t)$ and $u(t) = u(0) + f(t)N_u(t)$ , where $f(t) = 1 - e^{-t}$ satisfies $f(0)=0$ . This avoids numerical errors associated with soft constraints.
Loss Function: The network is trained by minimizing a composite loss function $L = L_{model} + L_{control} + L_{const} + L_{reg}$ $L = L_{m o d e l} + L_{co n t r o l} + L_{co n s t} + L_{r e g}$ :
- $L_{model}$ : Enforces the system dynamics (the master equation) by minimizing the residual $\|\dot{x} - A(\lambda, u)x\|^2$ . Time derivatives are computed via automatic differentiation.
- $L_{control}$ : Drives the system toward a target state $x_d$ (constant or time-dependent), weighted by a factor $\eta$ .
- $L_{const}$ : Incorporates additional physical constraints, such as probability conservation ( $Tr[\rho]=1$ ) or specific state populations.
- $L_{reg}$ : An $l_2$ -norm regularization term on network weights to prevent overfitting and encourage smooth, physically feasible control functions.
Optimization: The Adam optimizer is used to minimize the loss function. Training points are randomly perturbed during iterations to improve convergence.

Key Contributions and Results
The authors demonstrate the efficacy of this PINN framework through three distinct examples:

Two-Level System (TLS) for Gibbs State Preparation:
- Task: Drive a dissipative two-level system to a specific Gibbs (mixed) state.
- Result: The PINN discovered a control field $\xi(t)$ that achieved a fidelity of $F=0.99$ with the target state. The steady-state approach error was within 1% for all density matrix components.
- Comparison: The PINN solution outperformed an analytically optimized constant control solution, achieving better fidelity and demonstrating the ability to find non-trivial time-dependent controls.
Three-Level $\Lambda$ -System for State-to-State Transfer:
- Task: Transfer population from state $|1\rangle$ to $|2\rangle$ via a lossy intermediate state $|3\rangle$ while minimizing population in $|3\rangle$ and the pulse area (energy).
- Result: The PINN generated smooth control pulses ( $\Omega_p, \Omega_s$ ) achieving high fidelity ( $p_2 \approx 0.97$ ) in a significantly shorter transfer time ( $t_f = 2.0$ ) compared to standard methods like STIRAP ( $t_f = 35$ ), STIREP, and Inverse Engineering.
- Energy Efficiency: The control pulses exhibited the smallest pulse area ( $A=7.3$ ) compared to other methods, indicating lower energy consumption.
- Robustness: The PINN maintained high performance under two-photon detuning ( $\delta/2\pi = 0.2$ ) without retraining, whereas standard methods like STIRAP degrade significantly under such detuning. The method also successfully adapted to different initial states (coherent vs. mixed) without architectural changes.
Scalability and Generalization:
- Four-Level System: The framework successfully handled a four-level system, managing cross-talk to the new state and minimizing population in the lossy intermediate state, demonstrating scalability beyond the three-level case.
- Large Systems: The method was extended to non-interacting and interacting qubit systems (up to 5 qubits, $2^5=32$ levels). The PINN achieved final fidelities of 99.8% (non-interacting) and 99.1% (interacting), successfully minimizing the expectation value of the final Hamiltonian.

Significance and Claims
The paper claims that the proposed PINN architecture offers a flexible, data-free paradigm for quantum control that jointly solves the forward dynamics of open quantum systems and the inverse design of control functions. The authors highlight several advantages over standard techniques:

Adaptability: The method adapts to changes in physical parameters, initial conditions, and noise channels without requiring retraining or architectural modifications.
Efficiency: It achieves high-fidelity state transfer in shorter times with lower energy consumption (smaller pulse areas).
Robustness: The discovered control sequences are robust against parameter variations (e.g., detuning) that typically degrade standard protocols like STIRAP.
Generality: The approach is applicable to various quantum platforms, including atomic, optical, and solid-state systems, and scales to larger Hilbert spaces.

The authors conclude that PINNs represent a promising tool for complex quantum control tasks, including adiabatic quantum computing, quantum gate manipulation, and the initialization of entangled states in many-body systems.

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