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Arbitrary state preparation in quantum harmonic oscillators using neural networks

This paper presents a neural network-based methodology that enables the rapid, non-iterative preparation of arbitrary high-dimensional quantum states in harmonic oscillators coupled to an auxiliary qubit, achieving high fidelities by directly mapping target states to laser control parameters.

Original authors: Nicolas Parra-A, Vladimir Vargas-Calderón, Herbert Vinck-Posada

Published 2026-06-29

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CC BY 4.0

Original authors: Nicolas Parra-A, Vladimir Vargas-Calderón, Herbert Vinck-Posada

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 have a very delicate, invisible musical instrument called a Quantum Harmonic Oscillator. Think of it like a tiny, perfect bell that can vibrate at different levels of energy. In the world of quantum computing, we want to "ring" this bell in very specific, complex patterns to store information. These patterns are called quantum states.

The problem is that ringing this bell exactly the way you want is incredibly hard. Usually, if you want a new pattern, you have to spend hours or days calculating the perfect sequence of taps to get it right. It's like trying to tune a radio by guessing the frequency for every single song you want to hear.

The Big Idea: A "Magic" Predictor
This paper introduces a new way to do this using an Artificial Intelligence (AI) brain, specifically a type called a Neural Network.

Instead of calculating the solution from scratch every time, the researchers trained this AI brain once. Now, if you tell the AI, "I want the bell to ring in this specific pattern," it instantly knows exactly how to tap the bell to get there. It doesn't need to think or calculate; it just gives the answer immediately.

How It Works: The DJ and the Assistant
To make the bell ring correctly, the researchers use a clever trick involving two things:

The Bell (The Oscillator): The main instrument we want to control.
The Assistant (An Auxiliary Qubit): A tiny helper particle that acts like a second instrument.

They use lasers to "tap" both the bell and the assistant. But here's the catch: they don't just tap them randomly. They change the phase (the timing or rhythm) of these taps very quickly, like a DJ scratching a record or a conductor changing the tempo of an orchestra in split seconds.

The AI's job is to be the Conductor.

Input: You give the AI the sheet music (the target state you want).
Output: The AI instantly writes out the exact sequence of tempo changes and taps needed for both the bell and the assistant.
The Result: The bell ends up playing the exact note you asked for, and the assistant returns to being quiet.

The "Training" vs. "Playing" Analogy

The Old Way (Optimization): Imagine you are a chef trying to bake a perfect cake. For every new customer who wants a cake, you have to spend three hours testing ingredients, baking, tasting, and adjusting the recipe. This is slow and expensive.
The New Way (Neural Network): Imagine you spend one week in the kitchen testing thousands of recipes and writing them all down in a giant cookbook. Once the cookbook is done, if a customer orders a cake, you just look up the recipe and bake it in 10 minutes. You don't need to test anything new.

In this paper, the "cookbook" is the trained Neural Network. The "baking" is the actual experiment on the quantum computer.

What Did They Achieve?
The researchers tested this AI "Conductor" on different sizes of "bells":

Simple Bell (Qubit): They could ring it with 99.99% accuracy.
Medium Bell (Qutrit): They achieved 99.5% accuracy.
Complex Bell (Qudit, 4 levels): They reached 98.9% accuracy.

These numbers are incredibly high, meaning the AI is almost perfect at predicting the right taps.

Why Is This a Big Deal?
The main advantage is speed and efficiency.

Old Method: If you wanted to prepare 1,000 different quantum states, you might have to wait days or weeks because you have to solve a complex math problem for each one.
New Method: Once the AI is trained, it can spit out the instructions for 1,000 different states in a matter of seconds. It takes about 100 milliseconds (a blink of an eye) to get the instructions for a new state.

The Limits
The paper notes that this works best for systems that can be simulated on a computer. As the "bells" get more complex (more levels), the AI needs to be bigger and smarter, and it takes more "practice" (training data) to get it right. Also, in the real world, things like noise or imperfect lasers might make the results slightly less perfect than in the computer simulation, but the method is still very promising.

In Summary
This paper shows that we can train an AI to be a master conductor for quantum instruments. Instead of struggling to figure out how to play a new song every time, the AI instantly tells us the exact rhythm and timing needed to make the quantum system sing the right note, making quantum computing faster and more practical.

Technical Summary: Arbitrary State Preparation in Quantum Harmonic Oscillators Using Neural Networks

Problem Statement
Preparing specific quantum states is a fundamental prerequisite for quantum algorithms, machine learning, and simulations. While various methods exist to determine the physical interactions required to transition a system from an initial state to a target state (e.g., Krotov's methods, quantum steering, reinforcement learning), they typically require solving a separate optimization problem for each individual target state. This per-instance optimization becomes computationally prohibitive for applications requiring rapid or repeated state preparation across many different targets, such as in quantum machine learning. Furthermore, analytical techniques often yield distinct equations for each target, limiting their practicality. The specific challenge addressed in this work is the preparation of arbitrary states in $n$ -dimensional systems (qudits) realized via a truncated quantum harmonic oscillator (HO), a system that is not fully controllable under a finite set of steps without auxiliary interaction.

Methodology
The authors propose a general methodology utilizing a feed-forward neural network to predict control parameters for an HO coupled to an auxiliary qubit. The system is modeled as a Jaynes–Cummings-type interaction, where both the HO and the qubit are driven by lasers with time-dependent phase modulation.

Physical Model: The system evolves under a Hamiltonian where the qubit and HO interact with coupling constant $g$ . The control consists of a sequence of $N$ pulses with piecewise-constant phases ( $\phi_i$ for the HO, $\varphi_i$ for the qubit) and switch times ( $t_i$ ). The amplitudes are assumed constant and equal. The goal is to evolve the initial state $|0, 0\rangle$ to a separable state where the HO is in a desired target state $|\psi_{target}\rangle$ within a truncated $n$ -level manifold.
Neural Network Architecture: The network takes the target state as input and outputs the control parameters $\Theta = \{\phi_i, \varphi_i, t_i\}$ $Θ = {ϕ_{i}, φ_{i}, t_{i}}$ .
- Input Representation: The target state is encoded as the expectation values of the $d = n^2 - 1$ generators of the $SU(n)$ basis.
- Output: The network predicts the phases and switch times for a sequence of $N$ pulses.
Training Pipeline: The network is trained using stochastic gradient descent to minimize the average infidelity ( $1 - F$ ) over a dataset of random target states sampled from the Haar measure. The infidelity is defined as the overlap between the target state and the state obtained after tracing out the qubit degrees of freedom from the final evolved state.
Key Distinction: Unlike conventional optimal control methods that iterate to find parameters for a specific target, this framework shifts the computational burden to a one-time offline training phase. Once trained, the network performs a single forward pass to generate control parameters for any new target state within the trained dimension.

Key Contributions and Results
The paper presents numerical results for preparing arbitrary states in qubits ( $n=2$ ), qutrits ( $n=3$ ), and qudits ( $n=4$ ) using the proposed neural network approach.

Performance Metrics: The method achieves high average fidelities across 1000 random Haar-distributed target states:
- Qubits ( $n=2$ ): 99.99% fidelity (0.01% infidelity) using $N=9$ pulses.
- Qutrits ( $n=3$ ): 99.5% fidelity (0.5% infidelity) using $N=11$ pulses.
- Qudits ( $n=4$ ): 98.9% fidelity (1.1% infidelity) using $N=13$ pulses.
Pulse Sequence Analysis: The study confirms that increasing the number of pulses generally reduces infidelity, though an optimal point exists beyond which additional pulses yield diminishing returns. The learned phase-switching patterns are non-trivial and do not follow simple analytical rules, indicating the network learns complex control strategies to compensate for accumulated errors.
Visualization: For qubits, the trajectory on the Bloch sphere shows that early pulses may drive the system away from the target to intermediate states before final pulses align the trajectory. For qutrits and qudits, Wigner function analysis demonstrates that the network successfully reproduces the dominant phase-space support and nonclassical structures (e.g., interference patterns and negative regions) of the target states, including Bell states encoded in the oscillator subspace.

Significance and Claims
The authors claim that this work provides a practical solution for "on-demand" state preparation. The primary advantage is the elimination of per-instance optimization, reducing the online inference time to approximately 100 ms on a standard CPU. This contrasts sharply with iterative optimal control methods (like Krotov or GRAPE), which can take minutes to hours per target state.

The paper emphasizes that the method is particularly suited for scenarios requiring rapid encoding of information into quantum states, such as quantum machine learning workflows where the target state changes with every input. The approach is grounded in the theoretical controllability of Jaynes–Cummings dynamics within a truncated manifold (Pinna and Panati), ensuring that the predicted protocols correspond to physically preparable states within the model's constraints.

Limitations and Future Directions
The authors modestly note that the observed increase in infidelity with dimension $n$ reflects the growing complexity of the mapping, which can theoretically be mitigated by increasing network size and pulse count. However, a more fundamental constraint is the cost of generating training data via classical simulation, which scales poorly with the Hilbert space dimension. The paper acknowledges that for larger dimensions (e.g., bosonic codes requiring tens or hundreds of levels), direct simulation may become a bottleneck, necessitating future strategies like tensor networks or hardware-in-the-loop training. Additionally, the current results assume ideal conditions; future work must address open-system dynamics, decoherence, and hardware-specific non-idealities (e.g., finite bandwidth, phase jitter) to determine the method's robustness in experimental settings.

Technical Summary: Arbitrary State Preparation in Quantum Harmonic Oscillators Using Neural Networks

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