Accurate characterization of quantum states is crucial for the development and advancement of quantum technologies. Existing methods, including quantum state tomography and classical shadow estimation, often involve extensive experimental data gathered specifically for the state being characterized. This limitation hinders efficiency when dealing with multiple quantum states or when performing tasks like clustering or classification. The current research aims to address this challenge by developing a flexible neural network model that can be trained offline using simulated data from a representative set of states and measurements (a 'fiducial set'). This offline training allows the network to generalize to unseen quantum states sharing structural similarities with the training data, improving the efficiency of quantum state characterization in various applications. The ability to learn from simulated data eliminates the need for repeated experimental training for each new state, significantly accelerating the overall process.

Previous research has explored the use of neural networks for quantum state characterization, primarily focusing on training networks with experimental data from the target state. This approach, while effective, lacks flexibility and scalability when applied to multiple or diverse quantum states. The authors cite several relevant works using neural networks for state characterization tasks such as quantum state tomography, but highlight the limitation that these methods require retraining for each new quantum state. The current work draws inspiration from 'pretty good tomography' and generative query networks, adapting these concepts to create a more flexible and efficient learning framework.

The proposed model, Generative Query Network for Quantum states (GQNQ), is a neural network with two main components: a representation network and a generation network. The representation network processes measurement data (parametrization and outcome statistics) to produce a data-driven representation of the quantum state. The generation network then uses this representation to predict the outcome statistics of unperformed measurements. The network uses a generative query network architecture, adapted from classical image processing, to handle the task of learning the mapping from measurement data to outcome statistics. The training phase involves providing GQNQ with measurement data from a fiducial set of quantum states and measurements. This training can be performed offline using classically simulated data, which allows for faster and more efficient learning compared to methods that require experimental data for each state. The network parameters are optimized via gradient descent, using an Adam optimizer and batch gradient descent. The GQNQ does not require explicit parametrization of quantum states; it only needs to know which measurement data correspond to the same state. The methodology also includes details about the data generation procedures for various quantum systems (spin systems and harmonic oscillators), including the use of DMRG for simulating many-body systems. The testing phase evaluates the network's performance on unseen quantum states using metrics like classical fidelity. The paper details how the network can be adapted for online learning, where predictions are updated sequentially as new measurement data become available. The use of t-distributed Stochastic Neighbor Embedding (t-SNE) to visualize state representations and Gaussian Mixture Models (GMM) for clustering are also explained.

The paper demonstrates the effectiveness of GQNQ through extensive numerical experiments. For six-qubit systems (Ising and XXZ models, GHZ, and W states), high classical fidelities (close to 1) were achieved, even with noisy data from a limited number of measurements. The network's ability to generalize to unseen states within the same family is demonstrated by its performance across different parameter regimes and state types. In experiments with multi-qubit states (10, 20, and 50 qubits), GQNQ maintained high fidelities in ferromagnetic and antiferromagnetic regions of the Ising model, with smaller drops near the phase transition. Similar results were observed for the XXZ model, though fidelities were slightly lower in the XY phase due to increased quantum fluctuations. For continuous variable systems (harmonic oscillators), GQNQ performed well on squeezed thermal states, cat states, and GKP states, achieving high fidelities even in the presence of noise due to finite statistics or measurement errors. The state representations generated by GQNQ effectively cluster different types of states in unsupervised learning scenarios, as demonstrated by t-SNE visualizations and GMM clustering. Further, the state representations can be used in supervised learning models to classify states with high accuracy. Finally, the online learning capability of GQNQ is demonstrated by showing a gradual increase in fidelity with more incoming measurements in the case of cat states. The paper also shows that the network works fairly well even with non-informationally complete measurement sets.

The GQNQ model significantly advances quantum state characterization by offering a flexible and efficient approach. The ability to train offline using simulated data drastically reduces the time and resources required compared to methods relying on experimental data for each new state. The high fidelities achieved across diverse quantum systems and the successful application to clustering and classification demonstrate the model's robustness and versatility. The success in handling noisy and incomplete measurement data further enhances the practical applicability of the method. The results validate the intuition that generative models can capture the essential structural properties of quantum states, even with limited information. While the paper demonstrates strong performance, it also acknowledges that the effectiveness of GQNQ depends on the structural regularity of the quantum state family being considered; it performs less well on entirely arbitrary states. This dependence on state family structure is a key area for future research, as is a more thorough theoretical analysis of its capabilities and limitations.

This research presents GQNQ, a novel neural network architecture for flexible learning of quantum states. Its key innovation is the ability to train offline using simulated data, enabling efficient characterization of multiple quantum states and performing tasks such as clustering and classification. The high performance achieved in various numerical experiments highlights its potential to accelerate quantum technology development. Future work should focus on refining the criteria for determining which quantum state families are effectively learnable by GQNQ and potentially incorporating quantum-enhanced machine learning techniques to further improve performance.

The performance of GQNQ relies on the structural similarity between the training and test quantum states. The network might not perform well if the test states significantly differ from those in the training set. Additionally, while the paper demonstrates effectiveness with noisy data, further investigation into the robustness of the model against various types and levels of noise is warranted. Finally, a deeper theoretical understanding of the network's capacity and limitations is needed to guide future developments and applications.