Neural networks (NNs) representing quantum states are typically trained using Markov chain Monte Carlo based methods. However, unless specifically designed, such samplers only consist of local moves, making the slow-mixing problem prominent even for extremely simple quantum states. Here, we propose to use mode-assisted training that provides global information via the modes of the NN distribution. Applied to quantum state tomography using restricted Boltzmann machines, this method improves the quality of reconstructed quantum states by orders of magnitude. The method is applicable to other types of NNs and may efficiently tackle problems previously unmanageable.
Neural networks (NNs) representing quantum states are typically trained using Markov chain Monte Carlo based methods. However, unless specifically designed, such samplers only consist of local moves, making the slow-mixing problem prominent even for extremely simple quantum states. Here, we propose to use mode-assisted training that provides global information via the modes of the NN distribution. Applied to quantum state tomography using restricted Boltzmann machines, this method improves the quality of reconstructed quantum states by orders of magnitude. The method is applicable to other types of NNs and may efficiently tackle problems previously unmanageable.
The promise of quantum neural nets, which utilize quantum effects to model complex data sets, has made their development an aspirational goal for quantum machine learning and quantum computing in general. Here we provide methods of training quantum Boltzmann machines. Our work generalizes existing methods and provides additional approaches for training quantum neural networks that compare favorably to existing methods. We further demonstrate that quantum Boltzmann machines enable a form of partial quantum state tomography that further provides a generative model for the input quantum state. Classical Boltzmann machines are incapable of this. This verifies the long-conjectured connection between tomography and quantum machine learning. Finally, we prove that classical computers cannot simulate our training process in general unless BQP=BPP, provide lower bounds on the complexity of the training procedures and numerically investigate training for small nonstoquastic Hamiltonians.
Sampling problems are widely regarded as the task for which quantum computers can most readily provide a quantum advantage. Leveraging this feature, the quantum-enhanced Markov chain Monte Carlo [Layden , Nature 619, 282–287 (2023)] has been proposed recently, where sampling from a quantum computer is used as a proposal distribution and convergence to a target distribution is accelerated. However, guaranteeing convergence to the target distribution typically forces one to impose restrictive symmetry constraints on the quantum circuit, which makes it hard to design good proposal distributions and prevents making full use of the advantage of a quantum computer. We explore a hybrid quantum-classical Markov chain Monte Carlo framework that combines a quantum circuit with a generative neural sampler (GNS). The GNS is trained on quantum samples and acts as a classical surrogate to efficiently emulate quantum outputs, thereby lifting circuit constraints. We apply this method to Boltzmann sampling of spin glasses using proposals trained with a quantum approximate optimization algorithm circuit. This approach outperforms conventional methods, showing a polynomial speedup (over cubic) in spectral gap compared with uniform proposals. Notably, it maintains similar acceleration even without parameter optimization. These results establish the method as a viable sampling-based quantum algorithm for noisy intermediate-scale quantum devices and highlight its potential for solving practical problems with quantum computation.
Multi-qubit state tomography is a key problem in quantum information technology, which has been studied extensively. In this work, we focus on multi-qubit state tomography based on neural network estimators and typical conventional estimation approaches. For multi-qubit pure states, fully connected neural networks and restricted Boltzmann machine networks are designed, respectively, to carry out state tomography of 2-qubit (low-dimensional) systems and 5-qubit (high-dimensional) systems. Comparisons are made with maximum likelihood estimation and least squares estimation, where performance indicators are selected as reconstruction accuracy, time cost and the number of parameters. Simulation results indicate that intelligent approaches have significant advantages over conventional approaches for state tomography of low-dimensional systems; for high-dimensional systems, however, the conventional approach is still dominant when the measurement is complete, while the restricted Boltzmann machine network can achieve higher reconstruction accuracy when the measurement is incomplete.
In the present work, we present a study on Generative Neural Networks applied to the problem of Quantum State Tomography (QST). Our objective was to gain understanding of the use of various Neural Networks and obtain insights on how to apply them to the QST problem, in order to select an approach with promising application. To achieve this, we investigated various networks and explored two cases: Restricted Boltzmann Machines (RBMs) and Conditional Generative Adversarial Networks (cGANs); RBM was chosen for being a pioneer in the context of QST, while cGAN was selected due to its broad successful applicability in various problems, and having been the subject of a recent study in the field, representing, so far, the only application of this type of network to the QST problem. Our case study involved simulations of a 4-qubit quantum state and real data from an experiment with highly entangled 8-qubits. For the first case, we used a well-known Python package (QuCumber, employing an RBM), and also adapted an existing network (a cGAN) to our specific state, achieving successful reconstruction with relatively low training and processing time. However, in the second case, when applying the cGAN to the 8-qubit data, we did not succeed in the reconstruction, possibly due to the limited expressiveness of the network for this scenario or the high processing time required to train it, since we relied only on a CPU and limited computational resources. In our study, we discuss the relevance of these networks to the QST problem and speculate on the next interesting steps at the intersection between Quantum Information Theory and Machine Learning.
The purpose of quantum state tomography (QST) is to obtain a complete quantum state by reconstructing a density matrix from experimental data and therefore gives researchers a powerful tool to analyze complex synthetic quantum systems. In the past decades, many methods have been developed to improve the efficiency of quantum state tomography (QST), such as old methods: compress sensing on almost pure state and low-rank systems, new method: using restricted Boltzmann machine on systems of quantum many-body. However, they are both limited to some certain systems. In this paper, we introduce an universal method using deep neural networks based on stacked denoising autoencoder, which probably can help finish QST on all types of systems with a much smaller number of measurements and therefore is very efficient.
Abstract Quantum state tomography is a crucial technique for characterizing the state of a quantum system, which is essential for many applications in quantum technologies. In recent years, there has been growing interest in leveraging neural networks to enhance the efficiency and accuracy of quantum state tomography. However, versatile methods that are broadly applicable across diverse reconstruction scenarios remain relatively underexplored. In this paper, we present two neural network-based approaches for both pure and mixed quantum state tomography: Restricted Feature Based Neural Network and Mixed States Neural Network, evaluate its effectiveness in comparison to existing neural network-based methods. We demonstrate that our proposed methods can achieve state-of-the-art results in reconstructing mixed quantum states from experimental data. Our work highlights the potential of neural networks in facilitating the development of quantum technologies. Source code is publicly available at https://github.com/luutn2002/uni-qst