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Machine learning assisted quantum state tomography: a case-study using conditional generative adversarial networks

Silvio Jonas da Silva Júnior

Published May 29, 2024• 0 citations
Publisher
DOI
10.11606/d.43.2024.tde-12062025-180055

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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.
Quantum state tomography (QST) is a challenging task in intermediate-scale quantum devices. Here, we apply conditional generative adversarial networks (CGANs) to QST. In the CGAN framework, two dueling neural networks, a generator and a discriminator, learn multimodal models from data. We augment a CGAN with custom neural-network layers that enable conversion of output from any standard neural network into a physical density matrix. To reconstruct the density matrix, the generator and discriminator networks train each other on data using standard gradient-based methods. We demonstrate that our QST-CGAN reconstructs optical quantum states with high fidelity, using orders of magnitude fewer iterative steps, and less data, than both accelerated projected-gradient-based and iterative maximum-likelihood estimation. We also show that the QST-CGAN can reconstruct a quantum state in a single evaluation of the generator network if it has been pretrained on similar quantum states.
Generative adversarial networks represent a powerful tool for classical machine learning: a generator tries to create statistics for data that mimics those of a true data set, while a discriminator tries to discriminate between the true and fake data. The learning process for generator and discriminator can be thought of as an adversarial game, and under reasonable assumptions, the game converges to the point where the generator generates the same statistics as the true data and the discriminator is unable to discriminate between the true and the generated data. This Letter introduces the notion of quantum generative adversarial networks, where the data consist either of quantum states or of classical data, and the generator and discriminator are equipped with quantum information processors. We show that the unique fixed point of the quantum adversarial game also occurs when the generator produces the same statistics as the data. Neither the generator nor the discriminator perform quantum tomography; linear programing drives them to the optimal. Since quantum systems are intrinsically probabilistic, the proof of the quantum case is different from-and simpler than-the classical case. We show that, when the data consist of samples of measurements made on high-dimensional spaces, quantum adversarial networks may exhibit an exponential advantage over classical adversarial networks.