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.
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.
We apply deep-neural-network-based techniques to quantum state classification and reconstruction. Our methods demonstrate high classification accuracies and reconstruction fidelities, even in the presence of noise and with little data. Using optical quantum states as examples, we first demonstrate how convolutional neural networks (CNNs) can successfully classify several types of states distorted by, e.g., additive Gaussian noise or photon loss. We further show that a CNN trained on noisy inputs can learn to identify the most important regions in the data, which potentially can reduce the cost of tomography by guiding adaptive data collection. Secondly, we demonstrate reconstruction of quantum-state density matrices using neural networks that incorporate quantum-physics knowledge. The knowledge is implemented as custom neural-network layers that convert outputs from standard feed-forward neural networks to valid descriptions of quantum states. Any standard feed-forward neural-network architecture can be adapted for quantum state tomography (QST) with our method. We present further demonstrations of our proposed QST technique with conditional generative adversarial networks (QST-CGAN) [Ahmed et al., Phys. Rev. Lett. 127, 140502 (2021)]. We motivate our choice of a learnable loss function within an adversarial framework by demonstrating that the QST-CGAN outperforms, across a range of scenarios, generative networks trained with standard loss functions. For pure states with additive or convolutional Gaussian noise, the QST-CGAN is able to adapt to the noise and reconstruct the underlying state. The QST-CGAN reconstructs states using up to two orders of magnitude fewer iterative steps than iterative and accelerated projected-gradient-based maximum-likelihood estimation (MLE) methods. We also demonstrate that the QST-CGAN can reconstruct both pure and mixed states from two orders of magnitude fewer randomly chosen data points than these MLE methods. Our paper opens possibilities to use state-of-the-art deep-learning methods for quantum state classification and reconstruction under various types of noise.
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.
Abstract Quantum state tomography (QST) is a widely employed technique for characterizing the state of a quantum system. However, it is plagued by two fundamental challenges: computational and experimental complexity grows exponentially with the number of qubits, rendering experimental implementation and data post-processing arduous even for moderately sized systems. Here, we introduce gradient-descent (GD) algorithms for the post-processing step of QST in discrete- and continuous-variable systems. To ensure physically valid state reconstruction at each iteration step of the algorithm, we use various density-matrix parameterizations: Cholesky decomposition, Stiefel manifold, and projective normalization. These parameterizations have the added benefit of enabling a rank-controlled ansatz, which simplifies reconstruction when there is prior information about the system. We benchmark the performance of our GD-QST techniques against state-of-the-art methods, including constrained convex optimization, conditional generative adversarial networks, and iterative maximum likelihood estimation. Our comparison focuses on time complexity, iteration counts, data requirements, state rank, and robustness against noise. We find that rank-controlled ansatzes in our stochastic mini-batch GD-QST algorithms effectively handle noisy and incomplete data sets, yielding significantly higher reconstruction fidelity than other methods. Simulations achieving full-rank seven-qubit QST in under three minutes on a standard laptop, with 18 GB of RAM and no dedicated GPU, highlight that GD-QST is computationally more efficient and outperforms other techniques in most scenarios, offering a promising avenue for characterizing noisy intermediate-scale quantum devices. Our Python code for GD-QST algorithms is publicly available at github.com/mstorresh/GD-QST .
Quantum state and process tomography constitute essential diagnostic tools in quantum information science, yet their standard formulations suffer from prohibitive computational scaling as the number of qubits grows. In this work, we introduce a physics-constrained conditional generative adversarial network that bypasses iterative constrained inversion by directly learning a forward generative mapping conditioned on Pauli expectation values. The generator embeds a differentiable Cholesky layer at its output, which enforces Hermiticity, positive semidefiniteness, and unit trace by construction. Our experiments reveal that the strength of the L1 penalty critically governs the emergence of GHZ coherence during training: an excessively large penalty postpones the coherence onset and yields a prolonged low-fidelity plateau, whereas an intermediate value enables the fastest stable convergence. Moreover, for high-temperature thermal states, an over-complete measurement basis proves necessary to prevent sustained late-stage fluctuations. By extending the same Cholesky constraint to the Choi-matrix representation, the framework naturally accommodates quantum process tomography. For systems with n≥6 qubits, the exponential growth of the underlying 2n×2n density matrix remains the fundamental bottleneck; we discuss how integrating tensor-network structures can contain the per-iteration cost while preserving reconstruction fidelity. Altogether, these results suggest that physically constrained generative learning offers a scalable and amortizable pathway toward data-driven tomography for noisy intermediate-scale quantum devices.
Tremendous progress has been witnessed in artificial intelligence, where neural network backed deep learning systems have been used, with applications in almost every domain. As a representative deep learning framework, Generative Adversarial Network (GAN) has been widely used for generating artificial images, text-to-image or image augmentation across areas of science, arts and video games. However, GANs are computationally expensive, sometimes computationally prohibitive. Furthermore, training GANs may suffer from convergence failure and modal collapse. Aiming at the acceleration of use cases for practical quantum computers, we propose QuGAN, a quantum GAN architecture that provides stable convergence, quantum-states based gradients and significantly reduced parameter sets. The QuGANarchitecture runs both the discriminator and the generator purely on quantum state fidelity and utilizes the swap test on qubits to calculate the values of quantum-based loss functions. Built on quantum layers, QuGAN achieves similar performance with a 94.98% reduction on the parameter set when compared to classical GANs. With the same number of parameters, addition-ally, QuGAN outperforms state-of-the-art quantum based GANsin the literature providing a 48.33% improvement in system performance compared to others attaining less than 0.5% in terms of similarity between generated distributions and original data sets.
In recent years, deep learning-based quantum state estimation (QSE) has attracted extensive attention, and the generative adversarial network (GAN) is a popular model in the field of deep learning. In this paper, we apply GANs into the field of QSE and design a custom neural network layer which makes the original network outputs satisfy quantum state constraints. It is worth noting that our designed custom neural-network layers are applicable to arbitrary feed-forward neural networks used for QSE. GANs are trained and tested by randomly generated pure state density matrices. Simulation results show that the designed neural network estimator is capable of achieving a fidelity higher than 99% for both eigenstates and superposition states.