Quantum state tomography (QST) is the gold standard for estimating the unknown state of quantum systems but suffers from the longstanding problem of exponentially growing measurements. Recent deep learning approaches alleviate the problem by training neural networks in a data-driven manner without theoretical convergence guarantees. They lack reliability in tomography quality and are restricted by retraining when generalized to extensive quantum systems with varying realizations in different environments. To address this issue, we propose the first generalized deep learning method for QST, named HD-QST, that leverages hybrid denoising priors with well-established convergence guarantees to fit extensive quantum states and environments. Hybrid denoising priors are achieved with the convex combination of analytic low-rank denoiser and neural network-based smooth denoiser dynamically determined via reinforcement learning. We demonstrate in theory that, when characterizing a quantum system, HD-QST guarantees a geometric convergence rate under the relaxed condition that the true density matrix is the fixed point of any one denoiser rather than the shared fixed point of multiple denoisers. Extensive experiments show that, compared with existing methods, HD-QST consistently obtains superior fidelity for pure and mixed quantum states in both single-device simulations and cross-device applications. Remarkably, it achieves state-of-the-art fidelity in QST of real-world W states by trapped ions and precise evaluation and comparison on cross-device QST on IBM quantum computers.
Quantum state tomography (QST) is the gold standard for estimating the unknown state of quantum systems but suffers from the longstanding problem of exponentially growing measurements. Recent deep learning approaches alleviate the problem by training neural networks in a data-driven manner without theoretical convergence guarantees. They lack reliability in tomography quality and are restricted by retraining when generalized to extensive quantum systems with varying realizations in different environments. To address this issue, we propose the first generalized deep learning method for QST, named HD-QST, that leverages hybrid denoising priors with well-established convergence guarantees to fit extensive quantum states and environments. Hybrid denoising priors are achieved with the convex combination of analytic low-rank denoiser and neural network-based smooth denoiser dynamically determined via reinforcement learning. We demonstrate in theory that, when characterizing a quantum system, HD-QST guarantees a geometric convergence rate under the relaxed condition that the true density matrix is the fixed point of any one denoiser rather than the shared fixed point of multiple denoisers. Extensive experiments show that, compared with existing methods, HD-QST consistently obtains superior fidelity for pure and mixed quantum states in both single-device simulations and cross-device applications. Remarkably, it achieves state-of-the-art fidelity in QST of real-world W states by trapped ions and precise evaluation and comparison on cross-device QST on IBM quantum computers.
Abstract Quantum state tomography is one of the core techniques in quantum information science, aiming to reconstruct the density matrix of an unknown quantum state from measurement data. To address the limitations of conventional quantum state reconstruction methods in terms of noise resistance and computational efficiency, this work proposes an innovative Residual-Convolutional Quantum Denoising Autoencoder model (RC-QDAE). The results demonstrate that the model can reconstruct physical density matrices with fidelities comparable to those achieved by traditional full-data methods while requiring only 80% of the tomographic measurement data, thereby significantly alleviating the dependence on measurement resources in large-scale quantum system characterization. In three-qubit systems, the reconstruction speed of the proposed model achieves an improvement of three orders of magnitude over the maximum likelihood estimation method, effectively overcoming the excessive computational overhead inherent to traditional approaches. Moreover, the model incorporates a joint training mechanism with multiple typical noise types during the training phase, exhibiting stronger robustness when processing cross-noise data. Further evaluations show that under different noise intensities, the model consistently produces density matrices with higher fidelities than those obtained by conventional methods, highlighting its superior noise resistance and generalization capability.
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
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.
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 tomography (QST), the task of estimating an unknown quantum state given measurement outcomes, is essential to building reliable quantum computing devices. Whereas computing the maximum-likelihood (ML) estimate corresponds to solving a finite-sum convex optimization problem, the objective function is not smooth nor Lipschitz, so most existing convex optimization methods lack sample complexity guarantees; moreover, both the sample size and dimension grow exponentially with the number of qubits in a QST experiment, so a desired algorithm should be highly scalable with respect to the dimension and sample size, just like stochastic gradient descent. In this paper, we propose a stochastic first-order algorithm that computes an ε-approximate ML estimate in O((DlogD)/ε2) iterations with O(D3) per-iteration time complexity, where D denotes the dimension of the unknown quantum state and ε denotes the optimization error. Our algorithm is an extension of Soft-Bayes to the quantum setup.