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.
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.
Quantum state tomography (QST) is a fundamental technique in quantum information science, used to characterize the density matrix of an unknown quantum state. However, traditional QST methods are often computationally intensive, requiring a large number of measurements and sophisticated data analysis techniques. This paper proposes a novel approach utilizing self-organizing neural networks (SONNs) for direct quantum state reconstruction. The core idea is to train a SONN to learn the mapping between experimental measurement data and the corresponding density matrix, thereby bypassing the need for precise control over the quantum system and significantly reducing the computational burden. We detail the architecture of the SONN, the training process, and demonstrate the potential of this method to achieve accurate state reconstruction with fewer measurements. The proposed method offers a promising path towards more efficient and scalable QST, particularly for complex quantum systems.
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 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 compressed sensing is a fundamental technique for tomographic reconstruction of low-rank density matrices in informationally incomplete scenarios. However, when the available measurement data is insufficient, standard compressed sensing, like other algorithmic estimators such as maximum-likelihood, can yield imprecise or noisy estimates. To mitigate this, we propose a deep neural network-based post-processing to enhance the initial reconstruction. By treating the compressed sensing-estimated quantum state as a noisy input, the network performs a supervised denoising task. After the network is applied, a projection onto the space of feasible density matrices is performed to obtain an improved final state estimation. We demonstrate the effectiveness of our method through numerical experiments. Additionally, we explore iterative application of the inference process to further enhance performance, and we analyze the network’s generalization ability by testing its performance on states and noise models not seen during training.
The exact description of many-body quantum systems represents one of the major challenges in modern physics, because it requires an amount of computational resources that scales exponentially with the size of the system. Simulating the evolution of a state, or even storing its description, rapidly becomes intractable for exact classical algorithms. Recently, machine learning techniques, in the form of restricted Boltzmann machines, have been proposed as a way to efficiently represent certain quantum states with applications in state tomography and ground state estimation. Here, we introduce a practically usable deep architecture for representing and sampling from probability distributions of quantum states. Our representation is based on variational auto-encoders, a type of generative model in the form of a neural network. We show that this model is able to learn efficient representations of states that are easy to simulate classically and can compress states that are not classically tractable. Specifically, we consider the learnability of a class of quantum states introduced by Fefferman and Umans. Such states are provably hard to sample for classical computers, but not for quantum ones, under plausible computational complexity assumptions. The good level of compression achieved for hard states suggests these methods can be suitable for characterizing states of the size expected in first generation quantum hardware.
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.