Quantum state tomography (QST) is the art of reconstructing an unknown quantum state through measurements. It is a key primitive for developing quantum technologies. Neural network quantum state tomography (NNQST), which aims to reconstruct the quantum state via a neural network ansatz, is often implemented via a basis-dependent cross-entropy loss function. State-of-the-art implementations of NNQST are often restricted to characterizing a particular subclass of states, to avoid an exponential growth in the number of required measurement settings. To provide a more broadly applicable method for efficient state reconstruction, we present “neural-shadow quantum state tomography” (NSQST)—an alternative neural network-based QST protocol that uses infidelity as the loss function. The infidelity is estimated using the classical shadows of the target state. Infidelity is a natural choice for training loss, benefiting from the proven measurement sample efficiency of the classical shadow formalism. Furthermore, NSQST is robust against various types of noise without any error mitigation. We numerically demonstrate the advantage of NSQST over NNQST at learning the relative phases of three target quantum states of practical interest, as well as the advantage over direct shadow estimation. NSQST greatly extends the practical reach of NNQST and provides a novel route to effective quantum state tomography. Published by the American Physical Society 2024
Quantum state tomography (QST) is the art of reconstructing an unknown quantum state through measurements. It is a key primitive for developing quantum technologies. Neural network quantum state tomography (NNQST), which aims to reconstruct the quantum state via a neural network ansatz, is often implemented via a basis-dependent cross-entropy loss function. State-of-the-art implementations of NNQST are often restricted to characterizing a particular subclass of states, to avoid an exponential growth in the number of required measurement settings. To provide a more broadly applicable method for efficient state reconstruction, we present “neural-shadow quantum state tomography” (NSQST)—an alternative neural network-based QST protocol that uses infidelity as the loss function. The infidelity is estimated using the classical shadows of the target state. Infidelity is a natural choice for training loss, benefiting from the proven measurement sample efficiency of the classical shadow formalism. Furthermore, NSQST is robust against various types of noise without any error mitigation. We numerically demonstrate the advantage of NSQST over NNQST at learning the relative phases of three target quantum states of practical interest, as well as the advantage over direct shadow estimation. NSQST greatly extends the practical reach of NNQST and provides a novel route to effective quantum state tomography. Published by the American Physical Society 2024
We investigate the advantages of using autoregressive neural quantum states as ansatze for classical shadow tomography to improve its predictive power. We introduce a novel estimator for optimizing the cross-entropy loss function using classical shadows, and a new importance sampling strategy for estimating the loss gradient during training using stabilizer samples collected from classical shadows. We show that this loss function can be used to achieve stable reconstruction of GHZ states using a transformer-based neural network trained on classical shadow measurements. This loss function also enables the training of neural quantum states representing purifications of mixed states. Our results show that the intrinsic capability of autoregressive models in representing physically well-defined density matrices allows us to overcome the weakness of Pauli-based classical shadow tomography in predicting both high-weight observables and nonlinear observables such as the purity of pure and mixed states.
Current algorithms for quantum state tomography (QST) are costly both on the experimental front, requiring measurement of many copies of the state, and on the classical computational front, needing a long time to analyze the gathered data. Here, we introduce neural adaptive quantum state tomography (NAQT), a fast, flexible machine-learning-based algorithm for QST that adapts measurements and provides orders of magnitude faster processing while retaining state-of-the-art reconstruction accuracy. As in other adaptive QST schemes, measurement adaptation makes use of the information gathered from previous measured copies of the state to perform a targeted sensing of the next copy, maximizing the information gathered from that next copy. Our NAQT approach allows for a rapid and seamless integration of measurement adaptation and statistical inference, using a neural-network replacement of the standard Bayes’ update, to obtain the best estimate of the state. Our algorithm, which falls into the machine learning subfield of “meta-learning” (in effect “learning to learn” about quantum states), does not require any ansatz about the form of the state to be estimated. Despite this generality, it can be retrained within hours on a single laptop for a two-qubit situation, which suggests a feasible time-cost when extended to larger systems and potential speed-ups if provided with additional structure, such as a state ansatz.
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
Abstract Current algorithms for quantum state tomography (QST) are costly both on the experimental front, requiring measurement of many copies of the state, and on the classical computational front, needing a long time to analyze the gathered data. Here, we introduce neural adaptive quantum state tomography (NAQT), a fast, flexible machine-learning-based algorithm for QST that adapts measurements and provides orders of magnitude faster processing while retaining state-of-the-art reconstruction accuracy. As in other adaptive QST schemes, measurement adaptation makes use of the information gathered from previous measured copies of the state to perform a targeted sensing of the next copy, maximizing the information gathered from that next copy. Our NAQT approach allows for a rapid and seamless integration of measurement adaptation and statistical inference, using a neural-network replacement of the standard Bayes’ update, to obtain the best estimate of the state. Our algorithm, which falls into the machine learning subfield of “meta-learning” (in effect “learning to learn” about quantum states), does not require any ansatz about the form of the state to be estimated. Despite this generality, it can be retrained within hours on a single laptop for a two-qubit situation, which suggests a feasible time-cost when extended to larger systems and potential speed-ups if provided with additional structure, such as a state ansatz.
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 (QST) aiming at reconstructing the density matrix of a quantum state plays an important role in various emerging quantum technologies. Recognizing the challenges posed by imperfect measurement data, we develop a unified neural network (NN)-based approach for QST under constrained measurement scenarios, including limited measurement copies, incomplete measurements, and noisy measurements. Through comprehensive comparison with other estimation methods, we demonstrate that our method improves the estimation accuracy in scenarios with limited measurement resources, showcasing notable robustness in noisy measurement settings. These findings highlight the capability of NNs to enhance QST with constrained measurements.