Abstract We used deep learning techniques to construct various models for reconstructing quantum states from a given set of coincidence measurements. Through simulations, we have demonstrated that our approach generates functionally equivalent reconstructed states for a wide range of pure and mixed input states. Compared with traditional methods, our system offers the advantage of faster speed. Additionally, by training our system with measurement results containing simulated noise sources, the system shows a significant improvement in average fidelity compared with typical reconstruction methods. We also found that constraining the variational manifold to physical states, i.e., positive semi-definite density matrices, greatly enhances the quality of the reconstructed states in the presence of experimental imperfections and noise. Finally, we validated the correctness and superiority of our model by using data generated on IBM Quantum Platform, a real quantum computer.
Abstract We used deep learning techniques to construct various models for reconstructing quantum states from a given set of coincidence measurements. Through simulations, we have demonstrated that our approach generates functionally equivalent reconstructed states for a wide range of pure and mixed input states. Compared with traditional methods, our system offers the advantage of faster speed. Additionally, by training our system with measurement results containing simulated noise sources, the system shows a significant improvement in average fidelity compared with typical reconstruction methods. We also found that constraining the variational manifold to physical states, i.e., positive semi-definite density matrices, greatly enhances the quality of the reconstructed states in the presence of experimental imperfections and noise. Finally, we validated the correctness and superiority of our model by using data generated on IBM Quantum Platform, a real quantum computer.
Abstract We build a general quantum state tomography framework that makes use of machine learning techniques to reconstruct quantum states from a given set of coincidence measurements. For a wide range of pure and mixed input states we demonstrate via simulations that our method produces functionally equivalent reconstructed states to that of traditional methods with the added benefit that expensive computations are front-loaded with our system. Further, by training our system with measurement results that include simulated noise sources we are able to demonstrate a significantly enhanced average fidelity when compared to typical reconstruction methods. These enhancements in average fidelity are also shown to persist when we consider state reconstruction from partial tomography data where several measurements are missing. We anticipate that the present results combining the fields of machine intelligence and quantum state estimation will greatly improve and speed up tomography-based quantum experiments.
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.
We study the performance of efficient quantum state tomography methods based on neural-network quantum states using measured data from a two-photon experiment. Machine-learning-inspired variational methods provide a promising route towards scalable state characterization for quantum simulators. While the power of these methods has been demonstrated on synthetic data, applications to real experimental data remain scarce. We benchmark and compare several such approaches by applying them to measured data from an experiment producing two-qubit entangled states. We find that in the presence of experimental imperfections and noise, confining the variational manifold to physical states, i.e., to positive semidefinite density matrices, greatly improves the quality of the reconstructed states but renders the learning procedure more demanding. Including additional, possibly unjustified, constraints, such as assuming pure states, facilitates learning, but also biases the estimator.
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.
Reconstructing quantum states from measurement data represents a formidable challenge in quantum information science, especially as system sizes grow beyond the reach of traditional tomography methods. While recent studies have explored quantum machine learning (QML) for quantum state tomography (QST), nearly all rely on idealized assumptions, such as direct access to the unknown quantum state as quantum data input, which are incompatible with current hardware constraints. In this work, we present a QML-based tomography protocol that operates entirely on classical measurement data and is fully executable on noisy intermediate-scale quantum (NISQ) devices. Our approach employs a variational quantum circuit trained to reconstruct quantum states based solely on measurement outcomes. We test the method in simulation, achieving high-fidelity reconstructions of diverse quantum states, including GHZ states, spin chain ground states, and states generated by random circuits. The protocol is then validated on quantum hardware from IBM and IonQ. Additionally, we demonstrate accurate tomography is possible using incomplete measurement bases, a crucial step towards scaling up our protocol. Our results in various scenarios illustrate successful state reconstruction with fidelity reaching 90% or higher. To our knowledge, this is the first QML-based tomography scheme that has been implemented on real quantum processors using exclusively classical measurements. This work establishes the feasibility of QML-based tomography on current quantum platforms and offers a scalable pathway for practical quantum state reconstruction.
Quantum noise fundamentally limits the utility of near-term quantum devices, making error mitigation essential for practical quantum computation. While traditional quantum error correction codes require substantial qubit overhead and complex syndrome decoding, we propose a machine learning approach that directly reconstructs clean quantum states from noisy density matrices without additional qubits. We formulate quantum noise reduction as a supervised learning problem using a convolutional neural network (CNN) autoencoder architecture with a novel fidelity-aware composite loss function. Our method is trained and evaluated on a comprehensive synthetic dataset of 10,000 density matrices derived from random 5-qubit quantum circuits, encompassing five noise types (depolarizing, amplitude damping, phase damping, bit-flip, and mixed noise) across four intensity levels (0.05-0.20). The CNN successfully reconstructs quantum states across all noise conditions, achieving an average fidelity improvement from 0.298 to 0.774 (Δ = 0.476). Notably, the model demonstrates superior performance on complex mixed noise scenarios and higher noise intensities, with mixed noise showing the highest corrected fidelity (0.807) and improvement (0.567). The approach effectively preserves both diagonal elements (populations) and off-diagonal elements (quantum coherences), making it suitable for entanglement-dependent quantum algorithms. While phase damping presents fundamental information-theoretic limitations, our results suggest that CNN-based density matrix reconstruction offers a promising, resource-efficient alternative to traditional quantum error correction for NISQ-era devices. This data-driven approach could enable practical quantum advantage with fewer physical qubits than conventional error correction schemes require.