Quantum state tomography (QST) can be posed as an optimization problem, where the goal is to find the quantum state ρ that makes the observed data more likely. Obtaining the complete description of a quantum state is crucial for many quantum informational and computing tasks, and various approaches have been proposed to solve this fundamental problem. This thesis aims at presenting two models based on feedforward neural network architecture for QST. The first model is based on a multi-layer perceptron network, while the second one is based on autoregressive neural networks. We show that the perceptron model demonstrates less overfit compared to the standard iterative maximum likelihood model, however, the number of parameters in the network increases exponentially with respect to the system size. In contrast to the perceptron model, we show that the autoregressive model scales efficiently since the number of parameters grows polynomially instead of exponentially, making it suitable for QST of many-qubit systems. For the perceptron model, we presented a reconstruction of an engineered 4-qubit W state and an arbitrary optical superposition of up to 2 photons. On the other hand, using the autoregressive model, we performed QST of a 20-qubit engineered quantum state of trapped ions observed in an experiment. To the best of our knowledge, our results are the first of its kind in performing QST of complex and highly entangled out-of-equilibrium states produced by dynamics of an Ising-type Hamiltonian, engineered via laser fields on 1D trapped ion systems
Quantum state tomography (QST) can be posed as an optimization problem, where the goal is to find the quantum state ρ that makes the observed data more likely. Obtaining the complete description of a quantum state is crucial for many quantum informational and computing tasks, and various approaches have been proposed to solve this fundamental problem. This thesis aims at presenting two models based on feedforward neural network architecture for QST. The first model is based on a multi-layer perceptron network, while the second one is based on autoregressive neural networks. We show that the perceptron model demonstrates less overfit compared to the standard iterative maximum likelihood model, however, the number of parameters in the network increases exponentially with respect to the system size. In contrast to the perceptron model, we show that the autoregressive model scales efficiently since the number of parameters grows polynomially instead of exponentially, making it suitable for QST of many-qubit systems. For the perceptron model, we presented a reconstruction of an engineered 4-qubit W state and an arbitrary optical superposition of up to 2 photons. On the other hand, using the autoregressive model, we performed QST of a 20-qubit engineered quantum state of trapped ions observed in an experiment. To the best of our knowledge, our results are the first of its kind in performing QST of complex and highly entangled out-of-equilibrium states produced by dynamics of an Ising-type Hamiltonian, engineered via laser fields on 1D trapped ion systems
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.
Multi-qubit state tomography is a key problem in quantum information technology, which has been studied extensively. In this work, we focus on multi-qubit state tomography based on neural network estimators and typical conventional estimation approaches. For multi-qubit pure states, fully connected neural networks and restricted Boltzmann machine networks are designed, respectively, to carry out state tomography of 2-qubit (low-dimensional) systems and 5-qubit (high-dimensional) systems. Comparisons are made with maximum likelihood estimation and least squares estimation, where performance indicators are selected as reconstruction accuracy, time cost and the number of parameters. Simulation results indicate that intelligent approaches have significant advantages over conventional approaches for state tomography of low-dimensional systems; for high-dimensional systems, however, the conventional approach is still dominant when the measurement is complete, while the restricted Boltzmann machine network can achieve higher reconstruction accuracy when the measurement is incomplete.
Quantum state tomography (QST) is a crucial ingredient for almost all aspects of experimental quantum information processing. As an analog of the “imaging” technique in quantum settings, QST is born to be a data science problem, where machine learning techniques, noticeably neural networks, have been applied extensively. We build and demonstrate an optical neural network (ONN) for photonic polarization qubit QST. The ONN is equipped with built-in optical nonlinear activation functions based on electromagnetically induced transparency. The experimental results show that our ONN can determine the phase parameter of the qubit state accurately. As optics are highly desired for quantum interconnections, our ONN-QST may contribute to the realization of optical quantum networks and inspire the ideas combining artificial optical intelligence with quantum information studies.
A prerequisite to the successful development of quantum computers and simulators is precise understanding of the physical processes occurring therein, which can be achieved by measuring the quantum states that they produce.However, the resources required for traditional quantum state estimation scale exponentially with the system size, highlighting the need for alternative approaches.Here, we demonstrate an efficient method for reconstruction of significantly entangled multiqubit quantum states.Using a variational version of the matrix-product-state ansatz, we perform the tomography (in the pure-state approximation) of quantum states produced in a 20-qubit trapped-ion Ising-type quantum simulator, using the data acquired in only 27 bases, with 1000 measurements in each basis.We observe superior statereconstruction quality and faster convergence compared to the methods based on neural-network quantum state representations: restricted Boltzmann machines and feed-forward neural networks with autoregressive architecture.Our results pave the way toward efficient experimental characterization of complex states produced by the quench dynamics of many-body quantum systems.
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 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.