Characterizing correlations in a quantum system on the basis of the results of the projective measurements can be performed with different means including the calculation of the classical mutual information. Generally, estimating such information-entropy-based quantities requires having complete statistics of the system's states. Here we explore the possibility to reconstruct the classical mutual information and specific entropy of a quantum system with a neural network approach on the basis of a limited number of projective measurements. As a prominent example we consider the antiferromagnetic quantum Ising model in transverse and longitudinal magnetic fields which is in demand in both condensed matter physics and quantum computing. We show that the neural network approach gives reliable estimates of the classical mutual information even in the case of paramagnetic wave functions delocalized in the state space. In addition, the phase diagram of the considered quantum system is reconstructed with a special focus on discriminating various types of disordered states.
Characterizing correlations in a quantum system on the basis of the results of the projective measurements can be performed with different means including the calculation of the classical mutual information. Generally, estimating such information-entropy-based quantities requires having complete statistics of the system's states. Here we explore the possibility to reconstruct the classical mutual information and specific entropy of a quantum system with a neural network approach on the basis of a limited number of projective measurements. As a prominent example we consider the antiferromagnetic quantum Ising model in transverse and longitudinal magnetic fields which is in demand in both condensed matter physics and quantum computing. We show that the neural network approach gives reliable estimates of the classical mutual information even in the case of paramagnetic wave functions delocalized in the state space. In addition, the phase diagram of the considered quantum system is reconstructed with a special focus on discriminating various types of disordered states.
Quantum neural networks (QNNs) is a parameterized quantum circuit model, which can be trained by gradient-based optimizer, can be used for supervised learning, regression tasks, combinatorial optimization, etc. Although many works have demonstrated that QNNs have better learnability, generalizability, etc. compared to classical neural networks. However, as with classical neural networks, we still can't explain their working mechanism well. In this paper, we reveal the training mechanism of QNNs by mutual information. Unlike traditional mutual information in neural networks, due to quantum computing remains information conserved, the mutual information is trivial of the input and output of U operator. In our work, in order to observe the change of mutual information during training, we divide the quantum circuit (U operator) into two subsystems, discard subsystem (D) and measurement subsystem (M) respectively. We calculate two mutual information, I(Di : Mo) and I(Mi : Mo) (i and o means input or output of the corresponding subsystem), and observe their behavior during training. As the epochs increases, I(Di : Mo) gradually increases, this may means some information of discard subsystem is continuously pushed into the measurement subsystem, the information should be label-related. What's more, I(Mi : Mo) exist two-phase behavior in training process, this consistent with the information bottleneck anticipation. The first phase, I(Mi : Mo) is increasing, this means the measurement subsystem perform feature fitting. The second phase, I(Mi : Mo) is decreasing, this may means the system is generalizing, the measurement subsystem discard label-irrelevant information into the discard subsystem as many as possible. Our work discussed the working mechanism of QNNs by mutual information, further, it can be used to analyze the accuracy and generalization of QNNs.
We describe a direct method to estimate the bipartite mutual information of a classical spin system based on Monte Carlo sampling enhanced by autoregressive neural networks. It enables us to study arbitrary geometries of subsystems, and it can be generalized to classical field theories. We demonstrate it on the Ising model for four partitionings, including a multiply connected even-odd division. We show that the area law is satisfied for temperatures away from the critical temperature: the constant term is universal, whereas the proportionality coefficient is different for the even-odd partitioning.
Neural Networks (NNs) have received extensive attention and research due to their ability to extract and combine different features non-manually and to mine the internal relationships between features. In quantum mechanics, the entangled state simultaneously describes the classical and non-classical correlation between subsystems, so it can reveal important quantum phenomena such as non-locality, entanglement, etc., but its essence is the characterization of strong statistical relationships. Considering the superiority of the entangled state, this article attempts to use the entangled state to reconstruct the neurons of NNs to achieve that the network model can characterize the strong statistical relationship between the features, namely, the classical and non-classical correlation. Specifically, based on concurrence, a quantification method of entanglement, we propose a regularizer that can constrain a state vector to an entangled state, and apply it to the optimization process to ensure that the vector passed to the neuron is a legal entangled state. Finally, the entangled state is measured to obtain the output of the neuron. Experimental results show that our model is better than the baseline. Moreover, it performs well compared to the model inspired by quantum entanglement.
Abstract Along the way initiated by Carleo and Troyer [G. Carleo and M. Troyer, Science 355 (2017) 602], we construct the neural-network quantum state of transverse-field Ising model (TFIM) by an unsupervised machine learning method. Such a wave function is a map from the spin-configuration space to the complex number field determined by an array of network parameters. To get the ground state of the system, values of the network parameters are calculated by a Stochastic Reconfiguration (SR) method. We provide for this SR method an understanding from action principle and information geometry aspects. With this quantum state, we calculate key observables of the system, the energy, correlation function, correlation length, magnetic moment, and susceptibility. As innovations, we provide a high efficiency method and use it to calculate entanglement entropy (EE) of the system and get results consistent with previous work very well .
This thesis uses a quantity that is defined and justified by information theory -- mutual information -- to examine models of condensed matter systems. More precisely, it studies models which are made up out of ferromagnetically interacting spins. Quantum information theory often focuses on the ground state of such systems; we will however be interested in what happens at finite temperature. Using mutual information, which can be seen as a generalization of entanglement entropy to the finite-temperature case, we can study the different phases occurring in these models, and in particular the phase transitions between those. We examine broadly two different classes of models: classical spins on two-dimensional lattices, and fully-connected models of quantum-mechanical spin-1/2 particles. (Abstract abridged.)
We demonstrate quantum many-body state reconstruction from experimental data generated by a programmable quantum simulator by means of a neural-network model incorporating known experimental errors. Specifically, we extract restricted Boltzmann machine wave functions from data produced by a Rydberg quantum simulator with eight and nine atoms in a single measurement basis and apply a novel regularization technique to mitigate the effects of measurement errors in the training data. Reconstructions of modest complexity are able to capture one- and two-body observables not accessible to experimentalists, as well as more sophisticated observables such as the Rényi mutual information. Our results open the door to integration of machine learning architectures with intermediate-scale quantum hardware.