Deep tensor networks with matrix product operators

We introduce deep tensor networks, which are exponentially wide neural networks based on the tensor network representation of the weight matrices. We evaluate the proposed method on the image classification (MNIST, FashionMNIST) and sequence prediction (cellular automata) tasks. In the image classification case, deep tensor networks improve our matrix product state baselines and achieve 0.49% error rate on MNIST and 8.3% error rate on FashionMNIST. In the sequence prediction case, we demonstrate an exponential improvement in the number of parameters compared to the one-layer tensor network methods. In both cases, we discuss the non-uniform and the uniform tensor network models and show that the latter generalizes well to different input sizes.Comment: 9+2 pages, 8 figure

Heat transport in quantum harmonic chains with Redfield baths

We provide an explicit method for solving general markovian master equations for quadratic bosonic Hamiltonians with linear bath operators. As an example we consider a one-dimensional quantum harmonic oscillator chain coupled to thermal reservoirs at both ends of the chain. We derive an analytic solution of the Redfield master equation for homogeneous harmonic chain and recover classical results, namely, vanishing temperature gradient and constant heat current in the thermodynamic limit. In the case of the disordered gapped chains we observe universal heat current scaling independent of the bath spectral function, the system-bath coupling strength, and the boundary conditions.Comment: 17 pages, 3 figure

Positive unlabeled learning with tensor networks

Positive unlabeled learning is a binary classification problem with positive and unlabeled data. It is common in domains where negative labels are costly or impossible to obtain, e.g., medicine and personalized advertising. We apply the locally purified state tensor network to the positive unlabeled learning problem and test our model on the MNIST image and 15 categorical/mixed datasets. On the MNIST dataset, we achieve state-of-the-art results even with very few labeled positive samples. Similarly, we significantly improve the state-of-the-art on categorical datasets. Further, we show that the agreement fraction between outputs of different models on unlabeled samples is a good indicator of the model's performance. Finally, our method can generate new positive and negative instances, which we demonstrate on simple synthetic datasets.Comment: 12 pages, 5 figures, 4 table

Nonequilibrium Quantum Phase Transitions in the XY model: comparison of unitary time evolution and reduced density matrix approaches

We study nonequilibrium quantum phase transitions in XY spin 1/2 chain using the $C^*$ algebra. We show that the well-known quantum phase transition at magnetic field $h = 1$ persists also in the nonequilibrium setting as long as one of the reservoirs is set to absolute zero temperature. In addition, we find nonequilibrium phase transitions associated to imaginary part of the correlation matrix for any two different temperatures of the reservoirs at $h = 1$ and $h = h_{\rm c} \equiv|1-\gamma^2|$, where $\gamma$ is the anisotropy and $h$ the magnetic field strength. In particular, two nonequilibrium quantum phase transitions coexist at $h=1$. In addition we also study the quantum mutual information in all regimes and find a logarithmic correction of the area law in the nonequilibrium steady state independent of the system parameters. We use these nonequilibrium phase transitions to test the utility of two models of reduced density operator, namely Lindblad mesoreservoir and modified Redfield equation. We show that the nonequilibrium quantum phase transition at $h = 1$ related to the divergence of magnetic susceptibility is recovered in the mesoreservoir approach, whereas it is not recovered using the Redfield master equation formalism. However none of the reduced density operator approaches could recover all the transitions observed by the $C^*$ algebra. We also study thermalization properties of the mesoreservoir approach.Comment: 25 pages, 10 figure

Mean-field dynamics of an infinite-range interacting quantum system: chaos, dynamical phase transition, and localisation

We investigate the dynamical properties of the XY spin 1/2 chain with infinite-range transverse interactions and find a dynamical phase transition with a chaotic dynamical phase. In the latter, we find non-vanishing Lyapunov exponents and intermittent behavior signaled by periods of fast and slow entropy growth. Further, we study the XY chain with a local self-consistent transverse field and observe a localization phase transition. We show that localization stabilizes the chaotic dynamical phase.Comment: 8+6 pages, 15 figure

Grokking phase transitions in learning local rules with gradient descent

We discuss two solvable grokking (generalisation beyond overfitting) models in a rule learning scenario. We show that grokking is a phase transition and find exact analytic expressions for the critical exponents, grokking probability, and grokking time distribution. Further, we introduce a tensor-network map that connects the proposed grokking setup with the standard (perceptron) statistical learning theory and show that grokking is a consequence of the locality of the teacher model. As an example, we analyse the cellular automata learning task, numerically determine the critical exponent and the grokking time distributions and compare them with the prediction of the proposed grokking model. Finally, we numerically analyse the connection between structure formation and grokking.Comment: 31+10 pages, 22 figure

Transport properties of a boundary-driven one-dimensional gas of spinless fermions

We analytically study a system of spinless fermions driven at the boundary with an oscillating chemical potential. Various transport regimes can be observed: at zero driving frequency the particle current through the system is independent of the system's length; at the phase-transition frequency, being equal to the bandwidth, the current decays as n^{-alpha} with the chain length n, alpha being either 2 or 3; below the transition the scaling of the current is n^{-1/2}, indicating anomalous transport, while it is exponentially small exp{(-n/2xi)} above the transition. Therefore, by a simple change of frequency of the a.c. driving one can vary transport from ballistic, anomalous, to insulating.Comment: 9 pages, 10 figure

Lindblad master equation approach to superconductivity in open quantum systems

We consider an open quantum Fermi-system which consists of a single degenerate level with pairing interactions embedded into a superconducting bath. The time evolution of the reduced density matrix for the system is given by Linblad master equation, where the dissipators describe exchange of Bogoliubov quasiparticles with the bath. We obtain fixed points of the time evolution equation for the covariance matrix and study their stability by analyzing full dynamics of the order parameter.Comment: 7 pages, 2 pdf figure