24 research outputs found
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 algebra. We show that the well-known quantum phase transition at
magnetic field 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 and , where is the anisotropy and
the magnetic field strength. In particular, two nonequilibrium quantum
phase transitions coexist at . 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
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 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