16,220 research outputs found
Neural-Network Quantum States, String-Bond States, and Chiral Topological States
Neural-Network Quantum States have been recently introduced as an Ansatz for
describing the wave function of quantum many-body systems. We show that there
are strong connections between Neural-Network Quantum States in the form of
Restricted Boltzmann Machines and some classes of Tensor-Network states in
arbitrary dimensions. In particular we demonstrate that short-range Restricted
Boltzmann Machines are Entangled Plaquette States, while fully connected
Restricted Boltzmann Machines are String-Bond States with a nonlocal geometry
and low bond dimension. These results shed light on the underlying architecture
of Restricted Boltzmann Machines and their efficiency at representing many-body
quantum states. String-Bond States also provide a generic way of enhancing the
power of Neural-Network Quantum States and a natural generalization to systems
with larger local Hilbert space. We compare the advantages and drawbacks of
these different classes of states and present a method to combine them
together. This allows us to benefit from both the entanglement structure of
Tensor Networks and the efficiency of Neural-Network Quantum States into a
single Ansatz capable of targeting the wave function of strongly correlated
systems. While it remains a challenge to describe states with chiral
topological order using traditional Tensor Networks, we show that
Neural-Network Quantum States and their String-Bond States extension can
describe a lattice Fractional Quantum Hall state exactly. In addition, we
provide numerical evidence that Neural-Network Quantum States can approximate a
chiral spin liquid with better accuracy than Entangled Plaquette States and
local String-Bond States. Our results demonstrate the efficiency of neural
networks to describe complex quantum wave functions and pave the way towards
the use of String-Bond States as a tool in more traditional machine-learning
applications.Comment: 15 pages, 7 figure
A Bayesian Approach to Manifold Topology Reconstruction
In this paper, we investigate the problem of statistical reconstruction of piecewise linear manifold topology. Given a noisy, probably undersampled point cloud from a one- or two-manifold, the algorithm reconstructs an approximated most likely mesh in a Bayesian sense from which the sample might have been taken. We incorporate statistical priors on the object geometry to improve the reconstruction quality if additional knowledge about the class of original shapes is available. The priors can be formulated analytically or learned from example geometry with known manifold tessellation. The statistical objective function is approximated by a linear programming / integer programming problem, for which a globally optimal solution is found. We apply the algorithm to a set of 2D and 3D reconstruction examples, demon-strating that a statistics-based manifold reconstruction is feasible, and still yields plausible results in situations where sampling conditions are violated
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