117 research outputs found
Simulations of Quantum XXZ Models on Two-Dimensional Frustrated Lattices
We report recent progress in the study of a particular class of spin 1/2 XXZ
model on two-dimensional lattices with frustrated diagonal and unfrustrated
off-diagonal interactions. Quantum Monte Carlo simulations can be constructed
without a sign problem, however they require non-trivial algorithmic advances
in order to combat freezing tendencies. We discuss results obtained using these
techniques, in particular the discovery of unusual bulk quantum phases, studies
of quantum criticality, and the continuing search for exotic physics in these
models.Comment: 10 pages, 5 figures. Conference proceedings for Highly Frustrated
Magnetism 200
Latent Space Purification via Neural Density Operators
Machine learning is actively being explored for its potential to design,
validate, and even hybridize with near-term quantum devices. A central question
is whether neural networks can provide a tractable representation of a given
quantum state of interest. When true, stochastic neural networks can be
employed for many unsupervised tasks, including generative modeling and state
tomography. However, to be applicable for real experiments such methods must be
able to encode quantum mixed states. Here, we parametrize a density matrix
based on a restricted Boltzmann machine that is capable of purifying a mixed
state through auxiliary degrees of freedom embedded in the latent space of its
hidden units. We implement the algorithm numerically and use it to perform
tomography on some typical states of entangled photons, achieving fidelities
competitive with standard techniques.Comment: 5 pages, 3 figures, published versio
Learning Thermodynamics with Boltzmann Machines
A Boltzmann machine is a stochastic neural network that has been extensively
used in the layers of deep architectures for modern machine learning
applications. In this paper, we develop a Boltzmann machine that is capable of
modelling thermodynamic observables for physical systems in thermal
equilibrium. Through unsupervised learning, we train the Boltzmann machine on
data sets constructed with spin configurations importance-sampled from the
partition function of an Ising Hamiltonian at different temperatures using
Monte Carlo (MC) methods. The trained Boltzmann machine is then used to
generate spin states, for which we compare thermodynamic observables to those
computed by direct MC sampling. We demonstrate that the Boltzmann machine can
faithfully reproduce the observables of the physical system. Further, we
observe that the number of neurons required to obtain accurate results
increases as the system is brought close to criticality.Comment: 8 pages, 5 figure
Deep Learning the Ising Model Near Criticality
It is well established that neural networks with deep architectures perform
better than shallow networks for many tasks in machine learning. In statistical
physics, while there has been recent interest in representing physical data
with generative modelling, the focus has been on shallow neural networks. A
natural question to ask is whether deep neural networks hold any advantage over
shallow networks in representing such data. We investigate this question by
using unsupervised, generative graphical models to learn the probability
distribution of a two-dimensional Ising system. Deep Boltzmann machines, deep
belief networks, and deep restricted Boltzmann networks are trained on thermal
spin configurations from this system, and compared to the shallow architecture
of the restricted Boltzmann machine. We benchmark the models, focussing on the
accuracy of generating energetic observables near the phase transition, where
these quantities are most difficult to approximate. Interestingly, after
training the generative networks, we observe that the accuracy essentially
depends only on the number of neurons in the first hidden layer of the network,
and not on other model details such as network depth or model type. This is
evidence that shallow networks are more efficient than deep networks at
representing physical probability distributions associated with Ising systems
near criticality.Comment: 16 pages, 8 figures, 1 tabl
Entanglement at a Two-Dimensional Quantum Critical Point: a T=0 Projector Quantum Monte Carlo Study
Although the leading-order scaling of entanglement entropy is non-universal
at a quantum critical point (QCP), sub-leading scaling can contain universal
behaviour. Such universal quantities are commonly studied in non-interacting
field theories, however it typically requires numerical calculation to access
them in interacting theories. In this paper, we use large-scale T=0 quantum
Monte Carlo simulations to examine in detail the second R\'enyi entropy of
entangled regions at the QCP in the transverse-field Ising model in 2+1
space-time dimensions -- a fixed point for which there is no exact result for
the scaling of entanglement entropy. We calculate a universal coefficient of a
vertex-induced logarithmic scaling for a polygonal entangled subregion, and
compare the result to interacting and non-interacting theories. We also examine
the shape-dependence of the R\'enyi entropy for finite-size toroidal lattices
divided into two entangled cylinders by smooth boundaries. Remarkably, we find
that the dependence on cylinder length follows a shape-dependent function
calculated previously by Stephan {\it et al.} [New J. Phys., 15, 015004,
(2013)] at the QCP corresponding to the 2+1 dimensional quantum Lifshitz free
scalar field theory. The quality of the fit of our data to this scaling
function, as well as the apparent cutoff-independent coefficient that results,
presents tantalizing evidence that this function may reflect universal
behaviour across these and other very disparate QCPs in 2+1 dimensional
systems.Comment: 28 pages, 10 figure
Kernel methods for interpretable machine learning of order parameters
Machine learning is capable of discriminating phases of matter, and finding
associated phase transitions, directly from large data sets of raw state
configurations. In the context of condensed matter physics, most progress in
the field of supervised learning has come from employing neural networks as
classifiers. Although very powerful, such algorithms suffer from a lack of
interpretability, which is usually desired in scientific applications in order
to associate learned features with physical phenomena. In this paper, we
explore support vector machines (SVMs) which are a class of supervised kernel
methods that provide interpretable decision functions. We find that SVMs can
learn the mathematical form of physical discriminators, such as order
parameters and Hamiltonian constraints, for a set of two-dimensional spin
models: the ferromagnetic Ising model, a conserved-order-parameter Ising model,
and the Ising gauge theory. The ability of SVMs to provide interpretable
classification highlights their potential for automating feature detection in
both synthetic and experimental data sets for condensed matter and other
many-body systems.Comment: 8 pages, 6 figure
Machine learning phases of matter
Neural networks can be used to identify phases and phase transitions in
condensed matter systems via supervised machine learning. Readily programmable
through modern software libraries, we show that a standard feed-forward neural
network can be trained to detect multiple types of order parameter directly
from raw state configurations sampled with Monte Carlo. In addition, they can
detect highly non-trivial states such as Coulomb phases, and if modified to a
convolutional neural network, topological phases with no conventional order
parameter. We show that this classification occurs within the neural network
without knowledge of the Hamiltonian or even the general locality of
interactions. These results demonstrate the power of machine learning as a
basic research tool in the field of condensed matter and statistical physics.Comment: 18 pages, 8 figures, 1 tabl
Machine learning quantum states in the NISQ era
We review the development of generative modeling techniques in machine
learning for the purpose of reconstructing real, noisy, many-qubit quantum
states. Motivated by its interpretability and utility, we discuss in detail the
theory of the restricted Boltzmann machine. We demonstrate its practical use
for state reconstruction, starting from a classical thermal distribution of
Ising spins, then moving systematically through increasingly complex pure and
mixed quantum states. Intended for use on experimental noisy intermediate-scale
quantum (NISQ) devices, we review recent efforts in reconstruction of a cold
atom wavefunction. Finally, we discuss the outlook for future experimental
state reconstruction using machine learning, in the NISQ era and beyond.Comment: 14 pages, 4 figure
Spinon walk in quantum spin ice
We study a minimal model for the dynamics of spinons in quantum spin ice. The
model captures the essential strong coupling between the spinon and the
disordered background spins. We demonstrate that the spinon motion can be
mapped to a random walk with an entropy-induced memory in imaginary time. Our
numerical simulation of the spinon walk indicates that the spinon propagates as
a massive quasiparticle at low energy despite its strong coupling to the spin
background at the microscopic energy scale. We discuss the experimental
implications of our findings.Comment: 12 pages, 10 figure
Tripartite entangled plaquette state in a cluster magnet
Using large-scale quantum Monte Carlo simulations we show that a spin-
XXZ model on a two-dimensional anisotropic Kagome lattice exhibits a tripartite
entangled plaquette state that preserves all of the Hamiltonian symmetries. It
is connected via phase boundaries to a ferromagnet and a valence-bond solid
that break U(1) and lattice translation symmetries, respectively. We study the
phase diagram of the model in detail, in particular the transitions to the
tripartite entangled plaquette state, which are consistent with conventional
order-disorder transitions. Our results can be interpreted as a description of
the charge sector dynamics of a Hubbard model applied to the description of the
spin liquid candidate ,
as well as a model of strongly correlated bosonic atoms loaded onto highly
tunable {\it trimerized} optical Kagome lattices.Comment: 10 pages, 10 figure
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