10 research outputs found
Bayesian Modelling Approaches for Quantum States -- The Ultimate Gaussian Process States Handbook
Capturing the correlation emerging between constituents of many-body systems
accurately is one of the key challenges for the appropriate description of
various systems whose properties are underpinned by quantum mechanical
fundamentals. This thesis discusses novel tools and techniques for the
(classical) modelling of quantum many-body wavefunctions with the ultimate goal
to introduce a universal framework for finding accurate representations from
which system properties can be extracted efficiently. It is outlined how
synergies with standard machine learning approaches can be exploited to enable
an automated inference of the most relevant intrinsic characteristics through
rigorous Bayesian regression techniques. Based on the probabilistic framework
forming the foundation of the introduced ansatz, coined the Gaussian Process
State, different compression techniques are explored to extract numerically
feasible representations of relevant target states within stochastic schemes.
By following intuitively motivated design principles, the resulting model
carries a high degree of interpretability and offers an easily applicable tool
for the numerical study of quantum systems, including ones which are
notoriously difficult to simulate due to a strong intrinsic correlation. The
practical applicability of the Gaussian Process States framework is
demonstrated within several benchmark applications, in particular, ground state
approximations for prototypical quantum lattice models, Fermi-Hubbard models
and models, as well as simple ab-initio quantum chemical systems.Comment: PhD Thesis, King's College London, 202 page
A framework for efficient ab initio electronic structure with Gaussian Process States
We present a general framework for the efficient simulation of realistic
fermionic systems with modern machine learning inspired representations of
quantum many-body states, towards a universal tool for ab initio electronic
structure. These machine learning inspired ansatzes have recently come to the
fore in both a (first quantized) continuum and discrete Fock space
representations, where however the inherent scaling of the latter approach for
realistic interactions has so far limited practical applications. With
application to the 'Gaussian Process State', a recently introduced ansatz
inspired by systematically improvable kernel models in machine learning, we
discuss different choices to define the representation of the computational
Fock space. We show how local representations are particularly suited for
stochastic sampling of expectation values, while also indicating a route to
overcome the discrepancy in the scaling compared to continuum formulated
models. We are able to show competitive accuracy for systems with up to 64
electrons, including a simplified (yet fully ab initio) model of the Mott
transition in three-dimensional hydrogen, indicating a significant improvement
over similar approaches, even for moderate numbers of configurational samples.Comment: 15 pages, 5 figure
Impact of conditional modelling for universal autoregressive quantum states
We present a generalized framework to adapt universal quantum state
approximators, enabling them to satisfy rigorous normalization and
autoregressive properties. We also introduce filters as analogues to
convolutional layers in neural networks to incorporate translationally
symmetrized correlations in arbitrary quantum states. By applying this
framework to the Gaussian process state, we enforce autoregressive and/or
filter properties, analyzing the impact of the resulting inductive biases on
variational flexibility, symmetries, and conserved quantities. In doing so we
bring together different autoregressive states under a unified framework for
machine learning-inspired ans\"atze. Our results provide insights into how the
autoregressive construction influences the ability of a variational model to
describe correlations in spin and fermionic lattice models, as well as ab
initio electronic structure problems where the choice of representation affects
accuracy. We conclude that, while enabling efficient and direct sampling, thus
avoiding autocorrelation and loss of ergodicity issues in Metropolis sampling,
the autoregressive construction materially constrains the expressivity of the
model in many systems
Quantum Gaussian process state:A kernel-inspired state with quantum support data
We introduce the quantum Gaussian process state, motivated via a statistical
inference for the wave function supported by a data set of unentangled product
states. We show that this condenses down to a compact and expressive parametric
form, with a variational flexibility shown to be competitive or surpassing
established alternatives. The connections of the state to its roots as a
Bayesian inference machine as well as matrix product states, also allow for
efficient deterministic training of global states from small training data with
enhanced generalization, including on application to frustrated spin physics