Abstract
We present a novel, non-parametric form for compactly representing entangled many-body quantum states, which we call a `Gaussian Process State'. In contrast to other approaches, we define this state explicitly in terms of a configurational data set, with the probability amplitudes statistically inferred from this data according to Bayesian statistics. In this way the non-local physical correlated features of the state can be analytically resummed, allowing for exponential complexity to underpin the ansatz, but efficiently represented in a small data set. The state is found to be highly compact, systematically improvable and efficient to sample, representing a large number of known variational states within its span. It is also proven to be a `universal approximator' for quantum states, able to capture any entangled many-body state with increasing data set size. We develop two numerical approaches which can learn this form directly: a fragmentation approach, and direct variational optimization, and apply these schemes to the Fermionic Hubbard model. We find competitive or superior descriptions of correlated quantum problems compared to existing state-of-the-art variational ansatzes, as well as other numerical methods.
Highlights
Representing entangled quantum many-body states efficiently and compactly is a major challenge, with the need for tractable approaches underpinning a diverse set of fields including quantum computation, electronic or nuclear structure, and quantum chemistry
In order to describe the amplitudes of a quantum state, ΨðxÞ, evaluated for any many-body configuration x in the underlying Hilbert space, we work within a Bayesian inference framework known as Gaussian process (GP) regression
We introduce a Gaussian Process State (GPS) as the exponential of a GP estimator, where a configurational probability amplitude ΨgðxÞ is given as PHYS
Summary
Representing entangled quantum many-body states efficiently and compactly is a major challenge, with the need for tractable approaches underpinning a diverse set of fields including quantum computation, electronic or nuclear structure, and quantum chemistry. GLIELMO, RATH, CSÁNYI, DE VITA, and BOOTH for a random state where no structure exists in the form of the amplitudes, but it is known that physical many-body states exhibit much structure that can be effectively learnt in this approach, emerging from the underlying principles encoded in the Hamiltonian After this inferred wave function is defined and its limits discussed, we turn the idea into a practical and general method for quantum problems and demonstrate its accuracy and improvability in describing full N-body quantum correlations by application to the fermionic Hubbard model of the cuprates and strongly correlated solids. We show that an automatic selection of the relevant data required to capture the complexity of the many-body correlations allows for a compact description of this state compared to other approaches, with significant potential for a wide range of applications
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