Impact of conditional modelling for a universal autoregressive quantum state

Massimo Bortone, Yannic Rath, and George H. Booth

Department of Physics, King’s College London, Strand, London WC2R 2LS, United Kingdom

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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 ansätze. 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.

Computationally resolving interacting quantum particles, such as the electrons in a molecule, holds the promise to unlock many potential applications across a range of fields, from the design of new drugs to the discovery of exotic materials. However, this requires circumventing the exponential scaling of the quantum many-body wave function, the core mathematical object describing the behaviour of these electrons. Parameterizing these states with techniques inspired from the compression found from recent machine learning tools has emerged as a promising avenue for progress, with a broad range of applicability. This provides a surrogate model of the wave function with a far smaller number of parameters than the intractable number needed for a full description.

However, careful design of the surrogate model has important consequences in terms of the accuracy of the approximation and the efficiency of the optimization procedure. In this work we look under the hood at a particular class of these machine learning inspired states known as autoregressive models, which have been recently popularized by their success in image recognition, and advantageous sampling properties. We show how more general classes of states can inherit this property, and disentangle how different design choices affect the performance of these models.

Through our analysis and application to the ground states of a range of quantum many-body problems, we find that there is a cost to pay for the autoregressive property in terms of its ultimate flexibility in describing these states with a fixed number of parameters. With our work we hope to shine a light on important design choices required for the development of ever more powerful surrogate models for the wave function of interacting quantum particles.

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