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arXiv:2208.03264

Towards Antisymmetric Neural Ansatz Separation

Aaron Zweig, Joan Bruna

Published Aug 5, 2022• 1 citation
arXivHTMLPDF

Abstract

We study separations between two fundamental models (or \emph{Ansätze}) of antisymmetric functions, that is, functions $f$ of the form $f(x_{σ(1)}, \ldots, x_{σ(N)}) = \text{sign}(σ)f(x_1, \ldots, x_N)$, where $σ$ is any permutation. These arise in the context of quantum chemistry, and are the basic modeling tool for wavefunctions of Fermionic systems. Specifically, we consider two popular antisymmetric Ansätze: the Slater representation, which leverages the alternating structure of determinants, and the Jastrow ansatz, which augments Slater determinants with a product by an arbitrary symmetric function. We construct an antisymmetric function in $N$ dimensions that can be efficiently expressed in Jastrow form, yet provably cannot be approximated by Slater determinants unless there are exponentially (in $N^2$) many terms. This represents the first explicit quantitative separation between these two Ansätze.

DOI
10.48550/arxiv.2208.03264

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Given a function w completely antisymmetric in n variables, there may exist a set of n functions of one variable such that the given function is a Slater determinant in the latter. The first problem considered is that of obtaining a criterion for this to be the case for a given function. This problem is solved by considering the function w as a mapping of the space of functions in n − 1 variables onto the space of functions of one variable. A necessary and sufficient condition for the initial function to be a Slater determinant is then shown to be that the image space be n dimensional. This criterion is converted into practical algorithms which can be employed for the determination. The application of one of these yields the theorem that an arbitrary linear combination of the n + 1 Slater determinants in n variables formed from n + 1 one-variable functions can always be written as a single Slater determinant. It is further proved that if the image space of the mapping is m(>n) dimensional, the original function can be expressed as a linear combination of m!/(m − n)!n! Slater determinants in n variables formed from m one-variable functions. Playing an important role in the analysis is the product of the mapping described above by its adjoint (the product is simply related to Dirac's density matrix for a quantum mechanical system of identical particles) as well as the eigenvectors and eigenvalues of this Hermitian positive semidefinite mapping. The latter form a basis for a systematic approximation procedure for representing a given function by a single Slater determinant or by sums of Slater determinants formed from a particular number of one-variable functions, which yields results obtained previously by Löwdin. Problems of simultaneous approximation of sets of antisymmetric functions and possible physical applications to many-fermion systems are briefly discussed.
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Direct sampling from a Slater determinant is combined with an autoregressive deep neural network as a Jastrow factor into a fully autoregressive Slater-Jastrow ansatz for variational quantum Monte Carlo, which allows for uncorrelated sampling. The elimination of the autocorrelation time leads to a stochastic algorithm with provable cubic scaling (with a potentially large prefactor), i.e. the number of operations for producing an uncorrelated sample and for calculating the local energy scales like \mathcal{O}(N_s^3) 𝒪 ( N s 3 ) with the number of orbitals N_s N s . The implementation is benchmarked on the two-dimensional t-V t − V model of spinless fermions on the square lattice.
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