Ecient Solutions of Fermionic Systems using Articial Neural Networks Even M. Nordhagen Department of Physics and Njord Center University of Oslo N-0316 Oslo Norway

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Eﬃcient Solutions of Fermionic Systems using Artiﬁcial Neural Networks

Even M. Nordhagen

Department of Physics and Njord Center, University of Oslo, N-0316 Oslo, Norway

Jane M. Kim

Department of Physics and Astronomy and Facility for Rare Isotope Beams,

Michigan State University, East Lansing, MI 48824, USA

Bryce Fore and Alessandro Lovato

Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA

Morten Hjorth-Jensen

Department of Physics and Astronomy and Facility for Rare Isotope Beams,

Michigan State University, East Lansing, MI 48824, USA and

Department of Physics and Center for Computing in Science Education, University of Oslo, N-0316 Oslo, Norway

We discuss diﬀerences and similarities between variational Monte Carlo approaches that use con-

ventional and artiﬁcial neural network parameterizations of the ground-state wave function for

systems of fermions. We focus on a relatively shallow neural-network architectures, the so called

restricted Boltzmann machine, and discuss unsupervised learning algorithms that are suitable to

model complicated many-body correlations. We analyze the strengths and weaknesses of conven-

tional and neural-network wave functions by solving various circular quantum-dots systems. Results

for up to 90 electrons are presented and particular emphasis is placed on how to eﬃciently implement

these methods on homogeneous and heterogeneous high-performance computing facilities.

arXiv:2210.00365v1 [cond-mat.mes-hall] 1 Oct 2022

I. INTRODUCTION

Solving the Schr¨odinger equation for systems of many interacting bosons or fermions is classiﬁed as an NP-hard

problem due to the complexity of the required many-dimensional wave function, resulting in an exponential growth of

degrees of freedom. Reducing the dimensionalities of quantum mechanical many-body systems is an important aspect

of modern physics, ranging from the development of eﬃcient algorithms for studying many-body systems to exploiting

the increase in computing power. To write software that can fully utilize the available resources has long been known

to be an important aspect of these endeavors. Despite tremendous progress has been made in this direction, traditional

many-particle methods, either quantum mechanical or classical ones, face huge dimensionality problems when applied

to studies of systems with many interacting particles.

Over the last two decades, quantum computing and machine learning have emerged as some of the most promising

approaches for studying complex physical systems where several length and energy scales are involved. Machine

learning techniques and in particular neural-network quantum states [1] have recently been applied to studies of

many-body systems, see for example Refs. [2–9], in various ﬁelds of physics and quantum chemistry, with very

promising results. In many of these studies, one has obtained results that align well with exact analytical solutions

or are in close agreement with state-of-the-art quantum Monte Carlo calculations.

The variational and diﬀusion Monte Carlo algorithms are among the most popular and successful methods available

for ground-state studies of quantum mechanical systems. They both rely on a suitable ansatz for the ground-state of

the system, often dubbed trial wave function, which is deﬁned in terms of a set of variational parameters whose optimal

values are found by minimizing the total energy of the system. Devising ﬂexible and accurate functional forms for the

trial wave functions requires prior knowledge and physical intuition about the system under investigation. However,

for many systems we do not have this intuition, and as a result it is often diﬃcult to deﬁne a good ansatz for the

state function.

According to the universal approximation theorem, a deep neural network can represent any continuous function

within a certain error [10] — see also Refs. [1,11–13] for further discussions of deep leaning methods. Since the

variational state wave function in principle can take any functional form, it is natural to replace the trial wave

function with a neural network and treat it as a machine learning problem. This approach has been successfully

implemented in recent works, see for example Refs. [2,4,8,9,14], and forms the motivation for the present study.

Here, the neural network of choice was derived from so-called restricted Boltzmann machines, much inspired by the

recent contributions by Carleo et al., see for example Refs. [2,6]. Note that neural-networks representations of

variational states are more general, as they do not in principle require prior knowledge on the ground-state wave

function, thereby opening the door to systems that have yet to be solved. Particular attention however has to be

devoted to the symmetries of the problem, whose inclusion is critical to achieve accurate results [].

In this work, we will focus on systems of electrons conﬁned to move in two-dimensional harmonic oscillator sys-

tems, so-called quantum dots. These are strongly conﬁned electrons and oﬀer a wide variety of complex and subtle

phenomena which pose severe challenges to existing many-body methods. Due to their small size, quantum dots are

characterized by discrete quantum levels. For instance, the ground states of circular dots show similar shell structures

and magic numbers as seen for atoms and nuclei. These structures are particularly evident in measurements of the

change in electrochemical potential due to the addition of one extra electron. Here these systems will serve as our

test of the applicability of artiﬁcial neural network variational states, including restricted Boltzmann Machines.

The theoretical foundation and the methodology are explained in section II. The subsequent sections present our

results with an analysis of computational methods and resources. In the last section we present our conclusions and

perspectives for future work.

II. METHOD

For any Hamiltonian ˆ

Hand trial wave function ψT, the variational principle guarantees that the expectation value

of the energy ETis greater than or equal to the true ground state energy E0,

E0≤ET=hψT|ˆ

H|ψTi

hψT|ψTi.(1)

Thus approximate solutions to the time-independent Schr¨odinger equation can be obtained by choosing a careful

parameterization of the wave function and minimizing the energy ETwith respect to the parameters. Since the

integrals representing ETare normally high dimensional, it is most eﬃcient to evaluate them by means of Monte

Carlo methods

ET≈ hELi=1

i=1

EL(Ri),Ri∼ |ψT(R)|2.(2)

This involves collecting nsamples of conﬁgurations and averaging over the so-called local energies

EL(R) = 1

ψT(R)ˆ

HψT(R).(3)

We apply the variational Monte Carlo (VMC) method to various circular quantum dots systems. These are systems

of interacting electrons conﬁned to move in a two-dimensional harmonic oscillator well. The (scaled)[15] Hamiltonian

is given by

H=1

i

−∇2

i+ω2r2

i+X

j6=i

rij 

,(4)

where ωis the oscillator frequency, riis the distance between electron iand the origin, and rij is the distance between

electrons iand j. We will henceforth assume the total number of electrons Nto be even and the total spin of the

system to be zero.

A simple ansatz can be built starting from the analytical solutions to the non-interacting case. The harmonic

oscillator eigenfunctions are given by

φm,n(x, y)∝e−ω(x2+y2)Hm(√ωx)Hn(√ωy),(5)

where Hnare the Hermite polynomials of degree n. To constrain the antisymmetry of the many-body wave function,

products of the lowest N/2 spatial states and the two spin states ξ±(σ) are used as a basis for a Slater determinant

ψSD(R) = det hnφm,n(xi, yi)ξk(σi)oi,

where m, n, k label the single-particle state, ilabels the particle, and Rcontains all coordinates of the Nparticles.

As an aside, we do not include the spin projections σias explicit inputs to the wave function as we will describe how

to treat them separately in Section II.B. We then deﬁne a reference state by pulling the common exponential term

out of the determinant and inserting a single variational parameter α

ψRef(R;α) = e−αω Pi(x2

i+y2

i)det hnHm(√ωxi)Hn(√ωyi)ξk(σi)oi.(6)

Correlations among electrons can be handled by a Pad´e-Jastrow factor [16],

g(R;β) = exp 



i=1

j>i

aij rij

1 + βrij 

,(7)

where βis a variational parameter and

aij =(1/3 if σi=σj

1 if σi6=σj

in order for the Kato cusp condition to be satisﬁed [17]. The product of the Slater determinant and the Pad´e-Jastrow

factor is commonly named the Slater-Jastrow ansatz,

ψSlater-Jastrow(R;α, β) = ψRef(R;α)×g(R;β).(8)

A. Gaussian-binary restricted Boltzmann machine

There are many possible choices for a machine learning inspired wave function, but using an artiﬁcial neural network

is natural. Inspired by Ref. [2], our choice is to start from a restricted Boltzmann machine (RBM) conﬁgured for

1 1

w11

visible hidden

Figure 1. Architecture of a restricted Boltzmann machine. Inter-layer connections between the visible and the hidden layer

are represented by the black lines, where, for instance, the line connecting x1to h1represents the weight w11. The red lines

represent the visible biases, where the line going from the bias unit to the visible unit x3represents the bias weight a3. The

purple lines represent the hidden biases, where the line going from the bias unit to the hidden unit h3represents the bias weight

b3.

continuous inputs, illustrated in Fig. 1. The inputs x∈R2Nare the ﬂattened particle positions and interactions

between the particles are mediated by Hhidden binary nodes. After summing over all the possible values of the

hidden nodes, the marginal distribution of the inputs to the Gaussian-binary RBM takes the form

P(R;a,b,w) = exp −

i=1

(xi−ai)2

2σ2

i!H

j=1 "1 + exp bj+

i=1

xiwij

σ2

i!#.(9)

Here, a∈R2Nand b∈RHare the bias parameters of the input and hidden nodes, respectively. The weights between

the input and hidden nodes are w∈R2N×H, while σ∈R2Nare the widths of the Gaussian input nodes (not to be

confused with the spin projections). It is possible to train these widths by reparameterizing them as σi= exp(si),

but in this work all of the widths were ﬁxed to σ= 1/√ωand only the biases and weights are treated as variational

parameters. See Appendix A 1 for the derivation of the marginal probability.

Notice how the marginal distribution in Eq. (9) mimics the Gaussian parts of our aforementioned ans¨atze in Eqs. (6)

and (8). Based on such observations, our next step to is construct two corresponding ans¨atze

ψRBM(R;a,b,w) = P(R;a,b,w)×det hnHm(√ωxi)Hn(√ωyi)ξk(σi)oi,(10)

and

ψRBM+PJ(R;a,b,w, β) = P(R;a,b,w)×g(R;β)×det hnHm(√ωxi)Hn(√ωyi)ξk(σi)oi.(11)

The two trial wave functions above apply diﬀerent levels of physical intuition. While ψRBM does not contain speciﬁc

information about the electron-electron interactions, ψRBM+PJ contains a correlation factor that explicitly upholds the

cusp condition. Both ans¨atze contain knowledge about the required antisymmetry and the Gaussians in the marginal

distribution help localize the wave functions to satisfy the boundary conditions far from the oscillator well. Also,

as the marginal distribution is positive deﬁnite, these ans¨atze will never collapse into the bosonic state even if the

marginal distribution is not symmetric.

B. Code optimization

Parallel computing is an important part of our eﬀorts for developing an eﬃcient VMC solver. However, increasing

the available computational resources alone is often not suﬃcient. One should also consider developing sophisticated

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EcientSolutionsofFermionicSystemsusingArticialNeuralNetworksEvenM.NordhagenDepartmentofPhysicsandNjordCenter,UniversityofOslo,N-0316Oslo,NorwayJaneM.KimDepartmentofPhysicsandAstronomyandFacilityforRareIsotopeBeams,MichiganStateUniversity,EastLansing,MI48824,USABryceForeandAlessandroLovatoPhysicsDi...

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Ecient Solutions of Fermionic Systems using Articial Neural Networks Even M. Nordhagen Department of Physics and Njord Center University of Oslo N-0316 Oslo Norway

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