Diusion Monte Carlo using domains in conﬁguration space Roland Assaraf1Emmanuel Giner1Vijay Gopal Chilkuri2Pierre-François Loos2Anthony Scemama2and Michel Ca arel2 1Laboratoire de Chimie Théorique Sorbonne-Université Paris France

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Diﬀusion Monte Carlo using domains in conﬁguration space

Roland Assaraf,1Emmanuel Giner,1Vijay Gopal Chilkuri,2Pierre-François Loos,2Anthony Scemama,2and Michel Caﬀarel2, ∗

1Laboratoire de Chimie Théorique, Sorbonne-Université, Paris, France

2Laboratoire de Chimie et Physique Quantiques (UMR 5626), Université de Toulouse, CNRS, UPS, France

The sampling of the conﬁguration space in diﬀusion Monte Carlo (DMC) is done using walkers moving ran-

domly. In a previous work on the Hubbard model [Assaraf et al. Phys. Rev. B 60, 2299 (1999)], it was shown

that the probability for a walker to stay a certain amount of time in the same state obeys a Poisson law and that

the on-state dynamics can be integrated out exactly, leading to an eﬀective dynamics connecting only diﬀerent

states. Here, we extend this idea to the general case of a walker trapped within domains of arbitrary shape and

size. The equations of the resulting eﬀective stochastic dynamics are derived. The larger the average (trapping)

time spent by the walker within the domains, the greater the reduction in statistical ﬂuctuations. A numerical

application to the Hubbard model is presented. Although this work presents the method for (discrete) ﬁnite

linear spaces, it can be generalized without fundamental diﬃculties to continuous conﬁguration spaces.

I. INTRODUCTION

Diﬀusion Monte Carlo (DMC) is a class of stochastic

methods for evaluating the ground-state properties of quan-

tum systems. They have been extensively used in virtu-

ally all domains of physics and chemistry where the many-

body quantum problem plays a central role (condensed-matter

physics,1,2quantum liquids,3nuclear physics,4,5theoretical

chemistry,6etc). DMC can be used either for systems deﬁned

in a continuous conﬁguration space (typically, a set of parti-

cles moving in space) for which the Hamiltonian operator is

deﬁned in a Hilbert space of inﬁnite dimension or systems de-

ﬁned in a discrete conﬁguration space where the Hamiltonian

reduces to a matrix. Here, we shall consider only the dis-

crete case, that is, the general problem of calculating the low-

est eigenvalue and/or eigenstate of a (very large) matrix. The

generalization to continuous conﬁguration spaces presents no

fundamental diﬃculty.

In essence, DMC is based on stochastic power methods, a

family of well-established numerical approaches able to ex-

tract the largest or smallest eigenvalues of a matrix (see, e.g.,

Ref. 7). These approaches are particularly simple as they

merely consist in applying a given matrix (or some simple

function of it) as many times as required on some arbitrary

vector belonging to the linear space. Thus, the basic step of

the corresponding algorithm essentially reduces to successive

matrix-vector multiplications. In practice, power methods are

employed under more sophisticated implementations, such as,

e.g., the Lanczòs algorithm (based on Krylov subspaces)7

or Davidson’s method where a diagonal preconditioning is

performed.8When the size of the matrix is too large, matrix-

vector multiplications become unfeasible and probabilistic

techniques to sample only the most important contributions of

the matrix-vector product are required. This is the basic idea

of DMC. There exist several variants of DMC known under

various names: pure DMC,9DMC with branching,10 reptation

Monte Carlo,11 stochastic reconﬁguration Monte Carlo,12,13

etc. Here, we shall place ourselves within the framework of

pure DMC whose mathematical simplicity is particularly ap-

∗Corresponding author: caﬀarel@irsamc.ups-tlse.fr

pealing when developing new ideas. However, all the ideas

presented in this work can be adapted without too much dif-

ﬁculty to the other variants, so the denomination DMC must

ultimately be understood here as a generic name for this broad

class of methods.

Without entering into the mathematical details (which are

presented below), the main ingredient of DMC in order to per-

form the matrix-vector multiplications probabilistically is to

introduce a stochastic matrix (or transition probability matrix)

that generates stepwise a series of states over which statisti-

cal averages are evaluated. The critical aspect of any Monte

Carlo scheme is the amount of computational eﬀort required

to reach a given statistical error. Two important avenues to de-

crease the error are the use of variance reduction techniques

(for example, by introducing improved estimators14) or to im-

prove the quality of the sampling (minimization of the corre-

lation time between states). Another possibility, at the heart

of the present work, is to integrate out exactly some parts of

the dynamics, thus reducing the number of degrees of freedom

and, hence, the amount of statistical ﬂuctuations.

In previous works,15,16 it has been shown that the probabil-

ity for a walker to stay a certain amount of time in the same

state obeys a Poisson law and that the on-state dynamics can

be integrated out to generate an eﬀective dynamics connect-

ing only diﬀerent states with some renormalized estimators

for the properties. Numerical applications have shown that

the statistical errors can be very signiﬁcantly decreased. Here,

we extend this idea to the general case where a walker re-

mains a certain amount of time in a ﬁnite domain no longer

restricted to a single state. It is shown how to deﬁne an ef-

fective stochastic dynamics describing walkers moving from

one domain to another. The equations of the eﬀective dynam-

ics are derived and a numerical application for a model (one-

dimensional) problem is presented. In particular, it shows that

the statistical convergence of the energy can be greatly en-

hanced when domains associated with large average trapping

times are considered.

It should be noted that the use of domains in quantum

Monte Carlo is not new. Domains have been introduced within

the context of Green’s function Monte Carlo (GFMC) pio-

neered by Kalos17,18 and later developed and applied by Kalos

and others.19–22 In GFMC, an approximate Green’s function

arXiv:2210.11824v2 [cond-mat.str-el] 31 Dec 2022

that can be sampled is required for the stochastic propagation

of the wave function. In the so-called domain GFMC version

of GFMC introduced in Ref. 18 and 19 the sampling is real-

ized by using the restriction of the Green’s function to a small

domain consisting of the cartesian product of small spheres

around each particle, the potential being considered constant

within the domain. Fundamentally, the method presented in

this work is closely related to the domain GFMC, although

the way we present the formalism in terms of walkers trapped

within domains and derive the equations that may appear dif-

ferent. However, we show here how to use domains of arbi-

trary size, a new feature that greatly enhances the eﬃciency

of the simulations when domains are suitably chosen, as illus-

trated in our numerical application.

Finally, from a general perspective, it is interesting to em-

phasize that the present method illustrates how suitable com-

binations of stochastic and deterministic techniques lead to

a more eﬃcient and valuable method. In recent years, a

number of works have exploited this idea and proposed hy-

brid stochastic/deterministic schemes. Let us mention, for

example, the semi-stochastic approach of Petruzielo et al.,23

two diﬀerent hybrid algorithms for evaluating the second-

order perturbation energy in selected conﬁguration interac-

tion methods,24,25 the approach of Willow et al. for com-

puting stochastically second-order many-body perturbation

energies,26 or the zero variance Monte Carlo scheme for eval-

uating two-electron integrals in quantum chemistry.27

The paper is organized as follows. Section II presents the

basic equations and notations of DMC. First, the path inte-

gral representation of the Green’s function is given in Sub-

sec. II A. Second, the probabilistic framework allowing the

Monte Carlo calculation of the Green’s function is presented

in Subsec. II B. Section III is devoted to the use of domains

in DMC. We recall in Subsec. III A the case of a domain

consisting of a single state. The general case is then treated

in Subsec. III B. In Subsec. III C, both the time- and energy-

dependent Green’s function using domains are derived. Sec-

tion IV presents the application of the approach to the one-

dimensional Hubbard model. Finally, in Sec.V, some con-

clusions and perspectives are given. Atomic units are used

throughout.

II. DIFFUSION MONTE CARLO

A. Path-integral representation

As previously mentioned, DMC is a stochastic implemen-

tation of the power method deﬁned by the following operator:

T=1−τ(H−E1),(1)

where 1is the identity operator, τa small positive parameter

playing the role of a time step, Esome arbitrary reference

energy, and Hthe Hamiltonian operator. For any initial vector

|Ψ0iprovided that hΦ0|Ψ0i,0 and for τsuﬃciently small,

we have

lim

N→∞ TN|Ψ0i=|Φ0i,(2)

where |Φ0iis the ground-state wave function, i.e., H|Φ0i=

E0|Φ0i. The equality in Eq. (2) holds up to a global phase

factor playing no role in physical quantum averages. At large

but ﬁnite N, the vector TN|Ψ0idiﬀers from |Φ0ionly by an

exponentially small correction, making it straightforward to

extrapolate the ﬁnite-Nresults to N→ ∞.

Likewise, ground-state properties may be obtained at large

N. For example, in the important case of the energy, one can

project out the vector TN|Ψ0ion some approximate vector,

|ΨTi, as follows

E0=lim

N→∞

hΨ0|TN|HΨTi

hΨ0|TN|ΨTi.(3)

|ΨTiis known as the trial wave vector (function), and is chosen

as an approximation of the true ground-state vector.

To proceed further we introduce the time-dependent

Green’s matrix G(N)deﬁned as

G(N)

i j =hi|TN|ji.(4)

where |iiand |jiare basis vectors. The denomination “time-

dependent Green’s matrix” is used here since Gmay be

viewed as a short-time approximation of the (time-imaginary)

evolution operator e−NτH, which is usually referred to as the

imaginary-time dependent Green’s function.

Introducing the set of N−1 intermediate states, {|iki}1≤k≤N−1,

between each Tin TN,G(N)can be written in the following

expanded form

G(N)

i0iN=X

i1X

· · · X

iN−1

N−1

k=0

Tikik+1,(5)

where Ti j =hi|T|ji. Here, each index ikruns over all basis

vectors.

In quantum physics, Eq. (5) is referred to as the path-

integral representation of the Green’s matrix (or function).

The series of states |i0i,...,|iNiis interpreted as a “path” in

the Hilbert space starting at vector |i0iand ending at vector

|iNi, where kplays the role of a time index. Each path is as-

sociated with a weight QN−1

k=0Tikik+1and the path integral ex-

pression of Gcan be recast in the more suggestive form as

follows:

G(N)

i0iN=X

all paths |i1i,...,|iN−1i

N−1

k=0

Tikik+1.(6)

This expression allows a simple and vivid interpretation of

the solution. In the limit N→ ∞, the iNth component of the

ground-state wave function (obtained as limN→∞ G(N)

i0iN) is the

weighted sum over all possible paths arriving at vector |iNi.

This result is independent of the initial vector |i0i, apart from

some irrelevant global phase factor. We illustrate this funda-

mental property pictorially in Fig. 1. When the size of the

linear space is small, the explicit calculation of the full sums

involving MNterms (where Mis the size of the Hilbert space)

can be performed. In such a case, we are in the realm of what

one would call “deterministic” power methods, such as the

Lanczòs or Davidson approaches. If not, probabilistic tech-

niques for generating only the paths contributing signiﬁcantly

to the sums are to be used. This is the central theme of the

present work.

B. Probabilistic framework

To derive a probabilistic expression for the Green’s matrix,

we introduce a guiding wave function |Ψ+ihaving strictly pos-

itive components, i.e., Ψ+

i>0, in order to perform a similarity

transformation of the operators G(N)and T,

Ti j =

Ψ+

Ti j,¯

G(N)

i j =

Ψ+

G(N)

i j .(7)

Note that, thanks to the properties of similarity transforma-

tions, the path integral expression relating G(N)and T[see

Eq. (6)] remains unchanged for ¯

G(N)and ¯

Now, the key idea to take advantage of probabilistic tech-

niques is to rewrite the matrix elements of ¯

Tas those of a

stochastic matrix multiplied by some residual weights (here,

not necessarily positive), namely

Ti j =pi→jwi j.(8)

Here, we recall that a stochastic matrix is deﬁned as a matrix

with positive entries that obeys

pi→j=1.(9)

Using this representation for ¯

Ti j the similarity-transformed

Green’s matrix components can be rewritten as

G(N)

i0iN=X

i1,...,iN−1







N−1

k=0

pik→ik+1





N−1

k=0

wikik+1,(10)

which is amenable to Monte Carlo calculations by generating

paths using the transition probability matrix pi→j.

Let us illustrate this in the case of the energy as given by

Eq. (3). Taking |Ψ0i=|i0ias initial state, we have

E0=lim

N→∞ PiNG(N)

i0iN(HΨT)iN

PiNG(N)

i0iNΨTiN

.(11)

which can be rewritten probabilistically as

E0=lim

N→∞

QN−1

k=0wikik+1

(HΨT)iN

Ψ+

iN

QN−1

k=0wikik+1

ΨTiN

Ψ+

iN,(12)

where h...iis the probabilistic average deﬁned over the set of

paths |i1i,...,|iNioccurring with probability

Probi0(i1,...,iN)=

N−1

k=0

pik→ik+1.(13)

Using Eq. (9) and the fact that pi→j≥0, one can easily verify

that Probi0is positive and obeys

i1,...,iN

Probi0(i1,...,iN)=1,(14)

as it should.

The rewriting of ¯

Ti j as a product of a stochastic matrix times

some general real weight does not introduce any constraint

on the choice of the stochastic matrix, so that, in theory, any

stochastic matrix could be used. However, in practice, it is

highly desirable that the magnitude of the ﬂuctuations of the

weight during the Monte Carlo simulation be as small as pos-

sible. A natural solution is to choose a stochastic matrix as

close as possible to ¯

Ti j. This is done as follows.

Let us introduce the following operator

T+=1−τH+−E+

L1,(15)

where H+is the matrix obtained from Hby imposing the oﬀ-

diagonal elements to be negative

H+

i j =









Hi j,if i=j,

−Hi j,if i,j.(16)

Here, E+

L1is the diagonal matrix whose diagonal elements are

deﬁned as

(E+

L)i=PjH+

i jΨ+

Ψ+

.(17)

The vector E+

Lis known as the local energy vector associated

with Ψ+. By construction, the operator H+−E+

L1in the def-

inition of T+[see Eq. (15)] has been chosen to admit |Ψ+i

as a ground-state wave function with zero eigenvalue, i.e.,

H+−E+

L1|Ψ+i=0, leading to the relation

T+Ψ+=Ψ+.(18)

We are now in the position to deﬁne the stochastic matrix

pi→j=

Ψ+

T+

i j =









1−τhH+

ii −(E+

L)ii,if i=j,

τΨ+

Ψ+

iHi j≥0,if i,j.(19)

As readily seen in Eq. (19), the oﬀ-diagonal terms of the

stochastic matrix are positive, while the diagonal terms can

be made positive if τis chosen suﬃciently small via the con-

dition

τ≤1

maxiH+

ii −(E+

L)i

.(20)

The sum-over-states condition [see Eq. (9)],

pi→j=hi|T+|Ψ+i

Ψ+

=1,(21)

follows from the fact that |Ψ+iis eigenvector of T+, as evi-

denced by Eq. (18). This ensures that pi→jis indeed a stochas-

tic matrix.

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DiusionMonteCarlousingdomainsincongurationspaceRolandAssaraf,1EmmanuelGiner,1VijayGopalChilkuri,2Pierre-FrançoisLoos,2AnthonyScemama,2andMichelCaarel2,1LaboratoiredeChimieThéorique,Sorbonne-Université,Paris,France2LaboratoiredeChimieetPhysiqueQuantiques(UMR5626),UniversitédeToulouse,CNRS,UPS,Fra...

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Diusion Monte Carlo using domains in conﬁguration space Roland Assaraf1Emmanuel Giner1Vijay Gopal Chilkuri2Pierre-François Loos2Anthony Scemama2and Michel Ca arel2 1Laboratoire de Chimie Théorique Sorbonne-Université Paris France.pdf

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Diusion Monte Carlo using domains in conﬁguration space Roland Assaraf1Emmanuel Giner1Vijay Gopal Chilkuri2Pierre-François Loos2Anthony Scemama2and Michel Ca arel2 1Laboratoire de Chimie Théorique Sorbonne-Université Paris France

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