Generating Approximate Ground States of Molecules Using Quantum Machine Learning Jack Ceroni1 2Torin F. Stetina3 4Mária Kieferová5Carlos

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Generating Approximate Ground States of Molecules Using Quantum Machine

Learning

Jack Ceroni,

1, 2

Torin F. Stetina,

3, 4

Mária Kieferová,

Carlos

Ortiz Marrero,

6, 7

Juan Miguel Arrazola,

and Nathan Wiebe

8, 9

Xanadu, Toronto, ON, M5G 2C8, Canada

Department of Mathematics, University of Toronto, Toronto, ON, M5S 3E1, Canada

∗

Simons Institute for the Theory of Computing, Berkeley, CA, 94704, USA

Berkeley Quantum Information and Computation Center,

University of California, Berkeley, CA 94720, USA

†

Centre for Quantum Computation and Communication Technology,

Centre for Quantum Software and Information, University of Technology Sydney, NSW 2007, Australia

‡

AI & Data Analytics Division, Paciﬁc Northwest National Laboratory, Richland, WA 99354

Department of Electrical & Computer Engineering,

North Carolina State University, Raleigh, NC 27607

Department of Computer Science, University of Toronto, ON M5S 1A1, Canada

High Performance Computing Group, Paciﬁc Northwest National Laboratory, Richland, WA 99354

(Dated: January 3, 2023)

The potential energy surface (PES) of molecules with respect to their nuclear positions is a primary

tool in understanding chemical reactions from ﬁrst principles. However, obtaining this information

is complicated by the fact that sampling a large number of ground states over a high-dimensional

PES can require a vast number of state preparations. In this work, we propose using a generative

quantum machine learning model to prepare quantum states at arbitrary points on the PES. The

model is trained using quantum data consisting of ground-state wavefunctions associated with

diﬀerent classical nuclear coordinates. Our approach uses a classical neural network to convert the

nuclear coordinates of a molecule into quantum parameters of a variational quantum circuit. The

model is trained using a ﬁdelity loss function to optimize the neural network parameters. We show

that gradient evaluation is eﬃcient and numerically demonstrate our method’s ability to prepare

wavefunctions on the PES of hydrogen chains, water, and beryllium hydride. In all cases, we ﬁnd that

a small number of training points are needed to achieve very high overlap with the groundstates in

practice. From a theoretical perspective, we further prove limitations on these protocols by showing

that if we were able to learn across an avoided crossing using a small number of samples, then

we would be able to violate Grover’s lower bound. Additionally, we prove lower bounds on the

amount of quantum data needed to learn a locally optimal neural network function using arguments

from quantum Fisher information. This work further identiﬁes that quantum chemistry can be an

important use case for quantum machine learning.

I. INTRODUCTION

One of the most widely-studied uses of quantum com-

puters is the simulation and characterization of physical

systems. Signiﬁcant eﬀort has been dedicated to develop-

ing quantum algorithms for a variety of calculations across

many areas of physics, including condensed matter [

molecular physics [

], and quantum ﬁeld theory [

However, in recent years, quantum chemistry has emerged

as a leading application, and considerable focus has been

placed on using quantum devices to determine the prop-

erties of molecules and materials [

–

]. Most proposals

for quantum computational chemistry algorithms attempt

to solve the electronic structure problem, in which the

nucleus of a molecule is assumed to be ﬁxed, and the goal

∗jack.ceroni@mail.utoronto.ca

†torins@berkeley.edu

‡maria.kieferova@uts.edu.au

§carlos.ortizmarrero@pnnl.gov

¶nawiebe@cs.toronto.edu

is to compute the ground-state energy of the electronic

Hamiltonian. While the general problem of computing

the ground state of an arbitrary Hamiltonian is

QMA

complete [

], under certain conditions, such as when

given access to a suﬃciently high-ﬁdelity approximation

of the true ground state [

–

], quantum computers can

eﬃciently yield accurate ground states. The most studied

proposals for solving the electronic structure problem on

circuit-based quantum computers are the variational quan-

tum eigensolver (VQE) [

–

] for the noisy, intermediate

scale regime, and quantum phase estimation (QPE) [

]

for fault-tolerant quantum computers. Both techniques

yield an approximation of the ground state energy of a

Hamiltonian, as well as the ground state itself.

While these approaches are eﬀective in many scenarios,

their use comes with a considerable cost. For example,

for the FeMoCo molecule, a widely-used benchmark in

quantum computational chemistry, the best estimate of

QPE runtime for determining its ground state energy

is just under 4 days [

] on a fault-tolerant quantum

computer with millions of physical qubits. On the other

hand, VQE has the possibility of allowing for quantum

arXiv:2210.05489v3 [quant-ph] 2 Jan 2023

computational chemistry with fewer, noisier qubits, but

the number of measurements needed to estimate energies

is often signiﬁcant, making scaling of the algorithm to

large molecules a challenge [22–25].

The challenges for both QPE and VQE worsen even

further when they are considered in the context of solving

practical problems in quantum chemistry. The electronic

structure problem assumes a ﬁxed molecular conﬁguration,

and therefore a ﬁxed molecular Hamiltonian, of which

we compute the ground state. In order to determine

many dynamic or structural properties of molecules, such

as reaction barriers and optimal geometries, a molecule

must be studied in many diﬀerent conﬁgurations. In gen-

eral, characterizing this behaviour requires knowledge of

a family of ground states for a set of Hamiltonians param-

eterized by classical nuclear coordinates

(

). This is an

arduous task, which requires computing many diﬀerent

ground states with corresponding energies lying on a high-

dimensional potential energy surface. Running quantum

algorithms such as QPE or VQE for each conﬁguration in-

dependently would represent a signiﬁcant computational

cost even for molecules with a modest number of atoms.

We propose in this paper an alternative method for

computing ground states corresponding to a wide range

of molecular conﬁgurations, i.e., for reconstructing po-

tential energy surfaces of molecules. A key motivation

behind this work is that while the ﬁxed nuclei electronic

structure problem is already diﬃcult, the ultimate goal of

using quantum computers to compute accurate electronic

energies requires sampling over many diﬀerent nuclear

conﬁgurations in a proposed chemical reaction coordinate.

Therefore, the generation of many ground states, and

subsequently energies and other properties, is of central

interest in taking advantage of quantum computers for

designing new materials and technologies. Instead of

computing the ground states for many discrete molecular

conﬁgurations independently, our algorithm uses a limited

collection of data and, employing techniques from ma-

chine learning, builds a model that prepares the ground

state over some region in parameter space.

Our work connects with recent progress for the task

of learning from quantum mechanical data with both

classical and quantum methods, an increasingly active

area of research. Notable results include demonstration

that classical machine learning techniques are provably

eﬃcient for predicting and modelling certain properties

of quantum many-body systems [

], and that quantum

learning procedures can be more eﬃcient than classical

learning procedures for determining speciﬁc properties of

certain unknown quantum states and processes [

–

Our proposed algorithm shares some similarities with the

above works, with the key diﬀerence being that the output

of our model is a quantum state, rather than an estimate

of some observable quantity or a classical approximation

of a quantum state [

]. Therefore, algorithms of

the form we proposed can be used to extract arbitrary

ground state observables, or to output states that can

be used in other quantum computational procedures re-

quiring access to ground states. From the perspective of

Ref. [

], this model falls into the quantum-quantum or

“QQ” category of machine learning techniques: a quantum

model trained with quantum data, and complements the

growing literature on using quantum data and quantum

machine learning to understand quantum systems (in par-

ticular, quantum chemical systems). Existing examples of

QQ machine learning include learning excited states from

ground state [

], compression of quantum data [

], and

learning of parametrized Hamiltonians [

], which has

been applied to spin and molecular Hamiltonians under

the name quantum meta-learning [35].

The algorithm proposed in this work is a hybrid

classical-quantum generative model, in which we train a

classical neural network to yield parameters, which when

fed into a low-depth variational quantum circuit, approxi-

mate the corresponding ground state of

(

)for a range

of values of

. To train our model, we assume access to

quantum data: ground states of

(

)for a collection of

coordinates

{Ri}N

i=1

, which can be loaded into a quantum

computer. Since ground state preparation is a resource

intensive task, the amount of quantum data needed to

learn a model is a key metric for quantifying the eﬃciency

and the feasibility of the algorithm. Ideally, a model of

this form should generalize to new values of

not con-

tained in the training data. This allows us to generate

approximations of previously unseen molecular ground

states, at the more modest price of executing a shallow,

variational quantum circuit for some set of parameters

determined by a classical neural network.

To test these capabilities, we perform extensive numeri-

cal experiments for a collection of diﬀerent molecules, and

ﬁnd that even with few data points, there is good gen-

eralization to unseen geometries in the potential energy

surface. Ultimately, the aim of our proposal is to provide

a concrete ﬁrst step towards the development of practi-

cal techniques based on quantum machine learning for

alleviating the cost of computing ground states of a param-

eterized molecular Hamiltonian. Under this framework,

hard-to-obtain quantum data, originating from quantum

algorithms or physical experiments [

], is the resource

that we attempt to leverage.

It is important to note that the particular generative

model proposed is only one member of a large family

of quantum-classical machine learning architecture for

preparing ground states. Advanced models may utilize

more sophisticated choices of cost function, classical neu-

ral network architecture, initialization, and circuit con-

struction than those considered in this paper. The goal

of this work is to both discuss the general concept of

generative quantum machine learning applied to quantum

chemistry, as well as provide a concrete example of what

such a model would look like. As a result, we explore

both practicalities associated with the particular model,

including gradient sample complexity and numerical ex-

periments, as well as more general considerations about

abstract quantum state-learning procedures.

We begin in Sec. II by outlining a general architecture

and training strategy for a generative model which pre-

pares ground states of a parameterized Hamiltonian. In

Sec. III, we discuss the details of training, and provide

estimates on the sample complexity required for com-

puting gradients of the model, which is necessary for

optimization. In Sec. IV, to better understand the limits

of quantum generative models, we introduce theoretical

bounds related to data complexity of general quantum

state-learning algorithms, and interpret these results in

the context of our generative model. We conclude in

Sec. V by providing numerics to support the quality of

our model, demonstrating that one can eﬀectively learn

out-of-distribution ground states, and thus the resulting

potential energy surfaces, of the H

, H

, BeH

, and

H2O molecules, to a high degree of accuracy.

II. A GENERATIVE MODEL FOR PREPARING

ELECTRONIC GROUND STATES

Quantum chemistry has, in recent years, become

a major focus for the development of quantum algo-

rithms [

]. The speciﬁc problem that has domi-

nated most discussion surrounding quantum computing

in this space is the electronic structure problem, which

involves ﬁnding the minimum energy conﬁguration for a

molecule. This problem reduces to the problem of com-

puting the groundstate energy of a quantum system. The

electronic structure problem in a ﬁxed basis is, in general,

QMA

-Hard [

] Regardless, in many cases though a state

with suﬃcient overlap can be identiﬁed and the overlap

with the state can be computed within constant error



using a polynomial number of quantum gates [

The Hamiltonian for the electrons within the Born-

Oppenheimer approximation (which states that the nu-

clear motion is uncoupled with the electronic motion) can

be written as

H(R) = X

hpq(R)a†

paq+1

pqrs

hpqrs(R)a†

pa†

qaras,(1)

where

a†

and

are the fermionic creation and annihi-

lation operators acting on the p-th orbital, and

hpq(R) = Zdr φ∗

p(r) −∇2

2−X

|r−RI|!φq(r),

(2)

hpqrs(R) = Zdr1dr2

φ∗

p(r1)φ∗

q(r2)φr(r2)φs(r1)

|r1−r2|,(3)

are the one and two-electron integrals in the molecular

orbital basis,

φp

(

), yielded from the Hartree-Fock opti-

mization procedure. The one and two-electron integrals,

and subsequently the molecular orbitals depend implicitly

, which is the general nuclear coordinate associated

with the ﬁxed nuclear conﬁguration in 3-dimensional space.

We then utilize the Jordan-Wigner transform in order to

map the fermionic creation and annihilation operators to

qubit operators, which can be implemented in a quantum

circuit [42, 43].

The best known costs for estimating the groundstate

energy at a speciﬁc nucelar conﬁguration scales as [

]

(

Nλ/

)where

is the number of spin-orbitals used

in the model and

is a quantity that depends on the

details of the Hamiltonian and the basis set chosen. For

the case where a planewave basis is chosen an explicit

scaling can be achieved of

(

N3/

)[

]; however, such

an explicit scaling is diﬃcult to derive in a Gaussian

basis (like those commonly used in chemistry) but can be

numerically veriﬁed to scale polynomially in

for most

examples considered.

The challenge behind all of this is that the cost of

preparing these groundstates is diﬃcult. It requires per-

forming phase estimation to learn the groundstate en-

ergy within the eigenvalue gap, which leads to a cost of

(

N3/|E1−E0|

)for the case of planewaves. Such costs

for generating a quantum state at any point on the poten-

tial energy surface can be challenging for problems with

small gaps and so a question that naturally emerges is

whether there exist faster methods for preparing approx-

imate groundstates on quantum computers. In general,

we do not expect this to be possible without prior infor-

mation; however, when approaching chemistry we almost

never have a uniform prior. We are frequently met with

a host of prior experiments and often have systems that

are smooth enough so nearby queries yield information

about the quantum state in question. Our aim is to use

quantum machine learning as a paradigm to estimate

new states on the potential energy surface of a molecule.

We discuss how generative quantum machine learning to

approach this challenge below.

A. Generative QML Models for Chemistry

The generative model that we propose contains both a

classical and quantum component. We begin by ﬁxing a

parameterized quantum circuit

(

). Although the struc-

ture of

can vary, we assume that it can be expressed

as a series of parameterized rotations [17],

U(θ) =

j=1

e−iθjHj,(4)

where each

is Hermitian and chosen prior to learning.

Given a parameterized Hamiltonian

(

), the goal is

to learn a map

R7→ θ

(

), such that

(

))

|ψinit

(

)

approximates the ground state

|ψ0

(

)

(

)over some

range of

, where

|ψinit

(

)

is some pre-chosen initial

state. For the sake of simplicity,

(

)is taken to be

non-degenerate over

in the region considered. In order

FIG. 1: A diagram showing how a neural network is used to parameterize a map from coordinates Rto a state

|ψ(ν(R;γ))i. The goal is to ﬁnd the optimal parameters γ=γ∗such that when they are used in the generative

procedure, the resulting map from

|ψ

(

;

γ∗

))

approximates the map from

to the ground state

|ψ0

(

)

. This

allows for accurate extraction of ground state observables, such as ground state energies, from our model, over R.

to ﬁnd the mapping

R7→ θ

(

), we are given access to a

collection of quantum data, D, of the form

D=





Ri,



z }| {

|ψ0(Ri)i, . . . , |ψ0(Ri)i









i=1

,(5)

where each

is unique and we have access to

copies of

a state

|ψ0

(

)

. We assume a polynomially large number

of data, which should be contrasted with the exponentially

large number of possible grid points

that deﬁne a full

potential energy surface in high dimensions. We do not

explicitly consider the origin of the data set, but it can

be created with several diﬀerent methods, possibly by re-

peated application of QPE or VQE for the parameters

or even by transduction of quantum states yielded from

an experiment into a quantum device, as is envisioned

in Ref. [

]. We note that the model is not restricted

to returning parameterized ground states: given training

data, it can be used to construct an approximation of

any parameterized state. For example, one could imag-

ine using a similar procedure to produce a model of a

parameterized excited state

|ψj

(

)

. We focus on ground

state generation for the sake of concreteness, and due to

the fact that computing ground states is a problem of

practical importance.

Let

(

;

)denote a classical neural network, where

are the trainable parameters, which induces a func-

tion

R7→ ν

(

;

). Using the data

along with

we perform an optimization procedure which yields

γ∗

such that the output state

(

;

γ∗

))

|ψinit

(

)

approx-

imates

|ψ0

(

)

, for each

in the training data. Deﬁne

(

) :=

(

;

γ∗

)to be the output of the model. The goal

of such a training procedure is good generalization for all

values of R. In other words,

|ψ(ν(R;γ∗))i:= U(ν(R;γ∗))|ψinit(R)i

=U(θ(R))|ψinit(R)i,(6)

should approximate

|ψ0

(

)

for a wider range of

outside

of the training set. To ﬁnd the optimal parameters

γ∗

, we

minimize a cost function

(

)over the training data. In

this work, we choose

to be the average inﬁdelity of the

state produced by the model with each unique training

example (each state in

corresponds to a unique

More speciﬁcally, the cost function is

C(γ) = 1 −1

i=1 |hψ0(Ri)|ψ(ν(Ri;γ))i|2,(7)

where for suﬃciently expressive

, minimization of

will

result in

γ∗

such that

|ψ

(

;

γ∗

))

i≈|ψ0

(

)

for each

. One immediate beneﬁt of using the inﬁdelity cost

function is that it admits relatively simple gradients of

the form

∂C(γ)

∂γk

=−1

i=1 2Re [hψ0(Ri)|ψ(ν(Ri;γ))i]

a=1

∂ν(Ri;γ)a

∂γkDψ0(Ri)∂ψ(θ)

∂θaEθ=ν(Ri;γ)!,(8)

where

|ψ

(

)

(

)

|ψinit

(

)

for each term in the sum

and

is the output dimension of the classical neural

network, i.e, the number of parameters of the quantum

circuit (for a derivation, see Sec. III). Overlaps between

the training states, and the output/output derivatives of

the parameterized circuit

can be computed via sam-

pling, or amplitude estimation and the linear combination

of unitaries (LCU) method [

], while the derivatives

can be computed eﬃciently in the case of deep neural

networks via backpropagation [

]. With a procedure for

computing derivatives, it is then possible to minimize

via gradient-based optimization methods, such as gradient

descent or Adam [

]. The computational cost of comput-

ing gradients of the cost function in Eq.

(7)

is discussed

in detail in Section III.

We conclude this section by discussing one of the central

challenges in designing quantum machine learning models:

barren plateaus, and how the proposed model may be

robust to this issue. The barren plateaus problem is the

observation that many classes of parameterized quantum

circuits suﬀer from exponentially vanishing gradients over

large parts of the parameter space [

]. Barren plateaus

highlight the necessity to incorporate inductive biases into

parameterized quantum circuits based on the structure

of a particular problem, by choosing appropriate circuit

ansatzes, cost function, and initialization, among other

hyperparameters. As is discussed in Refs. [

(

)of

the general form in Eq.

(8)

is conducive to barren plateaus.

This issue can, in some cases, be resolved by choosing a

more sophisticated cost function [

]. However, before

swapping

for one of these functions, it is important to

note that any generative model tasked with returning a

molecular ground state can have as its initial guess an

approximation of the true ground state. In our work we

set

|ψinit

(

)

|ψHF

(

)

, where

|ψHF

(

)

is the Hartree-

Fock state at geometry

; an approximate solution to the

electronic Schrodinger equation which can be computed

eﬃciently on a classical device.

The parameterized quantum circuit

can then be

thought of as applying corrections to the Hartree-Fock

state. This choice makes the initialization of our model

far from random. In addition, since we have access to

the Hamiltonian

(

)of which we are attempting to

prepare the ground state, we can use this knowledge

to tailor our circuit ansatz to each particular molecule

considered, using the ADAPT-VQE algorithm [

] (see

Sec. V for details). Ref. [

] provides empirical evidence

that in many cases, adaptive circuits do not suﬀer from

the issue of barren plateaus. It is therefore reasonable

to hypothesize that initialization in the Hartree-Fock

state and an adaptively-prepared circuit will, in many

cases, constrain the model’s optimization to a region that

does not suﬀer from the barren plateau problem. This

conclusion is supported in the numerics (Sec. V), where

we observe good performance when using the inﬁdelity

cost function of Eq. (7) for training.

III. GRADIENT SAMPLING COMPLEXITY

While the model described in Section II is a hybrid

quantum-classical procedure, the part of the algorithm

that is executed on a quantum device can be reduced to

the calculation of gradients of the cost function

(

). As

a result, understanding the quantum sample complexity

of gradient calculations allows us to better understand the

cost of running the entire algorithm. The other important

aspect is the number of training steps required to ﬁnd

the optimal

γ∗

, but this task is much more challenging,

and likely varies considerably on a case-by-case basis.

In this section, we discuss the sample complexity re-

quired to compute gradients of the average inﬁdelity cost

function introduced in Section II. Recall that

|ψ

(

)

(

)

|ψinit

(

)

for some

. We can express the gradient

of the i-th overlap terms in Eq. (7) as

∂

∂γk|hψ0(Ri)|ψ(ν(Ri;γ))i|2

=∂

∂γkhψ0(Ri)|ψ(ν(Ri;γ))ihψ(ν(Ri;γ))|ψ0(Ri)i

=hψ(ν(Ri;γ))|ψ0(Ri)iDψ0(Ri)∂ψ(ν(Ri;γ))

∂γkE+h.c.

= 2Re hψ(ν(Ri;γ))|ψ0(Ri)iDψ0(Ri)∂ψ(ν(Ri;γ))

∂γkE

2Re"hψ(ν(Ri;γ))|ψ0(Ri)i

a=1

∂ν(Ri;γ)a

∂γk

×Dψ0(Ri)∂ψ(θ)

∂θaEθ=ν(Ri;γ)#.(9)

Now, note that



∂ψ(θ)

∂θa=∂

∂θa

U(θ)|ψinit(Ri)i

=∂

∂θa"NP

j=1

e−iθjHj#|ψinit(Ri)i

=−i"Y

j<a

e−iθjHjHaY

j≥a

e−iθjHj#|ψinit(Ri)i

=−i"Y

j<a

e−iθjHjHa Y

j<a

e−iθjHj!†

×Y

j<a

e−iθjHjY

j≥a

e−iθjHj#|ψinit(Ri)i

=−i"Y

j<a

e−iθjHjHa Y

j<a

e−iθjHj!†#|ψ(θ)i

=−iˆ

Ha(θ)|ψ(θ)i,(10)

where

(

) :=

Qj<a e−iθjHjHaQj<a e−iθjHj†

. Let

(

;

)). For brevity, we will henceforth refer

|ψ

(

;

))

|ψi

, and

|ψ0

(

)

|ψii

. Using Eq.

(9)

and Eq. (10), we get

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GeneratingApproximateGroundStatesofMoleculesUsingQuantumMachineLearningJackCeroni,1,2TorinF.Stetina,3,4MáriaKieferová,5CarlosOrtizMarrero,6,7JuanMiguelArrazola,1andNathanWiebe8,91Xanadu,Toronto,ON,M5G2C8,Canada2DepartmentofMathematics,UniversityofToronto,Toronto,ON,M5S3E1,Canada3SimonsInstitutefort...

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