Orbital Expansion Variational Quantum Eigensolver Enabling Ecient Simulation of Molecules with Shallow Quantum Circuit Yusen Wu1 2Zigeng Huang1yJinzhao Sun3Xiao Yuan4 5Jingbo B. Wang2and Dingshun Lv1z

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Orbital Expansion Variational Quantum Eigensolver:

Enabling Eﬃcient Simulation of Molecules with Shallow Quantum Circuit

Yusen Wu,1, 2, ∗Zigeng Huang,1, †Jinzhao Sun,3Xiao Yuan,4, 5 Jingbo B. Wang,2and Dingshun Lv1, ‡

1ByteDance Ltd., Zhonghang Plaza, No. 43, North 3rd Ring West Road, Haidian District, Beijing, China

2Department of Physics, The University of Western Australia, Perth, WA 6009, Australia

3Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom

4Center on Frontiers of Computing Studies, Peking University, Beijing 100871, China

5School of Computer Science, Peking University, Beijing 100871, China

In the noisy-intermediate-scale-quantum era, Variational Quantum Eigensolver (VQE) is a promis-

ing method to study ground state properties in quantum chemistry, materials science, and condensed

physics. However, general quantum eigensolvers are lack of systematical improvability, and achieve

rigorous convergence is generally hard in practice, especially in solving strong-correlated systems.

Here, we propose an Orbital Expansion VQE (OE-VQE) framework to construct an eﬃcient con-

vergence path. The path starts from a highly correlated compact active space and rapidly expands

and converges to the ground state, enabling simulating ground states with much shallower quantum

circuits. We benchmark the OE-VQE on a series of typical molecules including H6-chain, H10-ring

and N2, and the simulation results show that proposed convergence paths dramatically enhance the

performance of general quantum eigensolvers.

I. INTRODUCTION

Solving ground states of quantum systems is a nat-

ural application for quantum computers, which could

potentially innovate the study of quantum chemistry,

materials science, and many-body physics [1–5]. How-

ever, quantum techniques like quantum phase estimation

which promises accurate chemical simulations requiring

fault-tolerant quantum computers [6], is beyond current

quantum computers. In order to reduce the signiﬁcant

hardware demands required by universal quantum algo-

rithms, the variational quantum eigensolver (VQE) was

proposed [7–12] and demonstrated on noisy-intermediate

scale quantum (NISQ) devices [13–18]. While quantum

hardware continue to steadily advance, limitations on

deep high-ﬁdelity circuit still exist [19,20]. Therefore,

the demand of quantum algorithms that try to make full

use of the limited quantum resource is growing rapidly.

Particularly, one may think: how to eﬃciently design a

powerful quantum ansatz allowing accurate computation

with much shallower circuit depth. In other words, ﬁnd

an eﬃcient convergence path from the ﬁxed initial state

(such as Hartree-Fock state) to the exact ground state.

Actually, there are exponential large number of paths

between two states in the Bloch sphere, and the length

of the optimal convergence path deﬁnes the circuit com-

plexity of the ground state [21].

From diﬀerent perspectives, previous works consider

various strategies to reduce the quantum circuit depth

in solving quantum chemical problems, including qubit-

reduction methods [22–29], circuit depth optimization

methods [30–34] and unitary couple cluster with its vari-

ant methods [7,9–11,35,36]. Although such methods

∗The ﬁrst two authors contributed equally.

†huangzigeng@bytedance.com

‡lvdingshun@bytedance.com

can be utilized to reduce quantum circuit depth, they

suﬀer from a lack of systematic improvability when con-

structing a convergence path, that is the energy func-

tion may not decline steadily with the increase of quan-

tum (or classical) computational resources. For exam-

ple, the famous heuristic method ADAPT-VQE which

proposed in [30] may encounter the scenario where the

single gradient-based criteria cannot guarantee to ﬁnd

the global minimum. This because general unitary op-

erators lie in an extremely complex manifold, and ﬁxed

optimization strategy may fail.

Here, we propose an Orbital Expansion VQE (OE-

VQE) framework to provide an eﬃcient convergence path

in simulating molecular systems, which dramatically im-

proves the performance of a general quantum eigensolver

and reduces quantum measurement complexity. The pro-

posed convergence path is composed of two fundamen-

tal elements: (i) a good initial state inspired by chem-

ical insights and (ii) a systematic improvable conver-

gence direction to the ground state. In the proposed

OE-VQE framework, a specially initial state, which is

constructed from a very small active space composed by

orbitals with signiﬁcant correlations, is selected as the

starting point. The rest orbitals are reformulated and

then ranked by the low-scaling post-Hartree-Fock meth-

ods, such as the second-order (Moller-Plesset) perturba-

tion theory (MP2) [37]. The initial active space expands

steadily by iteratively appending ranked orbitals, and

a reasonable convergence direction is thus constructed.

Following this path, the accuracy of quantum eigensolver

will be systematically improved step by step.

The OE-VQE framework is demonstrated numerically

for typical systems including hydrogen chain, hydro-

gen ring and nitrogen molecules, which are challenging

strongly correlated systems in the dissociation distance.

The numerical results show much higher accuracy for

OE-VQE, which may only achievable with much deeper

circuits and larger measurement complexity for other

arXiv:2210.06897v1 [quant-ph] 13 Oct 2022

Frag

Environment Unentangled Environment

Frag

Bath

Ranked Extension Bath

Frag

Bath

(a) (b) (c)

Figure 1. Visualization of mentioned orbital sets in the Outline section. (a) There are LALOs are selected into the fragment

orbital set, and the rest of L−LAorbitals are named as environment set. General choices for fragment set are bonding orbitals

or the orbitals in atom valence shell. (b) There are LBLOs with fractional occupied number are assigned into the bath orbital

set. The fragment and bath set are termed as the Impurity. The (L−LA−LB)-scale unentangled-environment set contains

core orbitals and virtural orbitals. (c) The ranked unentangled environment orbitals are gradually appended into the bath

orbital set, until the bath set expands to the entire environment.

conventional quantum approaches. The proposed OE-

VQE framework lies in the chemical insight that intro-

duced as a novel dimension, paving the way for studying

and analysis more advanced quantum eigensolvers.

II. OUTLINE OF THE OE-VQE FRAMEWORK

To clearly deﬁne the whole framework, several funda-

mental deﬁnitions and notations needed to be clariﬁed

before providing details. Here, we utilize (p, q, r, s) to

represent arbitrary Localized molecular Orbitals (LO).

LO is a kind of orthogonal orbital basis that inher-

its chemical characteristic from Molecular Orbital basis,

meanwhile maintains a similar geometry structure to the

Atomic Orbital basis. Then the electron Hamiltonian

of a quantum chemical system under Born-Oppenheimer

approximation can be formed as:

He=Enuc +

p,q

dpq ˆa†

pˆaq+1

p,q,r,s

hpqrsˆa†

pˆa†

qˆasˆar,(1)

where Enuc is the nuclear repulsion energy, dpq (hpqrs)

represents single (double) electron integration, and ˆap

(ˆa†

p) denotes fermionic annihilation (creation) operator

to the p-th orbital.

The whole framework is composed of two fundamental

phases, namely ﬁnding a starting point and constructing

convergence directions.

Phase 1 starts from the density matrix DHF of

Hartree-Fock state. Firstly, LALOs (LAL) are se-

lected based on chemical insights. These LALOs are usu-

ally named as fragment and the rest orbitals are named

as environment. The bonding orbitals or the orbitals in

atom valence shell are common choices for the fragment

part. Then delete these LALOs from DHF and diagonal-

ize the rest of DHF to select LBorbitals with fractional-

occupied particle number. These LBorbitals have the

most signiﬁcant correlations to the fragment part, which

are usually named as bath, and these selected LA+LB

orbitals are named as Impurity. The remainder integral-

occupied and unoccupied orbitals in environment, termed

as core and virtual respectively, are unentangled with

the impurity at the Hartree-Fock theory level. There-

fore, core and virtual orbitals compose the unentangled

environment. The impurity orbitals form an initial active

space ˆ

Hsub(0), and its ground state |Ψ(0)iis the starting

point of the OE-VQE.

Phase 2 focuses on constructing convergence direc-

tions (quantum circuits) {Udir(Ns)}L−LA−LB

Ns=1 by leverag-

ing the rest of (L−LA−LB) unentangled-environment

with MP2 experience. Here, the systematical im-

provable directions are guided by a subspace hierarchy

{ˆ

Hsub(Ns)}L−LA−LB

Ns=1 , where each subspace ˆ

Hsub(Ns) is

determined by the particle number exchange between the

impurity and environment. Speciﬁcally, a Hartree-Fock

method implements on the impurity orbitals to provide

Locc occupied and (LA+LB−Locc) unoccupied orbitals

information. After that, perform the MP2 method on

impurity occupied and virtual in environment (impurity

unoccupied and core in environment) subspaces, and the

(L−LA−LB) unentangled environment orbitals will be

naturally ranked based on the particle change. Impurity

combined with the ﬁrst-Nsunentangled environment or-

bitals formulate a (LA+LB+Ns)-dimensional quantum

subspace ˆ

Hsub(Ns) which gradually convergences to ˆ

with the increase of Ns. The visualizations of fragment,

bath, environment, unentangled-environment and ranked

extension bath orbitals are illustrated as Fig. 1.

Denote N=L−LA−LB, for the number of orbitals

Ns∈[1, N], the proposed OE-VQE of electron Hamilto-

ℋ

Ψ0

Ψ ∗



dir(1)

ℋ

Ψ 0

Ψ 



1



dir()

sub 1

sub 

sub 



(b) (c)

(a)

Extension Bath

Use MP2 rank and generate

Extension Bath.

State Preparation

Parameters Update

Observables Measuring

VQE

Quantum Classical

Yes Converged?

Make Impurity

Make bath corresponding to

fragment.

+

⋮

Extension Bath Impurity

−1



+ 1

Build Subspace Build Subspace Hierarchy

Take impurity and extension

bath to make �

sub().

Framework

Input

Select one fragment

with chemical interest

Load Ansatz

dir(−1)

⋮

Convergence

Directions

()

++=?

=+ 1

Output

Ground state and energy

Yes

dir(−1)

dir()

grow

⋮

dir(−1)

dir()

Figure 2. (a) Outline of the OE-VQE framework: For the outside loop (green panel), a systematic improvable convergence path

is provided. The Impurity (fragment and bath) orbitals constructs a subspace ˆ

Hsub(0), and its ground state |Ψ(0)iis the starting

point provided by OE-VQE framework. The environment orbitals are iteratively appended to the Impurity, and a subspace

hierarchy {ˆ

Hsub(Ns)}N

Ns=1 is provided, where N=L−LA−LB. For the inside loop (red panel), each subspace determines

a convergence path (quantum circuit) Udir(Ns). The ﬁnal stage of the OE-VQE framework will be an approximation to the

ground state. (b) Visualization of the convergence path of a general quantum eigensolver without systematic improvability.

Starting from |Ψ(θ0)i, the quantum eigensolver ends up in |Ψ(θT) by following the path U1, U2, ..., UT. Here, Hrepresents the

whole Hilbert space. (c) Visualization of the convergence path of OE-VQE framework. The convergence direction is guaranteed

by the subspace hierarchy.

nians ˆ

Hecan be summarized into following steps:

1. Construct the subspace ˆ

Hsub(Ns);

2. Initialize the quantum state

|Ψ0(Ns)i=

Locc

m=1

ˆa†

m|0LA+LBi ⊗

l=1

ˆol|0Nsi.(2)

If l∈core, the operator ˆol= ˆa†

l, else ˆol=Il.

3. Generate |Ψ(Ns−1)i=Udir(Ns−1)|Ψ0(Ns)ias

the starting point in the Ns-th iteration. Here the

quantum circuit Udir(Ns−1) contains parameters

and corresponding operators in the (Ns−1)-th it-

eration;

4. Perform VQE programs on quantum device with

Hamiltonian ˆ

Hsub(Ns) and |Ψ(Ns−1)i, and using

the optimized fermionic operators and correspond-

ing parameters to update the convergence direction

Udir(Ns−1) 7→ Udir(Ns)

and output the ground state of ˆ

Hsub(Ns);

5. If Ns6=N, go back to step 1 and set Ns=Ns+ 1.

Finally, the ground state of ˆ

Hecan be approximated by

|Ψgi=Udir(N)|Ψ0(N)i.(3)

The whole procedure is visualized as Fig. 2.

III. TECHNICAL DETAILS OF OE-VQE

Here, elaborate technical details of OE-VQE frame-

work are provided. We ﬁrst introduce the method in

constructing the starting point, that is the ground state

of ˆ

Hsub(0). Then we introduce the workﬂow in showing

how to eﬃciently build a convergence path for quantum

eigensolvers.

A. Construct the Starting Point

This starts from the Hartree-Fock state

|Φei=Y

µ∈Nocc

ˆa†

µ|0⊗Li,(4)

which is a general reference of ground state for molecular

systems. Here Nocc represents occupied molecular orbital

set and {ˆa†

µ, µ ∈[|Nocc|]}represent creation operators.

Suppose a group of orthogonal Localized-Orbital (LO)

basis {ˆa†

k, k ∈[L]}has been selected via using Intrinsic

Atomic Orbital (IAO) methods [38], a L×Lreduced

density matrix

DHF

kl =hΦe|ˆa†

kˆal|Φei(5)

is obtained, in which (k, l) represent orbitals index in LO

basis. Suppose there are LALOs are selected to compose

the fragment subspace. To do this, the DHF should be

reorganized by moving each row and column in form of

DHF = DA

LA×LA

Dinter

LA×(L−LA)

Dinter†

LA×(L−LA)DB

(L−LA)×(L−LA)!,(6)

where DA

LA×LArepresents to the fragment subspace and

(L−LA)×(L−LA)represents to the environment. To fully

decouple the fragment DA

LA×LAwith the environment, it

is necessary to analyze the entanglement distribution in

environment orbitals. Speciﬁcally, the eigenvalues of den-

sity matrix DB

(L−LA)×(L−LA)provide the particle number

in the environment, that is

(L−LA)×(L−LA)=

L−LA

p=1

λp|BpihBp|=UBλBUB†.

(7)

Here the diagonal matrix λB=PL−LA

p=1 λp|pihp|,λp∈

[0,2] represents the particle number on a new environ-

ment orbital basis |Bpitransformed from LO basis by the

unitary operator UB. In our work, the eigenvector |Bpiis

termed as Unentangled Environment Orbital (UEO) ba-

sis. According to the distribution of λp, the environment

orbitals can be separated into three segments, including

LBbath orbitals, Lcore core orbitals and Lvir virtual or-

bitals. Obviously, the relationship

L=LA+LB+Lcore +Lvir

holds. The bath orbitals entangled with the fragment

will contribute all the particles between 0 and 2, while

core and virtual orbitals contain 2 particles and 0 par-

ticles, respectively. Due to MacDonald’s theorem [39],

the relationship LA≥LBholds, and a L×Lcoeﬃcient

transformation matrix can be formulated as

ULO7→UEO =IA

LA×LA0LA×(L−LA)

0LA×(L−LA)UB,(8)

where the last (L−LA) columns can be decomposed as

Ubath(LO7→UEO)

L×LB⊕Ucore(LO7→UEO)

L×Lcore ⊕Uvir(LO7→UEO)

L×Lvirt ,

(9)

and the ﬁrst LAcolumns are denoted as

IA

LA×LA

0LA×(L−LA)=Ufrag(LO7→UEO)

L×LA.(10)

Therefore, the impurity part is completely decoupled

with core and virtual orbitals in the Hartree-Fock level.

Denote the unitary matrix

Uimp(LO7→UEO)

L×(LA+LB)=Ufrag(LO7→UEO)

L×LA⊕Ubath(LO7→UEO)

L×LB,

(11)

then the initial subspace can be constructed by

Hsub(0) = Uimp(LO7→UEO)†

L×(LA+LB)ˆ

HeUimp(LO7→UEO)

L×(LA+LB),(12)

and the starting point |Ψ(0)iis the ground state of

Hsub(0). Details in calculating Eq. 12 are provided in

Appendix. C.

B. Construct Convergence Directions

The directions are a series of quantum circuits

{Udir(Ns)}N

Ns=1 which are guided by the subspace hier-

archy {ˆ

Hsub(Ns)}N

Ns=1, where N= (L−LA−LB). The

subspace hierarchy is described by impurity with Nsex-

tension bath orbitals. Here, extension bath orbitals are

ranked by their contributions to the particle number ex-

changing in occupied and unoccupied orbitals between

the impurity and unentangled environment. Therefore

the high-eﬀective MP2 is utilized to calculate the parti-

cle number exchanging, that is

δλenv =λvir

1, ..., λvir

Lvir ,2−λcore

1, ..., 2−λcore

Lcore ,(13)

where Lvir +Lcore =N,λvir

iand 2 −λcore

jrepresent par-

ticle number exchange from environment virtual orbitals

(core orbitals) to impurity occupied orbitals (unoccupied

orbitals), respectively. Rank the particle number changes

(from the largest to the smallest), these Norbitals are

reordered according to their entanglement contributions

to the impurity, and the corresponding coeﬃcient matrix

ULO7→MUEO =Uimp(LO7→MUEO)

L×(LA+LB),Uenv(LO7→MUEO)

L×N

(14)

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OrbitalExpansionVariationalQuantumEigensolver:EnablingEcientSimulationofMoleculeswithShallowQuantumCircuitYusenWu,1,2,ZigengHuang,1,yJinzhaoSun,3XiaoYuan,4,5JingboB.Wang,2andDingshunLv1,z1ByteDanceLtd.,ZhonghangPlaza,No.43,North3rdRingWestRoad,HaidianDistrict,Beijing,China2DepartmentofPhysics,TheU...

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Orbital Expansion Variational Quantum Eigensolver Enabling Ecient Simulation of Molecules with Shallow Quantum Circuit Yusen Wu1 2Zigeng Huang1yJinzhao Sun3Xiao Yuan4 5Jingbo B. Wang2and Dingshun Lv1z

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