Strongly Contracted N-Electron Valence State Perturbation Theory Using Reduced Density Matrices from a Quantum Computer Michal Krompiec1and David Mu noz Ramo1

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Strongly Contracted N-Electron Valence State Perturbation Theory Using Reduced

Density Matrices from a Quantum Computer

Michal Krompiec1, ∗and David Mu˜noz Ramo1

1Quantinuum, Partnership House, Carlisle Place, London SW1P 1BX, United Kingdom

(Dated: October 13, 2022)

We introduce QRDM-NEVPT2: a hybrid quantum-classical implementation of strongly-

contracted N-electron Valence State 2nd -order Perturbation Theory (SC-NEVPT2), in which the

Complete Active Space Conﬁguration Interaction (CASCI) step, capturing static correlation eﬀects,

is replaced by a simulation performed on a quantum computer. Subsequently, n-particle Reduced

Density Matrices (n-RDMs) measured on a quantum device are used directly in a classical SC-

NEVPT2 calculation, which recovers remaining dynamic electron correlation eﬀects approximately.

We also discuss the use of the cumulant expansion to approximate the whole 4-RDM matrix or only

its zeros. In addition to noiseless state-vector quantum simulations, we demonstrate, for the ﬁrst

time, a hybrid quantum-classical multi-reference perturbation theory calculation, with the quantum

part performed on a quantum computer.

Introduction – Hamiltonian simulation, the central

problem of computational quantum chemistry, is excep-

tionally well-deﬁned but has long been known to be, as

Dirac put it, “too complex to be solved” [1] in the gen-

eral case. Indeed, the exact solution (in a ﬁnite basis)

of the molecular electronic structure problem, i.e. Full

Conﬁguration Interaction (FCI), scales combinatorially

with basis size. Hence, the application of FCI is lim-

ited to small systems, even in its stochastic approximate

implementations [2]. Quantum computing has recently

emerged as a promising approach for large-scale accu-

rate electronic structure calculations, due to exponential

(or near-exponential) speedup of certain quantum algo-

rithms, such as Quantum Phase Estimation (QPE) com-

pared to classical solutions such as FCI [3–8].

Towards quantum multi-reference calculations – While

the motivation for the development of quantum algo-

rithms for computational chemistry seems to come from

the desire for an exact solution of the chemical Hamil-

tonian simulation problem (i.e. a quantum replacement

for FCI), exact diagonalization of the whole Hamiltonian

of a chemical system is in practice hardly ever pursued,

or even needed. The special case of ground-state ener-

gies of “single-reference” molecular systems (i.e. exhibit-

ing mainly weak correlations and having one dominant

conﬁguration in the Conﬁguration Interaction expansion)

can be very accurately calculated with Coupled Clus-

ter methods, such as CCSD(T), which scales with basis

size as O(N7) in the canonical implementation and only

O(N) in the DLPNO approximation [9]. The remaining

strongly correlated electronic systems can usually be de-

scribed by multi-reference or multi-conﬁgurational meth-

ods, where interactions within only a subspace of the

Hilbert space are calculated with high accuracy, while

interactions with remaining orbitals are treated only ap-

proximately. Thus, the orbitals are divided into two dis-

joint sets: the active orbitals and the inactive (core and

virtual) orbitals. A model Hamiltonian is deﬁned in the

reference (or model) subspace deﬁned by Slater determi-

nants generated by permutations of active orbitals and

accounts chieﬂy for the static electron correlation. An

expansion in which the model space includes all possible

distributions of electrons in the selected (active) orbitals

is called complete active space (CAS) [10,11]. In the

ﬁrst step of a multi-reference calculation, static correla-

tion eﬀects are introduced with a variationally optimized

reference wave function:

|Ψ0i=

µ=1

cµ|Ψµi(1)

Subsequently, dynamical correlation eﬀects are intro-

duced via a wave operator Ω acting on |Ψ0i[11]. The

action of Ω is often approximated at a lower level of the-

ory, for example perturbation theory truncated at 2nd

order, like in the popular CASPT2 [12] and NEVPT2

[13,14] methods. For large active spaces, the cost of

CASPT2 and NEVPT2 calculations is dominated by the

solution of the CAS problem which scales exponentially

with the size of the active space due to an exponential

scaling of the CI basis. By mapping the electronic oc-

cupation number vectors of length N to a quantum reg-

ister (i.e. to qubits), it becomes possible to express this

enormous CI basis via the 2Nbasis states of N qubits

[5]; the same mapping applied to the electronic Hamilto-

nian yields the corresponding qubit Hamiltonian. Deter-

mination of its the ground state via e.g. QPE requires

a number of steps only polynomial in N [5], suggesting

an exponential speedup with respect to the classical ap-

proach. When the cost of preparing the initial state with

non-negligible overlap with the ground state is factored

in, the quantum speedup is expected to be less than ex-

ponential but still potentially very large [15]. Hence, re-

placing the CAS-CI component of the calculation with

an eﬃcient quantum algorithm would allow application

of these techniques to very large active spaces, thereby

extending their applicability to extended, complex chem-

ical systems and eliminating the need for selection of ac-

arXiv:2210.05702v1 [quant-ph] 11 Oct 2022

tive orbitals.

NEVPT2(VQE,QSE) is a hybrid quantum-classical

implementation of uncontracted NEVPT2 recently pub-

lished by Tammaro et al. [16], in which the CAS calcula-

tion is replaced by the Variational Quantum Eigensolver

(VQE) [17,18]. After state preparation with VQE, n-

particle Reduced Density Matrices (RDMs) up to n= 4

are measured and Quantum Subspace Expansion (QSE)

[19] algorithm with single and double excitations as the

expansion operators is applied to determine all eigenvec-

tors and eigenvalues of the active space (Dyall) Hamilto-

nian (in the subspace deﬁned by the chosen expansion op-

erators). Finding all excited states via QSE, is expensive

and will likely become intractable for large active spaces,

potentially cancelling out any quantum advantage of the

state preparation step. Therefore, for complex multiref-

erence systems (for which quantum advantage is expected

in the CAS part of this workﬂow), NEVPT2(VQE,QSE)

will likely be prohibitively expensive and, if a limited

rank of expansion operators in QSE is used, inaccurate.

QRDM-NEVPT2 – In the present work, we introduce

QRDM-NEVPT2 : a hybrid quantum-classical ﬂavor of

SC-NEVPT2 [14,20] implemented in the quantum chem-

istry package InQuanto [21]. Following Tammaro et al.

[16], we apply VQE in place of the CAS-CI step, but in

contrast to their method, we do not rely on QSE but use

RDMs computed via measurement of expectation values

of RDM operators after state preparation with VQE.

Our workﬂow (see Fig. 1) starts with the usual classical

bootstrapping of a quantum simulation [18]: a mean-ﬁeld

calculation followed by an optional orbital transforma-

tion (e.g. localization), selection of the active space, con-

struction of a fermionic 2nd-quantized Hamiltonian and

Jordan-Wigner mapping [3,22] it into a qubit Hamilto-

nian. The state-preparation step in our implementation

consists of VQE [17,18], which also yields the expecta-

tion value of the active space Hamiltonian. Subsequently,

we use the Operator Averaging [17] method to jointly

measure the expectation values of the matrix elements

of spin-traced 1-, 2- and 3-RDM operators. In order to

reduce the number of required measurements, we parti-

tion the Pauli words deﬁning the RDM operators into

mutually commuting sets, each set corresponding to one

measurement circuit [23]. We make use of Z2symmetries

of the qubit Hamiltonian to mitigate errors via Parti-

tion Measurement Symmetry Veriﬁcation [24] and to re-

duce quantum resources via qubit tapering [25]. We note

that RDM operator matrix elements which violate any of

the Z2symmetries of the qubit Hamiltonian vanish and

therefore do not need to be measured. For systems with

more than 3 active electrons (i.e. where the 4-RDM does

not vanish), we estimate the spin-traced 4-RDM using

the cumulant approximation formula (see below). Once

the RDMs are determined, we compute the NEVPT2 en-

ergy using a modiﬁcation of Guo’s implementation [20] of

SC-NEVPT2 [14], where we replaced RDMs calculated

from CAS-CI or DMRG wave functions with those ob-

tained from our algorithm.

The cumulant approximation of 4-RDM – The spin-

traced excitation operators are deﬁned by

q= ˆa†

p↑ˆaq↑+ ˆa†

p↓ˆaq↓(2a)

Epr

qs = ˆa†

p↑ˆa†

r↑ˆas↑ˆaq↑+ ˆa†

p↑ˆa†

r↓ˆas↓ˆaq↑

+ˆa†

p↓ˆa†

r↑ˆas↑ˆaq↓+ ˆa†

p↓ˆa†

r↓ˆas↓ˆaq↓,(2b)

etc. The spin-traced 4-RDM (Γprtv

qsuw) is deﬁned as

Γprtv

qsuw =hΨ0|ˆ

Eprtv

qsuw |Ψ0i(3)

PySCF’s [20,26] and Orca’s [27] implementations of

NEVPT2 make use of a related tensor instead, dubbed

4-particle Pre-Density Matrix (4-PDM, γprtv

qsuw) by Guo

et al. [27]. PDM is deﬁned with the same creation and

annihilation operators as the corresponding RDM, but

applied in a diﬀerent order:

γprtv

qsuw =hΨ0|ˆ

qˆ

sˆ

uˆ

w|Ψ0i(4)

Hence, the 4-RDM can be computed easily from 4-PDM

and lower RDMs, via normal-ordering of the operators in

4-PDM [27].

4-RDM and 4-PDM are tensors consisting of N8ele-

ments, where N is the number of spatial orbitals, hence

their measurement on a quantum device would be very

expensive. Approximation of higher-order RDMs via

the cumulant expansion (i.e. setting the highest-order

cumulant to 0) is well known [28,29], but an analo-

gous expression for spin-traced RDMs has been intro-

duced by Kutzelnigg et al. [30] only in 2010. Their

formula connecting the spin-traced 4-particle density cu-

mulant ΛP1P2P3P4

Q1Q2Q3Q4with the spin-traced 4-particle RDM

ΓP1P2P3P4

Q1Q2Q3Q4and lower-order terms reads:

ΛP1P2P3P4

Q1Q2Q3Q4= ΓP1P2P3P4

Q1Q2Q3Q4(1) −ΓP1

Q1ΛP2P3P4

Q2Q3Q4(4)

−ΓP1

Q1ΓP2

Q2ΛP3P4

Q3Q4(6) −ΛP1P2

Q1Q2ΛP3P4

Q3Q4(3)

2nΓP1

Q1ΛP2P3P4

Q1Q3Q4(12) + ΓP1

Q2ΓP2

Q1ΛP3P4

Q3Q4(6)

+ΓP1

Q1ΓP2

Q3ΛP3P4

Q2Q4(24) + ΛP1P2

Q1Q3ΛP3P4

Q2Q4(12)o

−1

4nΓP1

Q2ΓP2

Q3ΛP3P4

Q1Q4(24) + ΓP1

Q3ΓP2

Q4ΛP3P4

Q1Q2(12)o

−1

12nΛP1P2

Q3Q4−ΛP1P2

Q4Q3ΛP3P4

Q1Q2−ΛP3P4

Q2Q1

+3 ΛP1P2

Q3Q4+ ΛP1P2

Q4Q3ΛP3P4

Q1Q2+ ΛP3P4

Q2Q1o

ΓP1

Q1ΓP2

Q2ΓP3

Q3ΓP4

Q4(1) + 1

2ΓP1

Q2ΓP2

Q1ΓP3

Q3ΓP4

Q4(6)

−1

4ΓP1

Q2ΓP2

Q3ΓP3

Q1ΓP4

Q4(8) −1

4ΓP1

Q3ΓP2

Q4ΓP3

Q1ΓP4

Q2(3)

8ΓP1

Q2ΓP2

Q3ΓP3

Q4ΓP4

Q1(6).(5)

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StronglyContractedN-ElectronValenceStatePerturbationTheoryUsingReducedDensityMatricesfromaQuantumComputerMichalKrompiec1,andDavidMu~nozRamo11Quantinuum,PartnershipHouse,CarlislePlace,LondonSW1P1BX,UnitedKingdom(Dated:October13,2022)WeintroduceQRDM-NEVPT2:ahybridquantum-classicalimplementationofstro...

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Strongly Contracted N-Electron Valence State Perturbation Theory Using Reduced Density Matrices from a Quantum Computer Michal Krompiec1and David Mu noz Ramo1

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