Chemically Aware Unitary Coupled Cluster with ab initio Calculations on System Model H1 A Refrigerant Chemicals Application

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Chemically Aware Unitary Coupled Cluster with ab initio

Calculations on System Model H1: A Refrigerant Chemicals’

Application

October 27, 2022

I. T. Khan, M. Tudorovskaya, J. J. M. Kirsopp, D. Mu˜

noz Ramo

Quantinuum, Terrington House, 13–15 Hills Road, Cambridge CB2 1NL, United Kingdom

P. Warrier, D. K. Papanastasiou, R. Singh

Honeywell Advanced Materials, 20 Peabody St, Buﬀalo, NY 14210, United States

Abstract

Circuit depth reduction is of critical importance for quantum chemistry simulations on current and near

term quantum computers. This issue is tackled by introducing a chemically aware strategy for the Unitary

Coupled Cluster ansatz. The objective is to use the chemical description of a system to aid in the synthesis

of a quantum circuit. We combine this approach with two ﬂavours of Symmetry Veriﬁcation for the

reduction of experimental noise. These method enable the use of System Model H1 for a 6-qubit Quantum

Subspace Expansion calculation. We present (i) calculations to obtain methane’s optical spectra; (ii) an

atmospheric gas reaction simulation involving [CH·

3— H — OH]‡. Using our chemically aware unitary

coupled cluster state-preparation strategy in tandem with state of the art symmetry veriﬁcation methods,

we improve device yield for CH4at 6-qubits. This is demonstrated by a 90% improvement in two-qubit

gate count and reduction in relative error to 0.2% for electronic energy calculated on System Model H1.

Keywords: Quantum Computation ·Unitary Coupled Cluster ·UV/ VIS Spectra ·Quantum Subspace Expansion

The simulation of molecules and materials using quantum

chemistry methods is a well established ﬁeld, with applica-

tions in many scientiﬁc and industrial areas of interest [1].

However, there is awareness about the shortcomings of per-

forming these simulations on classical machines [2]. The

main workhorse of classical simulations, Density Functional

Theory, fails to capture the qualitative behaviour of chemi-

cals with strong correlation [3]. One naturally turns to wave

function methods, such as Hartree-Fock (HF), Coupled Clus-

ter (CC) theory and Conﬁguration Interaction (CI) methods.

However, the steep memory requirements of these techniques

have limited their practicality in the study of complex chem-

icals. This issue also plagues classical excited state heuris-

tics that provide CI- or CCSD-like accuracy. For instance,

Equation of Motion Coupled Cluster (EOM-CCSD) scales

as O(N6), where Nis the number of spin orbitals [4].

Quantum computers provide an alternative path to enable

the use of wave function methods on problems of practical

interest. In particular, two state of the art algorithms have

been proposed, the Variational Quantum Eigensolver (VQE)

and Quantum Subspace Expansion (QSE) algorithms, as the

earliest candidates for scalable alternatives to classical wave

function methods [5, 6, 7]. VQE approximates the ground-

state of a Hamiltonian variationally, and QSE estimates ex-

cited states non-variationally. Despite the rapid develop-

ment and deployment of quantum resources, progress has

been limited due to quantum hardware being unable to meet

performance requirements of these algorithms. Neverthe-

less, the exploration of quantum computing applications for

chemistry is necessary for the progression of computational

chemistry, and also serve as important benchmarks for to-

day’s hardware and its relevance to the chemicals industry.

Quantum algorithms for chemistry require a large amount

of quantum resources for the state-preparation component of

these simulations. Broadly, two families of state-preparation

methods have been introduced in literature: Unitary Cou-

pled Cluster (UCC), and hardware-eﬃcient ans¨atze. With

UCC, the number of variational circuit parameters and the

2-qubit gate depth scale as O(N4), leading to circuits beyond

the capability of today’s machines [8]. Hardware-eﬃcient

methods require less coherent resources, but suﬀer from the

“barren plateau” problem [9]. Initial eﬀort has been focused

on improving the 2-qubit depth scaling for the UCC ansatz.

Notably, adaptive methods [10], circuit recompilation [11],

and unique UCC circuit synthesis approaches [12]. Adaptive

methods are unattractive. For example, ADAPT-VQE re-

quires additional measurements on top of regular VQE [13].

This encourages us to explore a unique UCC circuit synthesis

approach that improved the 2-qubit gate count by describing

Fermionic UCC spatial orbital to spatial orbital excitations

as hard-core Bosonic operators [14, 15, 16]. In tandem, there

has been additional work on discarding UCC excitations by

using symmetry ﬁltering as a priori [17, 18], also resulting in

lower two-qubit gate count.

One particular example where quantum computing can help

is the simulation of refrigerants. The design of novel re-

arXiv:2210.14834v1 [quant-ph] 26 Oct 2022

frigerants has proven challenging due to trade-oﬀs in key

molecular properties such as global warming potential and

ozone depletion potential, whilst also considering other prop-

erties such as toxicity, ﬂammability, and stability. As such,

molecular simulations have become increasingly used for in-

vestigating the structure-activity relationship of candidate

refrigerants [19]. As a test case, we consider methane (CH4)

and its reaction with the hydroxyl radical (OH·),

CH4+ OH·→[CH·

3—H—OH]‡→CH·

3+ H2O.(1)

CH4’s atmospheric properties have been thoroughly studied

[20] and can provide guidelines in the application of quantum

computers for search of new environment-friendly refriger-

ants. In order to characterise this process, one also needs to

calculate a series of energies corresponding to the products,

methyl radical (CH·

3) and water (H2O). To estimate the re-

action barrier that governs the kinetics of the reaction, we

also simulate the transition state, [CH·

3—H—OH]‡.

In Section II, we give a detailed overview combining exci-

tation ﬁltering based on Z2symmetries, hard-core Boson

representation, and favourable two-qubit gate cancellation

(via Pauli-gadget synthesis scheme of Ref. [21]). Section III

is devoted to the results obtained with our state-preparation

strategy on System Model H1, powered by Honeywell. We

focus on the resource reduction, and consider noisy calcu-

lations for CH4’s optical spectra. Finally, we complete our

investigation with a simulation of [CH·

3—H—OH]‡and other

reaction participants. We also apply symmetry veriﬁcation

to our calculations [22]. The two symmetry veriﬁcation tech-

niques we use are Partition Measurement Symmetry Veri-

ﬁcation (PMSV) and Mid-circuit Measurement Symmetry

Veriﬁcation (MMSV) [23].

1 Methods

1.1 Chemically Aware Unitary Coupled

Cluster

For our state-preparation strategy, we use Jordan-Wigner

Encoding (JWE) to map Trotterised Fermionic exponents

to Pauli operators acting on qubits [24, 25]. We refer to the

by-product sequence of Pauli-Zoperations as JWE-strings,

which consequently increase the eﬀective k-locality of UCC

exponents. Our spin orbitals and therefore the qubit reg-

ister have alpha-beta ordering (each even-odd indexed spin

orbital corresponds to a spatial orbital). Appendix A has

more information on the UCC state-preparation method and

conventional circuit decompositions (Individual and Com-

muting Sets synthesis). The steps of the chemically aware

strategy are as follows:

I. Symmetry Filtering: Filter the set of excitations com-

posing the ansatz via use of molecular symmetry to identify

forbidden terms. We used two techniques:

(a) Deﬁning Z2symmetries to check commutation

against the UCC excitation operators [26, 27, 17];

(b) A point group symmetry ﬁltering method for CCSD

adapted to be used for UCCSD [28].

II. Spatial Orbital to Spatial Orbital: There are three

steps to compactly describe a pair of electrons excited be-

tween spatial orbitals. For this method, we necessarily

change the ordering of the excitations to beneﬁt from the

two-qubit gate savings associated with spatial to spatial

UCC excitation.

(a) Specify Double Occupied Spatial Orbitals:

Only doubly occupied (|1i) and virtual spatial

orbitals (|0i) are considered. For example, the

Hartree-Fock state |111000ideﬁning molecular spin

orbital occupation, would be processed to return

|100iin the molecular spatial orbital occupation.

These occupations are mapped to the even-indexed

qubits on the circuit, |100000i. Single occupied spa-

tial orbitals are ignored.

(b) Hard-core Boson Representation: Operations

that excite a pair electrons from and to the same

spatial orbital can be synthesised more eﬃciently.

These excitations are of the type:

ˆa†

2pˆa2qˆa†

2p+1ˆa2q+1 −h.c., (2)

where q < p, and both variables track the spatial

orbital index. Applying JWE results in 8 unique

Pauli exponents over 4-qubits.

These excitations lack JWE-strings, signifying zero

parity exchange as these adjacent electrons hop be-

tween diﬀerent spatial orbitals. It can be seen that

these excitations act on spin-orbitals, but excite and

de-excite electrons between spatial orbitals. These

adjacent electrons travel together, yet cannot oc-

cupy the same spatial orbital with another pair of

electrons. Equation 2 can also be expressed as,

b†

pˆ

bq−h.c., (3)

where ˆ

bdenotes a Hard-core Bosonic operation.

Eq. 3 can be re-expressed by using the equiva-

lence between Hard-core Bosons and Pauli opera-

tions [29, 30],

2nˆ

Yqˆ

Xp−ˆ

Xqˆ

Ypo.(4)

We relabel the indices of equation 4 from p→2p

and q→2q. Each of these spatial orbital to spatial

orbital excitations act on 2 qubits and require 2 two-

qubit gates.

gates on relevant even-odd qubits. Each even-

indexed qubit corresponds to the alpha spin-orbital

of a spatial orbital, and similarly each beta spin-

orbital is represented by an odd-indexed qubit 3.

single occupation of a spatial orbital is included by

initializing the relevant alpha-index qubit to the |1i

state.

III. Commuting sets of Pauli strings: The remaining

double and single excitations are synthesised. Each double

excitation contains 8 Pauli-sub terms and these terms natu-

rally form a commuting set. We synthesise these excitations

in commuting sets with tket, resulting in 14 two-qubit gates

Fig. 1. Schematic showing three major steps of the chemically aware ucc state-preparation strategy. Step (i) uses symmetry as a

priori to discard excitations. Step (ii) is a compact synthesis scheme for spatial to spatial excitations. Step (iii) uses tket to

synthesise generic double and single excitations by commuting sets to maximize pauli-string cancellation. Each single or double

UCC excitation is naturally a commuting set of Pauli-strings.

at best. Increasing the length of JWE-strings increases the

number of two-qubit gates for the corresponding decompo-

sition. Circuits for single fermionic excitations contain at

minimum four two-qubit gates and act over 3 qubits. Both

two-qubit gate count and number of qubits grow as the num-

ber of Pauli-Zstrings increase.

2 Results

All the circuits in this paper are prepared using Quantin-

uum’s quantum chemistry package InQuanto [31, 32]. The

integrals to obtain the relevant chemistry Hamiltonians were

obtained using an InQuanto extension to the open-source

chemistry software package, PySCF, known as InQuanto-

PySCF [33]. InQuanto provides the tools provide the pre- &

post-processing logic to perform quantum computation, and

more importantly to map the results from quantum com-

putation back to the Chemistry problem. The UCC state-

preparation techniques used in this paper are available in

InQuanto. NGLView is used to visualize chemical struc-

tures and molecular orbitals via an InQuanto interface. We

use the open-source ADCC library to compute benchmark

optical spectra data at ADC-2 level of theory, [34].

For this investigation, we used system model H1 devices and

emulators [35, 36]. The IBMQ qasm simulator was used to

perform noiseless state-vector calculations, simulations with

ﬁnite sampling noise and zero quantum noise. We use tket to

synthesise, optimize and retarget quantum circuits to enable

execution on a H1 hardware [37, 38].

All our benchmark calculation are non-variational operator

averaging calculations (Hamiltonian Averaging or QSE). The

ground-state parameters for our state-preparation circuits

are obtained via a noiseless VQE qasm simulation.

2.1 Improvements due to the Chemically

Aware Strategy

We investigate the two-qubit gate count across three diﬀer-

ent methods, chemically aware, commuting sets, and indi-

vidual UCCSD synthesis. We use CH4at equilibrium geom-

etry as a benchmark system, alongside the D2point group

to describe the molecular orbital symmetry [39]. Fig. 2a

reports an improvement by approximately 81% (commuting

sets) and 95% (individual) in two-qubit gate count for var-

ious active-spaces ranging from 4 qubits to 18 qubits. By

neglecting symmetry (C1point group), we reduce the eﬃ-

cacy of chemically aware to reduce two-qubit gate count,

rendering only a small improvement due to the compact

spatial-orbital-only excitation synthesis. Fig. 2b does not

report any improvement in overall scaling with chemically

aware compared to commuting sets synthesis for CH4with

C1symmetry.

ADAPT-VQE improves upon individual synthesis by iter-

atively constructing a compact ansatz. At the cost of in-

creasing the total number of measurements, ADAPT dis-

cards both symmetry-forbidden and minimally contributing

symmetry-allowed excitations. As a consequence, ADAPT

circuits have a lower two-qubit gate count. With the chem-

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ChemicallyAwareUnitaryCoupledClusterwithabinitioCalculationsonSystemModelH1:ARefrigerantChemicals'ApplicationOctober27,2022I.T.Khan,M.Tudorovskaya,J.J.M.Kirsopp,D.Mu~nozRamoQuantinuum,TerringtonHouse,13{15HillsRoad,CambridgeCB21NL,UnitedKingdomP.Warrier,D.K.Papanastasiou,R.SinghHoneywellAdvancedMate...

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Chemically Aware Unitary Coupled Cluster with ab initio Calculations on System Model H1 A Refrigerant Chemicals Application

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