CO2 capture is critical to solving global warming. Amine-based solvents are extensively used to chemically absorb CO2. Thus, it is crucial to study the chemical absorption of CO2 by amine-based solvents to better understand and optimize CO2 capture processes. Here, we use quantum computing algorithms to quantify molecular vibrational energies and reaction pathways between CO2 and a simplified amine-based solvent model—NH3. Molecular vibrational properties are important to understanding kinetics of reactions. However, the molecule size correlates with the strength of anharmonicity effect on vibrational properties, which can be challenging to address using classical computing. Quantum computing can help enhance molecular vibrational calculations by including anharmonicity. We implement a variational quantum eigensolver (VQE) algorithm in a quantum simulator to calculate ground state vibrational energies of reactants and products of the CO2 and NH3 reaction. The VQE calculations yield ground vibrational energies of CO2 and NH3 with similar accuracy to classical computing. In the presence of hardware noise, Compact Heuristic for Chemistry (CHC) ansatz with shallower circuit depth performs better than Unitary Vibrational Coupled Cluster. The “Zero Noise Extrapolation” error-mitigation approach in combination with CHC ansatz improves the vibrational calculation accuracy. Excited vibrational states are accessed with quantum equation of motion method for CO2 and NH3. Using quantum Hartree–Fock (HF) embedding algorithm to calculate electronic energies, the corresponding reaction profile compares favorably with Coupled Cluster Singles and Doubles while being more accurate than HF. Our research showcases quantum computing applications in the study of CO2 capture reactions.

Quantum computing will play an important role in future innovations in computational chemistry. Quantum computers offer exponential speed-up to solve certain problems that are intractable for classical computers. For example, quantum computers in principle can reduce the simulation complexity for quantum many-body systems from exponential down to just polynomial.1–3 

The variational quantum eigensolver (VQE) is one of the most promising quantum computing tools for applications in computational chemistry of near-term quantum computers.4–7 VQE is a hybrid quantum-classical algorithm, which allows the eigenvalues of the Hamiltonian of fermionic or bosonic systems to be quantified.4 It relies on a quantum simulator or computer to estimate an energy expectation value, while relying on a classical optimizer to suggest improvements of the ansatz.4,8

For calculating the electronic energies of molecules, the combination of VQE algorithm with chemistry-inspired ansatz has been applied to evaluate the electronic ground and excited states of simple molecules with up to a few atoms.9–12 A typical VQE setup for the electronic energies of a reaction system with CO2 and NH3 molecules requires 46 qubits even for the minimal Slater-type Orbital-3G (STO-3G) basis set (the number of qubits used is equal to the number of spin–orbitals of the specified basis set), making the algorithm computationally challenging on available quantum simulators or devices. A quantum embedding scheme known as Hartree–Fock (HF) embedding has been proposed to circumvent the difficulty in simulating such large system. In this framework, the electronic structures of the full system are broken into fragments consisting of the active space (AS) which defines a subset of active frontier orbitals and the environment. Each part is described quantum mechanically with the environment treated with the HF method while employing a high-level quantum mechanical description for the AS. The time-independent Schrödinger equation of the AS is connected to the exchange-correlation embedding potential of the environment such that a new Hamiltonian is defined for the full system. Reduction of qubit resources is achieved since the VQE computation is restricted to the AS while the environment is treated with the HF classical solver. The algorithm was previously validated for the symmetric dissociation of H2O, which involves non-equilibrium geometry structures, yielding an energy profile in close agreement with the reference full configuration interaction (FCI) calculations when a sufficiently large AS was used.13 The dissociation of N2 and O2 was also chosen as additional systems to benchmark the algorithm since both molecules show strong correlation character when the bond is stretched. The HF embedding simulated potential energy curves were found to reach near FCI accuracy.

For assessing vibrational energies of the molecules, quantum computing algorithms can also be utilized to enhance the calculations of vibrational properties.14–16 Molecular vibrational properties are important in understanding the mechanisms and kinetics of chemical reactions. The anharmonicity or couplings of molecular vibrational modes are critical for polyatomic molecules. For example, full infrared (IR) and Raman spectra can be computed by including anharmonic corrections.17,18 The thermodynamic quantities, such as entropy and heat capacity, are greatly affected by anharmonicity.19 The larger the molecule, the stronger the effect of anharmonicity on vibrational properties.20,21 However, the significant anharmonicity makes determining the vibrational structure classically intractable for large molecules.22 Based on the second quantization representation of the molecular Hamiltonian,23 the VQE calculation of vibrational structures has qubit encoding of vibrational levels and enables the expansion of the potential energy surface to contain n-body coupling terms.14,24 The VQE algorithm has been demonstrated to solve the vibrational Hamiltonian with the effect of anharmonic vibrations for triatomic molecules, such as CO2, H2O, SO2, and NO2.14,15,25 In addition, the vibrational excitation energies can also be calculated with the quantum equation of motion (qEOM)24 approach.

For simulating reaction systems, both the electronic and vibrational energies of the molecules need to be assessed to obtain the chemical reaction energies. Here, we benchmark the performance of quantum computing in the study of chemical reactions for CO2 capture application. CO2 capture research is one of the low hanging fruits for the application of quantum computing, and is considered to be one of the near-term opportunities.3 Given the urgent need to address global warming, CO2 capture is considered a key pathway to reduce emissions from burning fossil fuels.26 Amine-based solvents are often used to capture CO2 by simple chemical reactions.27 To simplify the calculation, we use NH3 as a model solvent for the CO2 capture reaction. The reaction is as follows: NH3 + CO2 = NH2COOH. This work aims to utilize the VQE algorithm to calculate ground-state vibrational energies and construct the reaction energy profile of the reaction between CO2 and NH3 in the gas phase. We also compute vibrational excited energies with qEOM approach for CO2 and NH3. We evaluate the performance of HF embedding in the calculation of potential energies along the reaction pathway. In addition to computing vibrational energies, we also compare the performance of two ansatzes with different circuit depths on a quantum simulator: Unitary Vibrational Coupled Cluster (UVCC)15 and Compact Heuristic for Chemistry (CHC)14 ansatzes. Current noisy intermediate-scale quantum (NISQ) computers require quantum algorithms with shallow circuit depths to mitigate hardware noise. Hence, we compared the performance of both ansatzes in quantum noise simulations. We also investigated the effectiveness of the Zero Noise Extrapolation (ZNE) error-mitigation method.28 CHC has a shallower circuit depth than UVCC and thus higher accuracy in the presence of simulated quantum errors with and without error mitigation.

The HF embedding calculations for electronic energies were performed using the Qiskit Nature platform,29 a Python package that interfaces a quantum computing framework with existing classical quantum chemistry software PySCF30,31 to generate classical data such as electronic integrals in the atomic orbital basis.29 For the AS portion, we used a Unitary Coupled Cluster Ansatz with single and double excitations (UCCSD) to represent the electronic wavefunction, and the Jordan–Wigner scheme for fermion to qubit mapping.32 VQE simulations were carried out on the Qiskit statevector simulator and the minimal STO-3G basis set was employed to further reduce the quantum requirement.33 The gradient-based Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization method was used for the optimization of the VQE parameters.34,35 The reaction coordinate was computed using plane-wave DFT with the Perdew–Burke–Enzerhoff (PBE) functional and the all-electron, frozen core, projected augmented wave (PAW) method in VASP.36 The self-consistently simulated valence electrons were H 1s electron, and C, N, O 2s, and 2p electrons. The minimum-energy pathway across a density functional theory (DFT) potential energy surface was obtained from a Climbing Image Nudged Elastic Band (CI-NEB) calculation using seven movable images.37 HF embedding energies are compared with those obtained from HF and Coupled Cluster Singles and Doubles (CCSD) calculations. Our approach for ascertaining the performance of quantum algorithm for simple concerted reactions and more complex ones that involve bond rupture and formation follows previous works.38,39

Figure 1 shows the steps to compute ground state energy using variational quantum eigensolver (VQE) algorithm.40 

Fig. 1.

Ground state energy calculation flowchart using variational quantum eigensolver algorithm.

Fig. 1.

Ground state energy calculation flowchart using variational quantum eigensolver algorithm.

Close modal

The ground state vibrational energy is obtained by solving the nuclear Schrödinger equation under the Born–Oppenheimer approximation:

Hvib|Ψn=En|Ψn.
(1)

The anharmonicity effect can be included in our calculation by adding higher-order potential energy terms in the so-called Watson Hamiltonian (neglecting vibro-rotational couplings),

Hvib(Q1,,QL)=12Ll=12Ql2+V(Q1,,QL).
(2)

We describe the many-body vibrational potential using vibrational self-consistent field (VSCF) method,14,41

V(Q1,,QL)=12ijkijQiQj+16ijkkijkQiQjQk+124ijklkijklQiQjQkQl,
(3)

where V is potential energy, and Qi is vibrational internal coordinate. In this study, we describe many-body potential energy with expansion to the fourth order as shown in the equation. Four-body expansion can obtain an accuracy of about 1–2 cm−1. The terms kij, kijk, and kijkl are anharmonic force field constants. These constants are calculated by semi-numerical differentiation of the analytical Hessian as implemented in Gaussian software.42 Molecular geometries are first optimized based on quantum mechanics at the B3LYP/cc-pVTZ level of theory using Gaussian software.42 

In second quantization, nuclear Hamiltonian is described as follows:

hvibSQ=l=1Lkl,hlNlϕkl|T(Ql)+V[l](Ql)|ϕhlakl+ahl+l<mLkl,hlNlkm,hmNmϕklϕkm|V[l,m](Ql,Qm)|ϕhlϕhmαkl+αkm+αhlαhm+,
(4)

where VlQl and Vl,m(Ql,Qm) represent the variation of the potential energy upon displacements along one lth normal coordinate (Ql) and both lth and mth coordinates (Ql,Qm) from the equilibrium position correspondingly.14,43

V[l](Ql)=V(Q1eq,,Ql1eq,Ql,,QLeq),
(5)
V[l,m](Ql,Qm)=V(Q1eq,,Ql,Qm,,QLeq)V[l](Ql)V[m](Qm).
(6)

The many-body basis functions are encoded within the n-mode second quantization as occupation-number vectors (ONVs) as follows:14,23

ϕk1(Q1)ϕkL(QL)|011k10N1,011k20N2,011kL0NL.
(7)

The nuclear wavefunction can be expressed as follows:23 

|Ψ=k1=1N1kL=1NLCk1,,kLϕk1(1)(Q1)ϕkL(L)(QL),
(8)

where each mode l is described by a basis of Nl modals (basis functions).

Creation and annihilation operators per mode l and per basis function kl are defined as follows:

akl|,010kl0Nl,=|,010kl0Nl,akl|,010kl0Nl,=0,akl|,010kl0Nl,=|,010kl0Nl,akl|,010kl0Nl,=0.
(9)

The Hamiltonian is encoded onto qubits using the “direct mapping” method.15,16 The number of modals Nl for a given vibrational mode l are represented by a Nl-qubit register. We note that a more compact mapping, such as binary mapping, can reduce the number of qubits to encode the vibrational Hamiltonian.14,15,44 Vibrational wavefunction can be represented with L× N (direct mapping) or L× log(N) (compact mapping) qubits for molecules with L vibrational modes.44 However, such compact mapping has a higher circuit depth and number of terms in the Hamiltonian than direct mapping due to the more complex representation of the elementary raising and lowering operators.14,15 For example, the Hamiltonian contains O(LkNk) (direct mapping) or O(LkN2k) (compact mapping) terms (k: potential energy expansion order).15 The increased circuit depth will result in larger hardware errors in NISQ devices. Thus, we use direct mapping method for quantum computing of vibrational energies. The qubit numbers for CO2, NH3, and NH2COOH molecules are listed in Table I. Figure 2 shows the molecular structures of CO2, NH3, and NH2COOH.

Table I.

Quantum circuit resource estimation for the calculation of the ground-state vibrational energy of CO2, NH3, and NH2COOH with the UVCC and CHC approaches with single excitation. The number of CNOT gates (CX) is given for both approaches.

MoleculeModesModalsQubit NumberCX UVCCCX CHC
CO2 16 
NH3 12 24 12 
NH2COOH 15 30 60 30 
MoleculeModesModalsQubit NumberCX UVCCCX CHC
CO2 16 
NH3 12 24 12 
NH2COOH 15 30 60 30 
Fig. 2.

Molecular representations of (a) CO2, (b) NH3, (c) NH2COOH initial state (IS: NH3 + CO2), (d) NH2COOH transitional state 1 (TS1), (e) NH2COOH transitional state 2 (TS2), (f) NH2COOH final state 1 (FS1), and (g) NH2COOH final state 2 (FS2). Atom color code: hydrogen (silver), carbon (cyan), nitrogen (blue), and oxygen (red).

Fig. 2.

Molecular representations of (a) CO2, (b) NH3, (c) NH2COOH initial state (IS: NH3 + CO2), (d) NH2COOH transitional state 1 (TS1), (e) NH2COOH transitional state 2 (TS2), (f) NH2COOH final state 1 (FS1), and (g) NH2COOH final state 2 (FS2). Atom color code: hydrogen (silver), carbon (cyan), nitrogen (blue), and oxygen (red).

Close modal

We use the VQE algorithm to find the ground state of vibrational Hamiltonians. The accuracy of VQE algorithms is susceptible to quantum errors in noisy intermediate-scale quantum (NISQ) devices. Algorithms with shallow circuit depths can alleviate the effect of hardware noise by allowing quantum operations within the limited coherence time of the NISQ devices. Hardware-efficient ansatzes are, thus, developed to reduce the circuit depth needed to prepare the initial vibrational wavefunction.9,14 In this study, we use two ansatzes with different circuit depths: Unitary Vibrational Coupled Cluster (UVCC15) and Compact Heuristic for Chemistry (CHC14) ansatzes. The number of CNOT gates in the CHC ansatz circuit is one order of magnitude less than that of UVCC with double excitations.14 Thus, CHC with shallower circuit depth can help lessen the sensitivity to noise for better performance of VQE calculation in noisy simulations. To access the benefit of CHC, we compare the accuracy of VQE calculations against the classical diagonalization result with and without gate errors. Additionally, we apply the Zero Noise Extrapolation (ZNE) error mitigation technique28 to reduce the simulated noise effect. With the ZNE approach, the expectation values are computed with amplified noises by different factors (1, 3, and 5), and then, the noiseless results are extrapolated using linear extrapolation based on the measured expectation values. CHC and UVCC are implemented with single excitation. The ansatzes' CNOT gate numbers are shown in Table I. The constrained optimization by linear approximation (COBYLA) optimizer is utilized. All VQE calculations are performed by Qiskit.29 

The vibrational excitation energies are calculated with the quantum equation of motion (qEOM)24 approach by solving the pseudo-eigenvalue problem [Eq. (10)]. We use the qEOM operators with single and double excitations,

(MQQ*M*)(XnYn)=E0n(VWW*V*)(XnYn),
(10)

where

Mμανβ=0|[(Êμα(α)),Ĥ,(Êνβ(β))]|0,Qμανβ=0|[(Êμα(α)),Ĥ,(Êνβ(β))]|0,Vμανβ=0|[(Êμα(α)),Êνβ(β)]|0,Wμανβ=0|[(Êμα(α)),(Êνβ(β))]|0.

Figure 3 depicts the HF embedding predicted reaction profile using six spatial orbitals in the AS. We plotted the relative energy EEi along the reaction pathway, where the latter term is the initial state energy. The preliminary step occurs through a hydrogen transfer from NH3 to CO2 resulting in an O–H bond. Moreover, an N–C bond is formed as the molecules approach each other to produce a NH2–COOH pair. HF embedding found the energy of reaction to be uphill with an activation barrier of 0.031 Hartree (Ha). A bent CO2 molecule with significant interaction with both the transferring H atom and the remaining NH2 is observed for the transition state. Once the intermediate forms, the NH2–COOH pair reorients toward a more stable structure as the transferred H then flips away from the NH2 fragment. While this path is energetically downhill, it is found to be more highly activated with a barrier of 0.071 Ha.

Fig. 3.

Potential energy curve for addition reaction between CO2 and NH3 calculated by HF embedding, CCSD and HF. Atom colors: H: white, N: blue, C: gray, O: red.

Fig. 3.

Potential energy curve for addition reaction between CO2 and NH3 calculated by HF embedding, CCSD and HF. Atom colors: H: white, N: blue, C: gray, O: red.

Close modal

The results obtained with HF embedding were compared with standard HF and post-HF Coupled Cluster Singles and Doubles (CCSD) (Fig. 3). Comparison of HF embedding and post-HF CCSD calculations does not yield large discrepancies with the former within a few 10−2 Ha from the reference CCSD curve. Both methods predict that the initial NH2–COOH pair formation and its subsequent rearrangement are thermodynamically uphill and downhill, respectively. Additionally, the embedding scheme agrees with CCSD that the flipping process is the kinetically limiting step. Though CCSD yields an initial step that is transition-state-free, the positive reaction energy indicates that this process has an activation barrier. With reference to the CCSD results, the embedding scheme performs better than the conventional HF despite the drastic reduction of the number of qubits. In particular, the discrepancy is pronounced in the energy profile of the initial addition step. HF predicts this process to be transition-state-free but with a negative reaction energy, and thus a barrierless downhill process. By contrast, both CCSD and HF embedding predict this route as an activated process.

Figure 4 shows the evaluation result of ground state vibrational energies for CO2 in the energy unit of cm−1 (1 eV = 0.036 750 2 Hartree = 8065.73 cm−1) calculated by the VQE simulations. The performances of the UVCC and CHC ansatzes are similar for the ground state vibrational energy calculation of CO2. Convergence is reached after ∼40 iterations for UVCC and ∼65 iterations for CHC, as shown in Fig. 4. The calculated energies are compared with the exact solutions (Eexact) obtained by diagonalizing the vibrational Hamiltonian. Figure 4(b) shows that both ansatzes converge to the accuracy of 1 cm−1.

Fig. 4.

(a) Ground state vibrational energy and (b) accuracy convergence with number of iterations (eval count) for CO2 molecule. Eexact values are obtained by diagonalization of the vibrational Hamiltonian.

Fig. 4.

(a) Ground state vibrational energy and (b) accuracy convergence with number of iterations (eval count) for CO2 molecule. Eexact values are obtained by diagonalization of the vibrational Hamiltonian.

Close modal

As the molecule gets larger with an increasing number of atoms, VQE calculations with UVCC have higher accuracy than CHC. Figure 5 shows the result of ground state vibrational energies for NH3. UVCC converged to a higher accuracy of 9 cm−1 compared to that of CHC with 38 cm−1 after about 100 iterations. It is noted that the accuracy of the VQE calculations becomes less for NH3 compared to CO2. The difference between converged energy and exact diagonalization result is 1 cm−1 for CO2 [Fig. 3(b)] but increases to 9 cm−1 for NH3 [Fig. 5(b)] with the same UVCC ansatz. This decrease in accuracy might indicate that the VQE algorithm becomes less accurate for larger molecules such as NH3 compared to CO2.

Fig. 5.

(a) Ground state energy and (b) accuracy convergence with number of iterations (eval count) for NH3 molecule. Reference values (Eexact) are obtained by diagonalization of the vibrational Hamiltonian.

Fig. 5.

(a) Ground state energy and (b) accuracy convergence with number of iterations (eval count) for NH3 molecule. Reference values (Eexact) are obtained by diagonalization of the vibrational Hamiltonian.

Close modal

For the NH2COOH molecule, the VQE simulation with the UVCC ansatz converges to a lower energy than that with the CHC ansatz, as shown in Fig. 6. However, the classical diagonalization result for NH2COOH is not obtainable due to the large memory requirement on a classical computer.45 The size of the vibrational Hamiltonian matrix is k3N−6 × k3N−6 for a nonlinear molecule of N atoms with 3N − 6 degrees of freedom and a basis set of size k for each degree of freedom. Due to the exponential increase in matrix size with N atoms, classical diagonalization is only viable up to N = 5.45–47 The unfeasibility of the classical solution suggests the advantage of quantum computing over classical computing for the vibrational simulation of large molecules with atoms larger than five.

Fig. 6.

Ground state energy convergence with number of iterations (eval count) for NH2COOH molecule in its (a) initial state (IS), (b) transition state 1 (TS1), (c) transitional state 2 (TS2), (d) final state 1 (FS1), and (e) final state 2 (FS2).

Fig. 6.

Ground state energy convergence with number of iterations (eval count) for NH2COOH molecule in its (a) initial state (IS), (b) transition state 1 (TS1), (c) transitional state 2 (TS2), (d) final state 1 (FS1), and (e) final state 2 (FS2).

Close modal

The CHC ansatz with a shallower circuit depth has higher accuracy than the UVCC ansatz in the quantum simulator with the presence of hardware noise. We use a noise model to simulate the realistic condition of the quantum computer. The noise model includes depolarization error rates which are based on the average gate depolarization errors in the ibmq_almaden 20-qubit device [7 × 10−4, 1.4 × 10−3, and 2.2 × 10−2 for U2, U3, and CNOT gates, respectively].14Figure 7 shows the ground state vibrational energy results in the presence of simulated hardware noise for CO2 and NH3 molecules. The CHC circuit converges to better accuracy than the UVCC circuit for the CO2 molecule, with accuracies of 99 and 199 cm−1, respectively, as shown in Fig. 7(a). The same phenomenon is also present in the NH3 case. Figure 7(b) shows that the accuracy for CHC is 401 cm−1, which is higher than that of UVCC (644 cm−1) for the NH3 molecule. As CHC circuit has a smaller number of CNOT gates than UVCC (Table I), VQE calculations with CHC have lesser gate error and thus result in better accuracy. The higher accuracy of CHC compared to UVCC suggests the benefit of shallower depth circuits for near-term quantum computers where hardware noise is inevitable. It is noted that the noise model is not successful for larger molecules, such as NH2COOH, which require a higher number of qubits in the simulations.

Fig. 7.

Ground state energy convergence with number of iterations (eval count) for (a) CO2 and (b) NH3 molecules in the presence of simulated hardware noise and ZNE error-mitigation technique.

Fig. 7.

Ground state energy convergence with number of iterations (eval count) for (a) CO2 and (b) NH3 molecules in the presence of simulated hardware noise and ZNE error-mitigation technique.

Close modal

The VQE accuracy with CHC ansatz can increase by applying ZNE error-mitigation technique. Figure 7 shows that this ZNE approach improves the accuracy from 99 to 2 cm−1 for CO2 molecule, and from 401 to 141 cm−1 for NH3 molecule in vibrational ground state energy calculation by VQE algorithm with CHC ansatz. The ZNE method has little effect on the accuracy of VQE calculation with UVCC ansatz (from 199 to 192 cm−1 for CO2 and from 644 to 680 cm−1 for NH3). Our results suggest that the combination of shallow circuit depth and an effective error mitigation approach can help reduce the noise effect in real NISQ devices.

We use qEOM24 approach to calculate the vibrational excitation energies for CO2 and NH3 in noiseless simulations. The reference values are obtained by the classical diagonalization of the system Hamiltonian. The calculated results of vibrational energies are presented in Table II. At low-level excited states (first and second), the energies are computed with accuracy ∼4–6 cm−1 for CO2 with both CHC and UVCC ansatzes. For NH3, the accuracy is lower for CHC compared to UVCC. For example, at the first and second excited states, the accuracy for CHC is 87 and 117 cm−1, whereas the accuracy for UVCC is 57 and 38 cm−1, respectively. The better performance of UVCC on vibrational excited state assessment is also present in ground state calculation in noise-less simulation.

Table II.

Results of vibrational energy calculations for CO2, NH3, and NH2COOH molecules.

MoleculeQubit numberVibrational energy (cm−1)
Quantum computingClassical computing
Noiseless modelHardware noise modelError mitigation
CO2
Ground state 
2556 (CHC)
2556 (UVCC) 
2654 (CHC)
2754 (UVCC) 
2553 (CHC)
2747 (UVCC) 
2552 (DFT)
2555 (Exact) 
CO2
Excited states 
3220/3220 (CHC/UVCC)
3905/3905 (CHC/UVCC)
3931/3931 (CHC/UVCC) 
N/A N/A 3224 (Exact)
3910 (Exact)
3947 (Exact) 
NH3
Ground state 
12 7394 (CHC)
7423 (UVCC) 
7806 (CHC)
8103 (UVCC) 
7573 (CHC)
8112 (UVCC) 
7390 (DFT)
7432 (Exact) 
NH3
Excited states 
12 8357/8387 (CHC/UVCC)
8982/9011 (CHC/UVCC)
8983/9012 (CHC/UVCC)
9963/9991 (CHC/UVCC) 
N/A N/A 8444 (Exact)
9049 (Exact)
9051 (Exact)
10 056 (Exact) 
NH2COOH
IS 
30 11057 (CHC)
10 738 (UVCC) 
N/A N/A 10 416 (DFT) 
NH2COOH
TS1 
30 12043 (CHC)
11 656 (UVCC) 
N/A N/A 10 906 (DFT) 
NH2COOH
TS2 
30 8360 (CHC)
8219 (UVCC) 
N/A N/A 10 225 (DFT) 
NH2COOH
FS1 
30 11 412 (CHC)
11 218 (UVCC) 
N/A N/A 11 023 (DFT) 
NH2COOH
FS2 
30 11 433 (CHC)
11 311 (UVCC) 
N/A N/A 11 091 (DFT) 
MoleculeQubit numberVibrational energy (cm−1)
Quantum computingClassical computing
Noiseless modelHardware noise modelError mitigation
CO2
Ground state 
2556 (CHC)
2556 (UVCC) 
2654 (CHC)
2754 (UVCC) 
2553 (CHC)
2747 (UVCC) 
2552 (DFT)
2555 (Exact) 
CO2
Excited states 
3220/3220 (CHC/UVCC)
3905/3905 (CHC/UVCC)
3931/3931 (CHC/UVCC) 
N/A N/A 3224 (Exact)
3910 (Exact)
3947 (Exact) 
NH3
Ground state 
12 7394 (CHC)
7423 (UVCC) 
7806 (CHC)
8103 (UVCC) 
7573 (CHC)
8112 (UVCC) 
7390 (DFT)
7432 (Exact) 
NH3
Excited states 
12 8357/8387 (CHC/UVCC)
8982/9011 (CHC/UVCC)
8983/9012 (CHC/UVCC)
9963/9991 (CHC/UVCC) 
N/A N/A 8444 (Exact)
9049 (Exact)
9051 (Exact)
10 056 (Exact) 
NH2COOH
IS 
30 11057 (CHC)
10 738 (UVCC) 
N/A N/A 10 416 (DFT) 
NH2COOH
TS1 
30 12043 (CHC)
11 656 (UVCC) 
N/A N/A 10 906 (DFT) 
NH2COOH
TS2 
30 8360 (CHC)
8219 (UVCC) 
N/A N/A 10 225 (DFT) 
NH2COOH
FS1 
30 11 412 (CHC)
11 218 (UVCC) 
N/A N/A 11 023 (DFT) 
NH2COOH
FS2 
30 11 433 (CHC)
11 311 (UVCC) 
N/A N/A 11 091 (DFT) 

We used the VQE quantum computing algorithm to construct the reaction energy profile and calculate ground-state vibrational energies for the CO2 reaction with NH3. In the electronic energy calculations, the HF embedding predicted reaction profile is found to be in good agreement with CCSD and performs better than the conventional HF. In the molecular vibrational calculations, the quantum computing algorithm also helps enhance the calculation of vibrational energies by considering vibrational coupling or anharmonicity effects. We demonstrated that quantum-computed ground state energies had similar accuracy for CO2 and NH3 molecules compared to classically computed results using the traditional diagonalization method. The excited states for CO2 and NH3 are also evaluated with qEOM method. In noiseless simulations, the VQE algorithm with UVCC ansatz has higher accuracy than with CHC ansatz. However, we found that CHC performed better than UVCC ansatz circuit in the presence of hardware noise due to CHC's shallower circuit depth. In addition, the ZNE error-mitigation approach is more effective to increase the accuracy for VQE with CHC ansatz. For larger molecules, such as NH2COOH, more qubits and different ansatzes/circuits are needed to improve the accuracy of vibrational energy calculations. Moreover, the quantum computing calculations were able to solve for the ground state energy of NH2COOH while the exact diagonalization solution was not obtainable, which suggests the advantage of quantum computing over classical computing for the vibrational energy calculation of large molecules.

The authors would like to thank Drs. Hari Paudel, Scott Crawford, and Wissam Saidi for their useful inputs and valuable comments.

This work was performed in support of the National Energy Technology Laboratory (NETL) Laboratory Directed Research and Development (LDRD) program (No. 1024903). Research performed by Leidos Research Support Team staff was conducted under the RSS Contract No. 89243318CFE000003. We thank the computational resource of HPC centers at NETL and the University of Kentucky. M.T.N. was supported by AMO Summer Internships program sponsored by the U.S. Department of Energy (DOE)/EERE Advanced Manufacturing Office (AMO).

This research was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency hereof.

The authors have no conflicts to disclose.

Manh Tien Nguyen: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Yueh-Lin Lee: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Resources (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). Dominic Alfonso: Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Qing Shao: Resources (equal); Supervision (equal); Writing – review & editing (equal). Yuhua Duan: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
G.
Ortiz
,
J. E.
Gubernatis
,
E.
Knill
, and
R.
Laflamme
,
Phys. Rev. A
64
(
2
),
022319
(
2001
).
2.
H. P.
Paudel
,
M.
Syamlal
,
S. E.
Crawford
,
Y.-L.
Lee
,
R. A.
Shugayev
,
P.
Lu
,
P. R.
Ohodnicki
,
D.
Mollot
, and
Y.
Duan
,
ACS Eng. Au.
2
(
3
),
151
(
2022
).
3.
S.
Bush
,
Y.
Duan
,
B.
Gilbert
,
A.
Hussey
,
J.
Levy
,
D.
Miller
,
R.
Pooser
, and
M.
Syamlal
,
Fossil Energy Workshop on Quantum Information Science and Technology Summary Report
[
U. S. Department of Energy, National Energy Technology Laboratory (NETL)
,
Pittsburgh, PA
,
2020
].
4.
J.
Tilly
,
H.
Chen
,
S.
Cao
,
D.
Picozzi
,
K.
Setia
,
Y.
Li
,
E.
Grant
,
L.
Wossnig
,
I.
Rungger
 et al,
Phys. Rep.
986
,
1
(
2022
).
5.
G. S.
Barron
,
B. T.
Gard
,
O. J.
Altman
,
N. J.
Mayhall
,
E.
Barnes
, and
S. E.
Economou
,
Phys. Rev. Appl.
16
(
3
),
034003
(
2021
).
6.
W. M.
Kirby
and
P. J.
Love
,
Phys. Rev. Lett.
123
(
20
),
200501
(
2019
).
7.
S.-X.
Zhang
,
Z.-Q.
Wan
,
C.-K.
Lee
,
C.-Y.
Hsieh
,
S.
Zhang
, and
H.
Yao
,
Phys. Rev. Lett.
128
(
12
),
120502
(
2022
).
8.
O.
Higgott
,
D.
Wang
, and
S.
Brierley
,
Quantum
3
,
156
(
2019
).
9.
A.
Kandala
,
A.
Mezzacapo
,
K.
Temme
,
M.
Takita
,
M.
Brink
,
J. M.
Chow
, and
J. M.
Gambetta
,
Nature
549
(
7671
),
242
(
2017
).
10.
C.
Hempel
,
C.
Maier
,
J.
Romero
,
J.
McClean
,
T.
Monz
,
H.
Shen
,
P.
Jurcevic
,
B. P.
Lanyon
,
P.
Love
 et al,
Phys. Rev. X
8
(
3
),
031022
(
2018
).
11.
R. M.
Parrish
,
E. G.
Hohenstein
,
P. L.
McMahon
, and
T. J.
Martínez
,
Phys. Rev. Lett.
122
(
23
),
230401
(
2019
).
12.
N. H.
Stair
,
R.
Huang
, and
F. A.
Evangelista
,
J. Chem. Theory Comput.
16
(
4
),
2236
(
2020
).
13.
M.
Rossmannek
,
P. K.
Barkoutsos
,
P. J.
Ollitrault
, and
I.
Tavernelli
,
J. Chem. Phys.
154
(
11
),
114105
(
2021
).
14.
P. J.
Ollitrault
,
A.
Baiardi
,
M.
Reiher
, and
I.
Tavernelli
,
Chem. Sci.
11
(
26
),
6842
(
2020
).
15.
S.
McArdle
,
A.
Mayorov
,
X.
Shan
,
S.
Benjamin
, and
X.
Yuan
,
Chem. Sci.
10
(
22
),
5725
(
2019
).
16.
N. P. D.
Sawaya
,
T.
Menke
,
T. H.
Kyaw
,
S.
Johri
,
A.
Aspuru-Guzik
, and
G. G.
Guerreschi
,
npj Quantum Inf.
6
(
1
),
49
(
2020
).
17.
V.
Barone
,
M.
Biczysko
, and
J.
Bloino
,
Phys. Chem. Chem. Phys.
16
(
5
),
1759
(
2014
).
18.
E. L.
Sibert
,
J. Chem. Phys.
150
(
9
),
090901
(
2019
).
19.
B.
Njegic
and
M. S.
Gordon
,
J. Chem. Phys.
125
(
22
),
224102
(
2006
).
20.
T. L.
Nguyen
and
J. R.
Barker
,
J. Phys. Chem. A
114
(
10
),
3718
(
2010
).
21.
C.
Aieta
,
M.
Micciarelli
,
G.
Bertaina
, and
M.
Ceotto
,
Nat. Commun.
11
(
1
),
4348
(
2020
).
22.
N. P. D.
Sawaya
,
F.
Paesani
, and
D. P.
Tabor
,
Phys. Rev. A
104
(
6
),
062419
(
2021
).
23.
O.
Christiansen
,
J. Chem. Phys.
120
(
5
),
2140
(
2004
).
24.
P. J.
Ollitrault
,
A.
Kandala
,
C.-F.
Chen
,
P. K.
Barkoutsos
,
A.
Mezzacapo
,
M.
Pistoia
,
S.
Sheldon
,
S.
Woerner
,
J. M.
Gambetta
 et al,
Phys. Rev. Res
2
(
4
),
043140
(
2020
).
25.
N. P. D.
Sawaya
and
J.
Huh
,
J. Phys. Chem. Lett.
10
(
13
),
3586
3591
(
2019
).
26.
J.
Gibbins
and
H.
Chalmers
,
Energy Policy
36
(
12
),
4317
(
2008
).
27.
G. T.
Rochelle
,
Science
325
(
5948
),
1652
(
2009
).
28.
K.
Temme
,
S.
Bravyi
, and
J. M.
Gambetta
,
Phys. Rev. Lett.
119
(
18
),
180509
(
2017
).
29.
G.
Aleksandrowicz
,
T.
Alexander
,
P.
Barkoutsos
,
L.
Bello
,
Y.
Ben-Haim
,
D.
Bucher
,
F. J.
Cabrera-Hernández
,
J.
Carballo-Franquis
,
A.
Chen
 et al (
2019
). “Qiskit: An open-source framework for quantum computing,”
Zenodo
.
30.
Q.
Sun
,
T. C.
Berkelbach
,
N. S.
Blunt
,
G. H.
Booth
,
S.
Guo
,
Z.
Li
,
J.
Liu
,
J. D.
McClain
,
E. R.
Sayfutyarova
 et al,
WIREs Comput. Mol. Sci.
8
(
1
),
e1340
(
2018
).
31.
Q.
Sun
,
J. Comput. Chem.
36
(
22
),
1664
(
2015
).
32.
B.
Bauer
,
S.
Bravyi
,
M.
Motta
, and
G. K.-L.
Chan
,
Chem. Rev.
120
(
22
),
12685
(
2020
).
33.
W. J.
Hehre
,
R.
Ditchfield
,
R. F.
Stewart
, and
J. A.
Pople
,
J. Chem. Phys.
52
(
5
),
2769
(
1970
).
34.
D. C.
Liu
and
J.
Nocedal
,
Math. Program.
45
(
1–3
),
503
(
1989
).
35.
D. F.
Shanno
,
Math. Comput.
24
(
111
),
647
(
1970
).
36.
G.
Kresse
and
J.
Furthmüller
,
Comput. Mater. Sci.
6
(
1
),
15
(
1996
).
37.
G.
Henkelman
,
B. P.
Uberuaga
, and
H.
Jónsson
,
J. Chem. Phys
113
(
22
),
9901
(
2000
).
38.
W.
Li
,
Z.
Huang
,
C.
Cao
,
Y.
Huang
,
Z.
Shuai
,
X.
Sun
,
J.
Sun
,
X.
Yuan
, and
D.
Lv
,
Chem. Sci.
13
(
31
),
8953
(
2022
).
39.
C.
Cao
,
J.
Hu
,
W.
Zhang
,
X.
Xu
,
D.
Chen
,
F.
Yu
,
J.
Li
,
H.-S.
Hu
,
D.
Lv
 et al,
Phys. Rev. A
105
(
6
),
062452
(
2022
).
40.
A.
Peruzzo
,
J.
McClean
,
P.
Shadbolt
,
M.-H.
Yung
,
X.-Q.
Zhou
,
P. J.
Love
,
A.
Aspuru-Guzik
, and
J. L.
O'Brien
,
Nat. Commun.
5
(
1
),
4213
(
2014
).
41.
J. M.
Bowman
,
Acc. Chem. Res.
19
(
7
),
202
208
(
1986
).
42.
M. J.
Frisch
,
G. W.
Trucks
,
H. B.
Schlegel
,
G. E.
Scuseria
,
M. A.
Robb
,
J. R.
Cheeseman
,
G.
Scalmani
,
V.
Barone
,
G. A.
Petersson
 et al,
Gaussian 16, Revision C.01
(
Gaussian, Inc
.,
Wallingford, CT
,
2016
).
43.
J.
Kongsted
and
O.
Christiansen
,
J. Chem. Phys
125
(
12
),
124108
(
2006
).
44.
L.
Veis
,
J.
Višňák
,
H.
Nishizawa
,
H.
Nakai
, and
J.
Pittner
,
Int. J. Quantum Chem.
116
(
18
),
1328
(
2016
).
45.
E.
Lötstedt
,
K.
Yamanouchi
,
T.
Tsuchiya
, and
Y.
Tachikawa
,
Phys. Rev. A
103
(
6
),
062609
(
2021
).
46.
A. G.
Császár
,
I.
Simkó
,
T.
Szidarovszky
,
G. C.
Groenenboom
,
T.
Karman
, and
A.
Van Der Avoird
,
Phys. Chem. Chem. Phys.
22
(
27
),
15081
(
2020
).
47.
A. G.
Császár
,
C.
Fábri
,
T.
Szidarovszky
,
E.
Mátyus
,
T.
Furtenbacher
, and
G.
Czakó
,
Phys. Chem. Chem. Phys.
14
(
3
),
1085
(
2012
).