A Simulation of Hydrazine Molecule’s Potential Energy Surface using Variational Quantum Eigensolver Algorithm

Quantum computing is a technology that utilizes the principles of quantum mechanics to perform complex computational processes. In this work, we use Qiskit Module from IBM to do quantum computational calculation using Variational Quantum Eigensolver (VQE) algorithm. VQE is a hybrid quantum-classical method that combines a quantum computer to measure energies and a classical computer to process the measurement results and update the parameters of the quantum computer. The purpose of VQE is to find the ground state energy of a chemical system. In the previous study, many of the VQE calculations have been done on simple molecules. So, in this study, we would like to use Hydrazine molecule as our object of VQE calculation. Furthermore, these results will be compared with the results from the classical calculation (MP2, CCSD(T), QCISD(T), and CASSCF) methods for testing the effectiveness of VQE using Unitary Coupled-Cluster Single and Double excitations (UCCSD) Ansatz. The quantum algorithm based on the UCCSD Ansatz led to a simplification of the algorithm by reducing the circuit depth. Then, the possibility to use active space approximation, can be used to reduce the quantum gates while trying to keep a good level of accuracy. In this study, we chose (2,2) and (4,4) active spaces. Based on the results, as we increase the size of the active space during the evaluation of the single-point energy, the estimated ground states obtained from the VQE algorithm yield nearly identical values. Conversely, in CASSCF calculations, expanding the active space introduces more energy corrections, thus making it more sensitive. Additionally, when examining potential energy surfaces, VQE demonstrates results that gradually align with CCSD(T) and QCISD(T) methods.


Introduction
Quantum computing is a technology that utilizes the principles of quantum mechanics to perform complex computational processes [1].The concepts and ideas of quantum computing themselves date back to the 1980s.In 1982, Richard Feynman suggested that since the universe works on the principles of quantum mechanics, we need quantum mechanics as the simulator if we want to simulate nature [2].A decade later, in 1994, Peter Shor [3] devised an algorithm capable of factorizing integers exponentially faster than the fastest-known classical algorithms.This advancement had the potential to disrupt cryptography, as factorization serves as the foundation for encryption.Beyond cryptography, quantum computing is anticipated to revolutionize various fields, including drug discovery, battery design, and the development of advanced materials [4].Today, major technology companies such as IBM, Google, and Microsoft are actively investing in and developing quantum computer technologies [5].
However, the computations that are being carried out in the currently available quantum processor are not yet fully quantum.In terms of their readiness, the number of qubits on the current quantum devices to execute a quantum algorithm is still limited to a hundred qubits and cannot be used effectively for algorithms with deep circuits [4] [6].Qubits are prone to errors and environmental noise that could cause decoherence.Ideally, a million of physical qubits are needed to perform quantum error correction (QEC).With those limitations, we can describe this era as the "noisy intermediate-scale quantum (NISQ) [7]" The primary goal during the NISQ era is to develop algorithms that leverage existing quantum technology to potentially harness quantum advantages.Aspuru-Guzik et al. [8] have proposed that quantum computers equipped with tens of noise-free quantum bits (qubits) could already surpass the computational limits of classical calculations for quantum chemistry, while Shor's algorithm demands thousands of fully functioning logical qubits [3][9].Based on this, it can be inferred that the earliest practical applications of quantum computers in the NISQ era will likely revolve around simulations and computations of quantum systems.
Among the promising quantum algorithms for the NISQ era is the Variational Quantum Eigensolver (VQE), which was first conceptualized and experimentally implemented by Peruzzo et al. [10].VQE employs a hybrid quantum-classical approach, utilizing a quantum computer for energy measurements and a classical computer for processing measurement results and adjusting quantum computer parameters.The primary objective of VQE is to determine the ground state energy of a chemical system, a crucial initial step in assessing the energetic properties of molecules and materials [11].However, because of existing limitations in hardware, VQE is currently limited to small molecule applications [4].
In prior studies [4][12] [13][14], many VQE calculations using a classical simulator were tested on simple molecules.In this study, we intend to focus on the Hydrazine molecule as our subject of calculation.Furthermore, we will compare these results with those obtained from classical calculations (MP2, CCSD(T), QCISD(T), CASSCF).

Variational Quantum Eigensolver (VQE)
Using VQE, we aim to find the ground state of a chemical system.The basic idea of VQE is to utilize a quantum computer to prepare a parameterized wavefunction (or ansatz) and measure the expectation value.The parameters of the wavefunction are given by the optimization process running on a classical computer.These two subroutines constitute the flow of the algorithm.

Electronic Structure Hamiltonian Representation
Before delving into the mathematical principle of VQE, it is better for us to know the problem being given in chemistry, that is, to find the solution of the many-body Schrödinger equation: with H as the Hamiltonian, |Ψ⟩ as our wavefunction, and E as the energy eigenvalue.Using the Born-Oppenheimer approximation, which neglect the motion of nucleus, the second quantization formulation of Hamiltonian can be expressed as: where, â † and â act as fermionic creation and annihilation operator respectively, and 10th Asian Physics Symposium (APS 2023) Journal of Physics: Conference Series 2734 (2024) 012052 Eq.3 and Eq.4 correspond to the one and two electron integrals, respectively.R i and r indicate the nuclear and electronic spatial coordinate, and σ represents the electron's spatial and spin coordinates.

Mathematical Principle VQE uses the Rayleigh-Ritz variational principle by parameterizing the trial wavefunction
We hope to find the lowest possible energy, E 0 , by trying to find the right values of parameters, ⃗ θ, which will change the wavefunction and consequently minimize the expected value.
To perform the minimization process on a quantum computer, we must first implement the ansatz as a series of parameterized quantum gates: U ( ⃗ θ) represents the parameterized unitary gates that construct our ansatz circuit, and |Ψ 0 ⟩ is our chosen initial state.The initial state consists of an N -qubit register, usually initialized as |0⟩ ⊗N .
Once the ansatz has been generated, the next step is to encode the Hamiltonian from Eq. 2 into a form that can be measured by a quantum computer using the qubit encoding method.With the Jordan-Wigner encoding [16], the Hamiltonian can then be re-expressed as: Ĥ = j h j Pj (7) where h j as a scalar coefficient, and Pj ∈ (I, X, Y, Z) denotes the Pauli Operator.By combining Eq.5, 6, and 7 we can write: Now, we can compute Eq.8 on a quantum computer.Each measurement will generate an energy value that will be optimized on a classical computer, which then gives the updated parameters to the quantum computer again.These processes will be repeated many times until the energy converges to a minimum.

Qubit Encoding Methods
Qubit encoding is a method used to transform the fermionic basis into the qubit basis, where each fermionic state corresponds to a qubit state.One of the simplest encoding processes is the Jordan-Wigner (JW) transformation.According to this method, an occupied orbital in the fermionic basis can be represented as |1⟩ in the qubit state, while an unoccupied orbital is denoted as the |0⟩ state.Mathematically, we can express this transformation as: M denotes the number of spin orbitals used in the simulation, while q p and f p tells the occupation of qubit and fermionic state, respectively.
The other encoding method called the Bravyi-Kitaev (BK) transformation [17] could also be used in the mapping process.This method is more efficient than JW transormation, although being more complicated.Both JW and BK transformation are supported in Qiskit [18].

Unitary Coupled Cluster Ansatz
In the unitary coupled cluster (UCC), the trial wavefunction can be written as: UCC are often used with the single and double cluster operators (Eq.11), known as the Unitary Coupled Cluster Single-Double excitations (UCCSD).T = T1 + T2 (11) where T = i Ti , and where, occ refers to the occupied orbitals in the initial state, while unocc refers to the unoccupied orbitals in the initial state |Ψ 0 ⟩.The quantum algorithm based on the UCCSD Ansatz led to a simplification of the algorithm by reducing the circuit depth [19].

Active Space Approximation
The size of the chosen basis set corresponds to the required number of qubits used and the length of the Pauli string.Consequently, the number of qubits required in quantum devices for larger molecules with extensive basis sets significantly increases, posing an intractable challenge within the current era of noisy intermediate-scale quantum (NISQ) devices.To reduce the qubit requirements and enable the use of larger basis sets without the need for additional qubits, we can employ the Active Space (AS) approximation [20], which divides the initial state into active (|Ψ A 0 ⟩) and inactive (|Ψ I 0 ⟩) orbitals.
This approximation suggests that the wavefunction is predominantly influenced by a few Slater determinants, which can be encapsulated by expanding the wavefunction within the active space.Therefore, we can assign the computation of inactive electrons to the classical HF (Hartree-Fock) driver, restricting the quantum computation to the AS [21].As a result, this allows for a reduction of the Hilbert space within the UCC ansatz.

|Ψ(
The choice of active space orbitals typically corresponds to the highest occupied molecular orbitals and the lowest unoccupied molecular orbitals within a system.

Simulation and Computational Detail
In this study, the simulation of VQE will be carried out using the Qiskit module from IBM. Qiskit is an open-source module developed by IBM for designing quantum circuits and algorithms.In addition, we also add classical algorithm computations such as MP2, CASSCF, CCSD(T), and QCISD(T) to benchmark the VQE results.All of the classical computations are done with Gaussian 09 [22].

Energy Calculations
The aim is to calculate the single-point energy and the Potential Energy Surface (PES) of the Hydrazine molecule and its conformations.The first step is to optimize the geometries using MP2 with the 6-31+G(d,p) basis, which is available in the PySCF driver for Qiskit's VQE application and Gaussian 09.Then, the basis and the optimized geometries will be used as a starting point for further energy calculations.The VQE and CASSCF calculations are performed with the (n,m) active space approximation, where 'n' is the number of active electrons, and 'm' is the number of active orbitals.In this study, we selected active spaces of (2,2) and (4,4), where the number of qubits equals twice the number of active orbitals.
In both Qiskit and Gaussian 09, the active space is chosen around the Fermi level.This choice assumes that the electrons come from the highest occupied orbitals within the initial guess determinant, while the additional orbitals needed for the active space are drawn from the lowest virtual orbitals of the initial guess.

VQE Simulation
The steps of VQE simulation are summarized as follows:

Classical Preprocessing
In this step, we use the PySCF package [23] to construct the molecular Hamiltonian.Then, we employ an active space approximation approach to reduce the complexity of the wavefunction.Afterward, we apply the Jordan-Wigner transformation method with the Qiskit module to map the fermions into the qubit Hamiltonian.

Quantum Simulation
In this step, we generate the trial wavefunction, and we use the UCCSD (Unitary Coupled Cluster Singles and Doubles) ansatz as our choice.The circuit and UCCSD generation [19] are provided by Qiskit.Once the wavefunction is prepared, we can compute the Hamiltonian.All calculations are performed using the IBM Qiskit Nature module [24] and Estimator.

Parameter Optimization
The parameters are updated using the SLSQP (Sequential Least Squares Quadratic Programming) optimizer provided by the Qiskit Optimizer.

Results and Discussion
In Table I, the total and relative energies from the single-point calculations are presented, with the total energy of the skew structure being set as the zero-energy reference.Geometry optimizations were performed using full second-order Møller-Plesset perturbation theory (MP2), which provides nearly accurate geometry calculations [25].Subsequently, we assess the effects of higher-order correlation corrections with QCISD(T) and CCSD(T) calculations, often referred to as the 'gold standard' when the correlations are not too strong [26].
In the VQE calculation, the computation is carried out under conditions where errors from measurement and environmental noise can be excluded.The estimated ground states obtained from the VQE algorithm within (2,2) and (4,4) active spaces yield nearly identical values.Conversely, in CASSCF calculations, expanding the active space introduces energy corrections that bring the relative energy towards the results of MP2 calculations.Compared to the outcomes of CASSCF methods, both the total and relative energies derived from VQE appear higher.In this context, it could be observed that CASSCF methods demonstrate better sensitivity in evaluating the energies of the molecule in a fixed position.Nevertheless, the application of the UCCSD Ansatz in this report results in relative energy trends from VQE that reasonably align with those of classical methods.
In Figure 2, 3, and 4, we compare the potential energy surface calculations from each method.The energy is calculated by changing the distance between one of the Hydrogen and Nitrogen atoms, with a 0.1 increment from 0.9 to 2.2 Angstroms.Then, we use the minimum value of each method as the zero energy reference to evaluate the relative energy of each method.
When comparing the VQE results with those of the CASSCF methods, we observe a qualitative similarity in trends up to a certain point.Beyond that point, the CASSCF trend    Results of Potential Energy Surface of Hydrazine (Trans).ascends, while the VQE results gradually approach those of CCSD(T) and QCISD(T), even when the active space of CASSCF is increased.The expansion of the active space makes the VQE approximation results closer to those obtained from CCSD(T) and QCISD(T) methods.Hence, it could be proposed that VQE is better suited for evaluating the energies associated with the changing position of the structure, thereby providing VQE with a feature related to electron interaction.

Conclusion
In this work, we examine the implementation of VQE algorithms and compare them with other classical methods.The quantum algorithm used is based on the UCCSD Ansatz with the active space approximation, simplifying the computational process and reducing the required number of qubits.When we expand the size of the active space during the evaluation of the singlepoint energy at a fixed position, the estimated ground states obtained from the VQE algorithm yield nearly identical values.Conversely, in CASSCF calculations, expanding the active space introduces more energy corrections, making it more sensitive.Additionally, when examining potential energy surfaces, VQE demonstrates results that gradually align with CCSD(T) and QCISD(T) methods.This implies that VQE might be more adept at evaluating the energies related to the structural changes, effectively capturing electron interaction.

Figure 1 .
Figure 1.Structures of Hydrazine molecule and its conformations.

Figure 2 .
Figure 2. Results of Potential Energy Surface of Hydrazine (Cis).

Figure 3 .
Figure 3. Results of Potential Energy Surface of Hydrazine (Skew).

Figure 4 .
Figure 4. Results of Potential Energy Surface of Hydrazine (Trans).

Table 1 .
Results of Single-point energy calculations