# Morse Potential on a Quantum Computer for Molecules and Supersymmetric Quantum Mechanics

Josh Apanavicius, Yuan Feng, Yasmin Flores, Mohammad Hassan, Michael McGuigan

MMorse Potential on a Quantum Computer forMolecules and Supersymmetric Quantum Mechanics

Josh Apanavicius, Yuan Feng, Yasmin Flores, Mohammad Hassan, Michael McGuigan ∗ Abstract

In this paper we discuss the Morse potential on a quantum computer. The Morsepotential is useful to describe diatomic molecules and has a ﬁnite number of boundstates which can be measured through spectroscopy. It is also a example of an exactlysoluble potential using supersymmetric quantum mechanics. Using the the supersym-metric quantum mechanics formalism one can derive a heirachy of Hamiltonians suchthat the ground state of the next rung on the heirarchy yeids the ﬁrst excited stateof the hamiltonian below it. Using this method one can determine all the states ofthe Morse potential by calculating all the ground states of the sequence of Hamiltoni-ans in the heirarchy. We use the IBM QISKit software together with the VariationalQuantum Eiegensolver (VQE) algorithm to calculate the ground state and ﬁrst excitedstate energy of the Morse potential and ﬁnd agreement with the exact expression for thebound state energies of the Morse Potential. We analyze diﬀerent optimizers to studythe numerical eﬀect on the calculations. Finally we perform quantum computations fordiatomic and triatomic molecules to illustrate the application of these techniques onnear term quantum computers and ﬁnd excellent agreement with experimental data. ∗ In author order: Indiana University, Pasadena City College, St. Joseph’s College, City College of NY,Brookhaven National Laboratory a r X i v : . [ qu a n t - ph ] F e b Introduction

The Morse potential was introduced in [1] to describe the bound state energies of diatomicmolecules. The potential also serves as example like the simple harmonic oscillator asan exactly soluble model in that one can analytically determine all the eigenstates andeigenvalues. The exact solvablity can be traced back to it’s relation to supersymmetricquantum mechanics (SusyQM)[2][3][4][5][6][7][8] , deﬁnition of ladder operators and a setof Hamiltonians that are related by change of parameters in the potential. The Morsepotential takes a diﬀerent form depending on whether one is studying diatomic moleculesor one is interested in the supersymmetric quantum mechanics connection. We will startwith the SusyQM form but will return to the diatomic molecule form in a later section.The use of the Variational Quantum Eigensolver (VQE) has been shown to be an eﬃcientquantum algorithm for the calculation of ground state energies on noisy intermediate scalequantum computers. Having an exact solution for the Morse potential gives us an excellentpoint of comparison and allows us to see what level of accuracy can be achieved on currentquantum computing hardware and software.This paper is organized as follows. In section 1 we give an introduction to the study ofthe Morse potential on a quantum computer. In section 2 we give the exact solutions tothe Morse potential and we go over the relation of the Morse potential to supersymmetricquantum mechanics, discuss the heirarchy of Hamiltonians for the Morse potential andhow this can be used to calculate all the bound states,and give the exact solutions tothe Morse potential. In section 3 we discuss the calculation of the bound states for theMorse potential using the IBM QISKit eigensolver applied to SUSY partner Hamiltonians,compute ground state and ﬁrst excited state energies and compare our results to the exactcalculation. In section 4 we discuss the application of the variational quantum eigensolverto calculating the ground state energies of realistic diatomic molecules and in section 5 wediscuss the calculations with Morse potentials of two variables that can be used to describetriatomic molecules. Finally in section 6 we state the main conclusions of the paper.

In it’s relation to supersymmetric quantum mechanics the Morse potential is written as: V − ( x, A ) = e − x − (2 A + 1) e − x + A (2.1)with A given by the Morse parameter. This potential is shown in Figure 1 for A = 5.When one studies diatomic molecules one writes the Morse potential in a diﬀerent form: V ( x ) = D ( e − a ( x − x ) − e − a ( x − x ) ) (2.2)with parameters D , a and x . 2 - - V M ( x ) Figure 1: Morse potential for A = 5 with ﬁve bound state energies 0 , , , , V − ( x ) = W ( x ) − W (cid:48) ( x ) V + ( x ) = W ( x ) + W (cid:48) ( x ) (2.3)where the superpotential W ( x ) is given by: W ( x ) = A − e − x (2.4)so that the plus partner potential is given by: V + ( x, A ) = e − x − (2 A − e − x + A (2.5)The partner Hamiltonians are: H − = p + V − ( x ) = p + W ( x ) − W (cid:48) ( x ) H + = p + V + ( x ) = p + W ( x ) + W (cid:48) ( x ) (2.6)where we have set 2 m = 1. The ladder operators are given by: a = ip + W ( x ) = ip + A − e − x a † = − ip + W ( x ) = − ip + A − e − x (2.7)Then the partner Hamiltonians can be realized as: H − = a † aH + = aa † (2.8)3 ( x ) Figure 2: Hierarchy of potentials associated with the Morse potential for A = 5.One can then form a sequence of hierarchy of Hamiltonians as H i = p + V i ( x ) where: V ( x ) = V − ( x, A ) V ( x ) = V + ( x, A ) V ( x ) = V + ( x, A −

1) + 2( A −

1) + 1 V ( x ) = V + ( x, A −

2) + 2( A −

1) + 1 + 2( A −

2) + 1 V ( x ) = V + ( x, A −

3) + 2( A −

1) + 1 + 2( A −

2) + 1 + 2( A −

3) + 1 (2.9)These are plotted in ﬁgure 2. Exact solutions to the Schrodinger equation can be obtainedusing the ladder operators for the Morse potential. The expressions are given by: witheigenfunctions: ψ − n ( x ) = e − x ( A − n ) e − e − x L (2 A − n ) n (2 e − x ) (2.10)with L ( k ) n ( y ) the associated Laguerre polynomials. with bound state energies: E n = A − ( A − n ) (2.11)For A = 5 these are given by: { , , , , } (2.12)which correspond to the ground state energies of the Hierarchy Hamiltonians in (2.9). In this section we describe the calculation of the ground state energy of the Morse potentialin the supersymmetric quantum mechanics formulation of the Morse potential. Before one4an set up the calculation to compute the ground state energies using the VQE one needsto perform a Hamiltonian mapping in terms of qubits. The ﬁrst step is to represent theHamiltonian as an N × N matrix using a discete quantum mechanics approximation to thequantum mechanical operators which would be inﬁnite dimensional for bosonic observables[9][10][11][12]. In this paper we will use three diﬀerent types of discrete Hamiltonians andcompare the results from each. Gaussian or Simple Harmonic Oscillator basis

This is a very useful basis based on the matrix treatment of the simple harmonic oscillatorwhich is sparse in representing the position and momentum operator. For the positionoperator we have: X osc = 1 √ √ · · · √ √ · · · √ √ N −

10 0 · · · √ N − (3.1)while for the momentum operator we have: P osc = i √ −√ · · · √ −√ · · · √ −√ N −

10 0 · · · √ N − (3.2)The Morse Hamiltonian H − is then H − = P osc + Exp ( − X osc ) − (2 A + 1) Exp ( − X osc ) + A I (3.3)where Exp refers to the Matrix exponential and I is the N × N identity matrix. Position basis

In the position basis the position matrix is diagonal but the momentum matrix is denseand constructed from the position operator using a Sylvester matrix F . In the positionbasis the position matrix is:( X pos ) j,k = (cid:114) π N (2 j − ( N + 1)) δ j,k (3.4)5nd the momentum matrix is: P pos = F † X pos F (3.5)where F j,k = 1 √ N e πi N (2 j − ( N +1))(2 k − ( N +1)) (3.6)The Morse Hamiltonian is formed from H − = P pos + Exp ( − X pos ) − (2 A + 1) Exp ( − X pos ) + A I (3.7)but in this case the matrix exponential is very simple as it is the exponential of a diagonalmatrix. Finite diﬀerence basis

This is the type of basis that comes up when realized diﬀerential equations in terms ofﬁnite diﬀerence equations. In this case the position operator is again diagonal but themomentum operator althogh not diagonal is still sparse. In the ﬁnite diﬀerence basis theposition matrix is: ( X fd ) j,k = (cid:114) N (2 j − ( N + 1)) δ j,k and the momentum-squared matrix is: P fd = N − · · · − − · · · − −

10 0 · · · − (3.8)The Morse Hamiltonian is then: H − = P fd + Exp ( − X fd ) − (2 A + 1) Exp ( − X fd ) + A I (3.9)Whatever basis one uses one needs to map the Hamiltonian to a an expression in termsof a sum of tensor products of Pauli spin matrices plus the identity matrix which arecalled Pauli terms. As there are four such matrices the maximum number of terms in thisexpansion is 4 n where n is the number of qubits, In most of our simulations the number ofqubits was ﬁxed at 4 so that the maximum number of Pauli terms was 256.6igure 3: A parameterized R y variational form with 4 qubits, full entanglement and adepth of 3. This circuit is the quantum computing analog of an ansatz wavefunction. Calculation of bound state energies using the VQE for supersymmetricquantum mechanics

The VQE algorithm is a semi-quantum algorithm which is based on the variational methodof quantum mechanics. The variational method allows one to calculate the upper bound ofthe ground state energy of a quantum system, without ever having to solve the Schrodingerequation. All that is needed is knowledge of the Hamiltonian, H, and for one to pick anyansatz wave function, ψ . Mathematically, the variational method says that for this ansatzwave function: E ≤ (cid:104) ψ | H | ψ (cid:105) ≡ (cid:104) H (cid:105) (3.10)This means that the expectation value of the Hamiltonian, in the ansatz state, will alwaysbe an upper bound on the true ground state energy. If the ansatz is chosen well enough, theexpectation value of the Hamiltonian can be made arbitrarily precise to the true groundstate energy. The VQE algorithm is an implementation of the variational method on aquantum computer. The idea is to create the ansatz wave function as a quantum circuitusing parameterized quantum gates. A quantum circuit which represents the ansatz wavefunction is called the variational form. An example of a variational form is shown in Figure3, where a series of parameterized R y gates are applied on each qubit, and each qubit isentangled to every other qubit using CN OT gates. This sort of entanglement is known asfull entanglement. This circuit is known as an R y variational form, where R y represents arotation of the qubit statevector about the y -axis on a Bloch sphere. The Hamiltonian isalso mapped to a quantum circuit using the fundamental X , Y , Z , and I gates. The VQEalgorithm ﬁrst sets arbitrary parameters for the variational form (in this case, arbitraryangles for each R y ) gate, then it calculates the expectation value of the Hamiltonian inthe current ansatz state using a quantum computer. It then updates the parameters andcalculates the expectation value again. Using a classical computer, it ﬁnds the cost functionbetween the current and previous expectation value, after which it uses a classical optimizerto optimize the parameters until the expectation value is minimized. The eﬃciency andaccuracy of results can be controlled by the depth, the entanglement, the variational formof the circuit, and optimizer used. The depth of the circuit represents how many times a7ariational form is repeated. For example, one unit of the R y variational form applies an R y gate to each qubit, then CNOT gates to entangle each qubit, then another set of R y gates to each qubit. This is shown in Figure 4. A depth of 3, as shown in Figure 3, repeatsthis pattern 3 times. A larger depth allows the variational form to generate a larger setof states; however, this comes at the cost of longer algorithm runtimes, as the number ofparameters to be optimized is also increased.Figure 4: One single unit (depth = 1) of an R y variational form.In calculating the bound state energies on the quantum computer it is convenient toscale the partner Hamiltonians by 1 /

2. Then the H − and H + Hamiltonians for the Morsepotential are written as: H − = p (cid:0) e − x − (2 A + 1) e − x + A (cid:1) H + = p (cid:0) e − x − (2 A − e − x + A (cid:1) (3.11)Performing a VQE calculation using IBM QISKit, we ﬁnd accurate results for the case A = 5 given by E = 0 . H − and E = 4 . H + in the oscillator basis. Thiscompares well with the exact values of E = 0 and E = 9 / A = 5. These resultsare shown in Table 1. The convergence plots for these calculations are shown in Figure5s. The VQE algorithm was then run on the H − Hamiltonian using diﬀerent optimizers.The result for each optimizer is shown in Table 2. The convergence plot for the variousoptimizers is shown in Figure 5. Note that is all cases the VQE result lies above the exactenergy value. This is because the variational method provides a lower bound of the energy E V QE ≥ E .Finally it is possible by re-scaling the x coordinate to write the Hamiltonion H − in theform H − = p (cid:18) A + 12 (cid:19) (cid:0) e − x − e − x + 1 (cid:1) − (cid:18) A + 14 (cid:19) (3.12)8asis Hamiltonian VQE Result No. Pauli TermsOscillator H − H + A = 5 using the oscillator basis. All of theHamiltonian were mapped to 4-qubit operators. The quantum circuit for each simulationutilized an R y variational form, with a fully entangled circuit of depth 3. The backendused was a statevector simulator. The Sequential Least SQuares Programming (SLSQP)optimizer was used, with a maximum of 600 iterations. The exact result for H − and H + for the Morse potential used was 0 and 4 . H − obtained using various optimizers.This form of the Morse potential similar to that used to describe diatomic molecules whichwe discuss in the next section. The Hamiltonian for the Morse potential describing diatomic molecules is given by [13]: H M = p m r + V M ( x ) (4.1)where the Morse potential is given by: V M = D mol ( e − ax − e − ax ) (4.2)with the reduced mass of the diatomic molecule given. by: m r = m m m + m (4.3)To study the Morse potential on a quantum computer it is more convenient to study theHamiltonian H = p λ e − x − e − x + 1) (4.4)9igure 5: Convergence plot for the various optimizers used in the VQE calculation of theMorse potential.This is related to the Morse Hamiltonian in a simple way. If the ground state energy ofthis Hamiltonian is ε the ground state energy of the molecule is E = ( ε − λ E mult (4.5)where λ and E mult as well as other parameters are listed for various molecules in Tables2, 3 and 4. In terms of ε we have ε = λ − x → ∞ ε ∞ = λ ε ε ∞ = 1 λ − λ (4.8)10igure 6: (Left) Convergence plot for the VQE for the minus partner Hamiltonian (Right)Convergence plot for the VQE for the plus partner Hamiltonian using the oscillator basis.In terms of the parameters of the molecule λ = 2 m r c D mol a (cid:126) c E mult = a (cid:126) c m r c (4.9)We plot some of the potentials in ﬁgures 7 and 8. Tables 3, 4 and 3 contain the resultsfor the VQE calculations using 4 qubits and the the oscillator basis. In all cases we foundexcellent agreement with the experimentally measured values. Convergence plots for theVQE calculations are given in ﬁgures 9 and 10 and all cases rapidly converge near theground state energy. 11olecule λ E mult No. bound states E ( eV ) E ( eV ) VQE H HCl

LiH CO O N λ / λ ε /ε ∞ ε ε VQE H HCl

LiH CO O N m r ( amu ) D mol ( eV ) a (10 m − ) H .50391 4.7446 1.9426 HCl .9796 4.618 1.869

LiH CO O N - - - - - V H2 ( x ) - - - - V HCl ( x ) Figure 7: (Left) Morse potential for the H molecule (Right) Morse potential for the HCl molecule. - - - - - V LiH ( x ) - - - - - - V CO ( x ) Figure 8: (Left) Morse potential for the

LiH molecule. (Right) Morse potential for the CO potential. 13igure 9: (Left) VQE convergence plot for H versus the number of optimizationsteps.(Middle) VQE convergence plot for HCl versus the number of optimization steps.(Right) VQE convergence plot for

LiH versus the number of optimization steps.Figure 10: (Left) VQE convergence plot for CO versus the number of optimizationsteps.(Middle) VQE convergence plot for O versus the number of optimization steps.(Right) VQE convergence plot for N versus the number of optimization steps. The Morse potential can be extended to treat triatomic molecules as well [14][15]. In thiscase the Morse potential depends on two coordinates x, y and Hamiltonian can be written14s : H = p x m + p y m − p x p y M + C (cid:16) − e − x/b (cid:17) + C (cid:16) − e − y/b (cid:17) (5.1)Another form of the Hamiltonian can be formed by changing variables x = ( x + x ) / √ y = ( x − x ) / √ m = m − mM m = m mM (5.3)In this form of the Hamiltonian can be written: H = p m + p m + C (cid:16) − e − ( x + x ) / √ b (cid:17) + C (cid:16) − e − ( x − x ) / √ b (cid:17) (5.4)A plot of this potential is given in Figure 12. For the case m = 1, M = 2, C = 10and b = √

20, we ﬁnd the ground state energy E = 0 . X = X osc ⊗ IX = I ⊗ X osc P = P osc ⊗ IP = I ⊗ P osc (5.5)and using 8 qubits for the deﬁnition of the Hamiltonian in Equation 5.1, we ﬁnd E =0 . H and H are presented in Table 6. The variational form for both simulations isshown in Figure 11.Figure 11: 8-qubit R y variational form of depth 3 used in the VQE calculation of the Morsepotential for triatomic molecules. 15amiltonian VQE Result No. Pauli Terms H H H and H .The parameters in each equation were set as m = 1, M = 2, C = 10 and b = √

20. BothHamiltonians were mapped to an 8-qubit operator. The quantum circuit for each simulationutilized an R y variational form, with a fully entangled circuit of depth 3. The backendused was a statevector simulator. The SLSQP optimizer was used, with a maximum of 600iterations. The exact ground state energy for both Hamiltonians is E = 0 . - - - - - - Figure 12: (Left) 3D plot of a form of the Morse potential that can be applied to triatomicmolecules (Right) Contour plot of a form of the Morse potential that can be applied totriatomic molecules.

In this paper we investigated using a quantum computer to simulate Hamiltonians using theMorse potential for supersymmetric quantum mechanics, diatomic molecules and triatomicmolecules. In all cases we found excellent agreement with the exact answers using IBMQISKit and its statevector simulator. It will be interesting to extend these calculationsto other molecules and combine these methods with electronic structure calculations onnear term quantum computers which can determine the parameters of the Morse potential[16][17]. This approach was used in [18] to compute thermodynamic observables of the16olecular systems. Finally it is important to study other variational forms as well as errorand noise mitigation on near term devices associated with the calculations of the Morsepotential to gain further understanding of the applicability of this eﬀective Hamiltonianapproach for molecules. The eﬀective Hamiltonian approach may prove quite useful formolecular systems for which electronic structure calculations are too expensive on existingquantum hardware.

References [1] P. M. Morse, “Diatomic Molecules According to the Wave Mechanics. 2. VibrationalLevels,” Phys. Rev. , 57 (1929). doi:10.1103/PhysRev.34.57[2] E. Witten, “Dynamical Breaking of Supersymmetry,” Nucl. Phys. B , 513 (1981).doi:10.1016/0550-3213(81)90006-7[3] F. Cooper, A. Khare and U. Sukhatme, “Supersymmetry and quantum mechanics,”Phys. Rept.251