Variational quantum algorithm for molecular geometry optimization

Classical algorithms for predicting the equilibrium geometry of strongly correlated molecules require expensive wave function methods that become impractical already for few-atom systems. In this work, we introduce a variational quantum algorithm for finding the most stable structure of a molecule by explicitly considering the parametric dependence of the electronic Hamiltonian on the nuclear coordinates. The equilibrium geometry of the molecule is obtained by minimizing a more general cost function that depends on both the quantum circuit and the Hamiltonian parameters, which are simultaneously optimized at each step. The algorithm is applied to find the equilibrium geometries of the ${\mathrm{H}}_2,$ ${\mathrm{H}}_3{}^{+},$ ${\mathrm{BeH}}_2,$ and ${\mathrm{H}}_2\mathrm{O}$ molecules. The quantum circuits used to prepare the electronic ground state for each molecule were designed using an adaptive algorithm where excitation gates in the form of Givens rotations are selected according to the norm of their gradient. All quantum simulations are performed using the PennyLane library for quantum differentiable programming. The optimized geometrical parameters for the simulated molecules show an excellent agreement with their counterparts computed using classical quantum chemistry methods.

Figure 1

Workflow of the variational quantum algorithm to find the equilibrium geometry of a molecule. The circuit parameters $\theta$ and the nuclear coordinates $x$ entering the Hamiltonian are jointly optimized. The iterative optimization is performed until a loop condition is satisfied. This could be a maximum number of iterations or a given convergence tolerance for the energy or the maximum component of the nuclear gradient.

Figure 2

Atomic structures and geometrical parameters of the simulated molecules. The parameters $d$ and $\varphi$ denote the bond length and angle, respectively. The color code for the elements is white for hydrogen, green for beryllium, and red for oxygen.

Figure 3

Variational circuits to prepare the electronic ground state of the ${\mathrm{H}}_3{}^{+}$ molecule including (a) all single- and double-excitation operations preserving the total-spin projection of the HF state, (b) only the relevant excitation operations selected by the adaptive method. The squares indicate the qubits the operations act on.

Figure 4

(a) Convergence of the ground-state energy $E_{\mathrm{VQE}}$ and (b) the bond length $d$ for the ${\mathrm{H}}_3{}^{+}$ molecule as the circuit parameters and the nuclear coordinates are jointly optimized by the variational quantum algorithm. Values of the energy are reported relative to the analogous FCI value.