Improving the Qubit-Efficiency of Quantum Algorithms for the Electronic Structure Problem Using Orbital Optimization

Solving the time-independent Schrödinger equation for the electronic structure Hamiltonian in quantum chemistry is hoped to be one problem where quantum computers may provide an early advantage over classical computers. The basic intuition behind this is that quantum computers are able to prepare states with an exponentially large number of probability amplitudes with a linear number of qubits, whereas classical methods must introduce approximations and heuristics to avoid the need to store and perform operations on such exponentially large states. We address two challenges that occur within the context of developing algorithms for solving the electronic structure problem on quantum computers: 1. developing methods which not only find the ground state, but also excited states and; 2. contending with the basis set truncation error which requires the use of large numbers of qubits using conventional methods. We first develop the quantum Orbital Minimization Method (qOMM) and show through numerical simulations using Qiskit that it is able to converge much more quickly to a set of low-lying excited states than another method, the Subspace Search Variational Quantum Eigensolver (SSVQE) which has appeared in the literature in recent years. We then develop the optimal orbital variational quantum eigensolver (OptOrbVQE) algorithm and numerically simulate it using Qiskit to show that it can often achieves lower basis set truncation error in the ground state energy than methods using larger, conventional basis sets. We then generalize this method to find excited states in optimized basis sets and demonstrate analogous results to the ground state case through numerical simulations in Qiskit.