Efficient quantum circuits for quantum computational chemistry

Abstract

The VQE (variational quantum eigensolver) is a hybrid classical-quantum algorithm that can determine the energy eigenvalues of molecules by solving the electronic structure problem. Compared to other purely quantum algorithms, the VQE requires shallower quantum circuits and is more noise tolerant, which makes it a potential application for early NISQ (noisy intermediate-scale quantum) computers. The VQE works by minimizing the expectation value of a molecular Hamiltonian with respect to a parametrized ansatz state. The majority of ansatz states, so far considered by the scientific community, like the UCCSD (unitary coupled-cluster single and double), are derived from classical algorithms, and correspond to a combination of electron (fermionic) excitations applied to an initial guess ansatz state. Therefore, constructing efficient circuits that perform the action of fermionic excitations is critical for the practical realization of the VQE. Here, we introduce the concept of a "qubit excitation", which compared to a fermionic excitation, does not account for fermionic anti-commutation relations. We then construct circuits, optimized in terms of $CNOT$ gates, that perform such single and double qubit excitations. Finally, we modify the functionality of these circuits to account for fermionic anti-commutation relations. In this way we obtain circuits for single and double fermionic excitations that reduce the number of $CNOT$ gates used, by factors of $2$ and $8$, respectively, compared to circuits constructed with the standard stair-case $CNOT$ structures.