  The variational quantum eigensolver (VQE) is currently the flagship algorithm
for solving electronic structure problems on near-term quantum computers. This
hybrid quantum/classical algorithm involves implementing a sequence of
parameterized gates on quantum hardware to generate a target quantum state, and
then measuring the expectation value of the molecular Hamiltonian. Due to
finite coherence times and frequent gate errors, the number of gates that can
be implemented remains limited on current quantum devices, preventing accurate
applications to systems with significant entanglement, such as strongly
correlated molecules. In this work, we propose an alternative algorithm (which
we refer to as ctrl-VQE) where the quantum circuit used for state preparation
is removed entirely and replaced by a quantum control routine which
variationally shapes a pulse to drive the initial Hartree-Fock state to the
full CI target state. As with VQE, the objective function optimized is the
expectation value of the qubit-mapped molecular Hamiltonian. However, by
removing the quantum circuit, the coherence times required for state
preparation can be drastically reduced by directly optimizing the pulses. We
demonstrate the potential of this method numerically by directly optimizing
pulse shapes which accurately model the dissociation curves of the hydrogen
molecule (covalent bond) and helium hydride ion (ionic bond), and we compute
the single point energy for LiH with four transmons.
