  Quantum simulation of chemical systems is one of the most promising near-term
applications of quantum computers. The variational quantum eigensolver, a
leading algorithm for molecular simulations on quantum hardware, has a serious
limitation in that it typically relies on a pre-selected wavefunction ansatz
that results in approximate wavefunctions and energies. Here we present an
arbitrarily accurate variational algorithm that instead of fixing an ansatz
upfront, this algorithm grows it systematically one operator at a time in a way
dictated by the molecule being simulated. This generates an ansatz with a small
number of parameters, leading to shallow-depth circuits. We present numerical
simulations, including for a prototypical strongly correlated molecule, which
show that our algorithm performs much better than a unitary coupled cluster
approach, in terms of both circuit depth and chemical accuracy. Our results
highlight the potential of our adaptive algorithm for exact simulations with
present-day and near-term quantum hardware.
