  Quantum chemistry is a near-term application for quantum computers. This
application may be facilitated by variational quantum-classical algorithms
(VQCAs), although a concern for VQCAs is the large number of measurements
needed for convergence, especially for chemical accuracy. Here we introduce a
strategy for reducing the number of measurements (i.e., shots) by randomly
sampling operators $h_i$ from the overall Hamiltonian $H = \sum_i c_i h_i$. In
particular, we employ weighted sampling, which is important when the $c_i$'s
are highly non-uniform, as is typical in chemistry. We integrate this strategy
with an adaptive optimizer developed recently by our group to construct an
improved optimizer called Rosalin (Random Operator Sampling for Adaptive
Learning with Individual Number of shots). Rosalin implements stochastic
gradient descent while adapting the shot noise for each partial derivative and
randomly assigning the shots amongst the $h_i$ according to a weighted
distribution. We implement this and other optimizers to find the ground states
of molecules H$_2$, LiH, and BeH$_2$, without and with quantum hardware noise,
and Rosalin outperforms other optimizers in most cases.
