  Unitary coupled cluster (UCC), originally developed as a variational
alternative to the popular traditional coupled cluster method, has seen a
resurgence as a functional form for use on quantum computers. However, the
number of excitors present in the ansatz often presents a barrier to
implementation on quantum computers. Given the natural sparsity of
wavefunctions obtained from Quantum Monte Carlo methods, we consider here a
stochastic solution to the UCC problem. Using the Coupled Cluster Monte Carlo
framework, we develop cluster selection schemes that capture the structure of
the UCC wavefunction, as well as its Trotterized approximation, and use these
to solve the corresponding projected equations. Due to the fast convergence of
the equations with order in the cluster expansion, this approach scales
polynomially with the size of the system. Unlike traditional UCC
implementations, our approach naturally produces a non-variational estimator
for the energy in the form of the projected energy. For UCCSD in small systems,
we find this agrees well with the expectation value of the energy and, in the
case of two electrons, with full configuration interaction results. For the
larger N$_2$ system, the two estimators diverge, with the projected energy
approaching the coupled cluster result, while the expectation value is close to
results from traditional UCCSD.
