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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.