A quantum annealer exploits quantum effects to solve a particular type of optimization problem. The advantage of this specialized hardware is that it effectively considers all possible solutions in parallel, thereby potentially outperforming classical computing systems. However, despite quantum annealers having recently become commercially available, there are relatively few high-level programming models that target these devices. In this article, we show how to compile a subset of Prolog enhanced with support for constraint logic programming into a 2-local Ising-model Hamiltonian suitable for execution on a quantum annealer. In particular, we describe the series of transformations one can apply to convert constraint logic programs expressed in Prolog into an executable form that bears virtually no resemblance to a classical machine model yet that evaluates the specified constraints in a fully parallel manner. We evaluate our efforts on a 1095-qubit D-Wave 2X quantum annealer and describe the approach's associated capabilities and shortcomings. Under consideration in Theory and Practice of Logic Programming (TPLP).