  The Markov decision process is the mathematical formalization underlying the
modern field of reinforcement learning when transition and reward functions are
unknown. We derive a pseudo-Boolean cost function that is equivalent to a
K-spin Hamiltonian representation of the discrete, finite, discounted Markov
decision process with infinite horizon. This K-spin Hamiltonian furnishes a
starting point from which to solve for an optimal policy using heuristic
quantum algorithms such as adiabatic quantum annealing and the quantum
approximate optimization algorithm on near-term quantum hardware. In proving
that the variational minimization of our Hamiltonian is equivalent to the
Bellman optimality condition we establish an interesting analogy with classical
field theory. Along with proof-of-concept calculations to corroborate our
formulation by simulated and quantum annealing against classical Q-Learning, we
analyze the scaling of physical resources required to solve our Hamiltonian on
quantum hardware.
