K-Spin Hamiltonian for Quantum-Resolvable Markov Decision Processes

Abstract

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 arguing that the variational minimization of our Hamiltonian is approximately equivalent to the Bellman optimality condition for a prevalent class of environments 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.