Casting Rubik’s Group into a Unitary Representation for Reinforcement Learning

Abstract

Rubik’s Cube is one of the most famous combinatorial puzzles involving nearly 4.3 × 10¹⁹ possible configurations. However, only a single configuration matches the solved one. Its mathematical description is expressed by the Rubik’s group, whose elements define how its layers rotate. We develop a unitary representation of the Rubik’s group and a quantum formalism to describe the Cube based on its geometrical constraints. Using single particle quantum states, we describe the cubies as bosons for corners and fermions for edges. By introducing a set of four Ising-like Hamiltonians, we managed to set the solved configuration of the Cube as the global ground state for all the Hamiltonians. To reach the ground state of all the Hamiltonian operators, we made use of a Deep Reinforcement Learning algorithm based on a Hamiltonian reward. The Rubik’s Cube is successfully solved through four phases, each phase driven by a corresponding Hamiltonian reward based on its energy spectrum. We call our algorithm QUBE, as it employs quantum mechanics to tackle the combinatorial problem of solving the Rubik’s Cube. Embedding combinatorial problems into the quantum mechanics formalism suggests new possible algorithms and future implementations on quantum hardware.