Solving Rubik's Cube via Quantum Mechanics and Deep Reinforcement Learning

TL;DR

This paper introduces a novel approach to solving the Rubik's Cube by combining quantum mechanics formalism with deep reinforcement learning, representing cube states as quantum particles and using Hamiltonian-based rewards.

Contribution

It develops a quantum formalism for the Rubik's Cube and integrates it with deep reinforcement learning to solve the puzzle through Hamiltonian-based rewards.

Findings

Successfully modeled cube states using quantum particles

Solved the cube in four phases using Hamiltonian rewards

Proposed a framework for future quantum algorithms

Abstract

Rubik's Cube is one of the most famous combinatorial puzzles involving nearly $4.3 \times 10^{19}$ possible configurations. Its mathematical description is expressed by the Rubik's group, whose elements define how its layers rotate. We develop a unitary representation of such group and a quantum formalism to describe the Cube from its geometrical constraints. Cubies are describedby single particle states which turn out to behave like bosons for corners and fermions for edges, respectively. When in its solved configuration, the Cube, as a geometrical object, shows symmetrieswhich are broken when driven away from this configuration. For each of such symmetries, we build a Hamiltonian operator. When a Hamiltonian lies in its ground state, the respective symmetry of the Cube is preserved. When all such symmetries are preserved, the configuration of the Cube matches the solution of the game. To reach the ground state of all the Hamiltonian operators, we make use of a Deep Reinforcement Learning algorithm based on a Hamiltonian reward. The Cube is solved in four phases, all based on a respective Hamiltonian reward based on its spectrum, inspired by the Ising model. Embedding combinatorial problems into the quantum mechanics formalism suggests new possible algorithms and future implementations on quantum hardware.