Solving Rubik's cube via quantum mechanics and deep reinforcement learning

Rubik's cube is one of the most famous combinatorial puzzles involving nearly 4.3 x 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 described by 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 symmetries which 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.

Automated summary

This study explores the Rubik's cube through a quantum formalism, representing it as a unitary representation of the Rubik's group, and revealing the behavior of cubies as bosons and fermions. By constructing Hamiltonian operators, it shows that the symmetries of the cube are preserved when the Hamiltonian is in its ground state. By utilizing deep reinforcement learning algorithm and Hamiltonian rewards, it successfully solves the Rubik's cube configurations.