A Quantum Rubik's Cube and Unexpected Qudits

Many proposed quantum games simply sample randomness from some quantum process to make a classical game quantum. We propose a simple quantum game that is a variation of a Rubik’s cube known as a permutation puzzle. By requiring the pieces of the game to be either fermions or bosons we see the emergence of a rich structure that has no classical equivalent. In fact, with just a simple set of legal moves, we see that even the smallest puzzles allow an infinite number of board positions. We also show a method to easily encode qudits into the allowed states of the games and find a universal set of gates that are all described by legal moves in the game.