The basic rules seems simple enough: a quantum move is an equal superposition of one piece making 2 moves in a row and no move at all (no mention of their relative phase here). Therefore the expectation value for number of moves is still one, equal to the classical chess move. However, yours truly missed the subtler rules of entanglement in quantum chess upon initial viewing, especially how it was that [spoiler alert!] Paul Rudd eventually overcame Stephen Hawking. Interest somewhat piqued, my cursory Google search turned up unrelated versions of the game, thereby snuffing out said interest.
So I was pleased to come across this blog post that begins to explain the rules of quantum chess in detail, as well as a Kickstarter to fund a computerized creation of the game (I await the rules regarding entanglement). Apparently there’s this physicist, Spiros Michalakis, who keeps introducing quantum rules in other games, starting with MineCraft, and now he’s working with the Kickstarteree on the quantum version of chess (just shy of $3K with 10 days left to support it!).