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The game Quantum Chess, as seen in the video where Paul Rudd defeats Hawking, has no "set of rules" written down. You either have to do the in-game tutorial, watch a video on the game and pick them up.
So what are the rules of Quantum Chess?
This refers to the game (currently only available on steam) created by Chris Cantwell.
The board and pieces
Modifications to existing chess rules
Movement
Quantum moves
Superposition
When any quantum move is made, that piece exists in multiple places simultaneously. This is represented by multiple copies of the piece displayed, with appropriate meters to show the probability of finding the piece in that square. There is only one piece, but it can exist in one of multiple places.
If a piece "takes" one of the copies of another piece in superposition and has 100% probability of being available to take it, the piece is taken if it was in that square. Since there is a probability of the piece being in other squares, those copies of the piece remain, however there is now a chance that the piece has already been taken and is no longer on the board.
Entanglement
Special circumstances
The following "rules" are combinations of rules from above, however they are worth mentioning as useful tactics:
Quantum Trading
Schrödinger's King
Ghost capture
Quantum Castling
Double Castling
Here's a shorter answer than Aric's, based on my reading of Chris Cantwell's blog post "Quantum Chess" (February 2016). That post ends: "In the next post I’ll go into entanglement and a bit more on the quantum move!" But it seems that the author never actually came back to write any followup posts (possibly because there's not much more to say about the game ;)).
TLDR, I'm not aware of any "quantum" aspects to the game. It's more like "5D Chess with Multiverse Time Travel," except without the time-travel part. Here's the rules:
The game state consists of an arbitrarily large collection of ordinary chess positions. It may help to think of this collection of boards as shares of stock in a company. Each time you make a move, you can either:
Make an ordinary chess move. This modifies every active board in parallel. On boards where the chosen move is legal, the move happens. On boards where the chosen move would be illegal, impossible, or nonsensical, nothing happens — it's as if you said "pass" on those boards. ("Pass" is not legal in ordinary chess, but in this case, it's what happens.) On boards where the chosen move captures the opposing king, the board is deleted from the collection. (See What about check? below.)
Make a "quantum" chess move. This performs a "2-for-1 stock split" on the collection of active boards. Each old board gives rise to two new boards: one new board where the move takes place á là "Make an ordinary chess move" above, and one new board where it's as if you'd said "pass."
What about check? In short, Quantum Chess does not do "check." Kings are not forced to move out of check.
When the collection of active boards becomes empty, the game ends. Whoever made the last move (the move that captured the king on all remaining boards simultaneously) is declared the winner.
All the cute names in Aric's answer ("Schrodinger's King," "superposition," "entanglement," "double castling," etc) is just flavor text on top of these basic rules.
The Quantum Chess app displays the current game state by "flattening" the collection of board positions into numerical probabilities, square by square. For example, if on 47% of active boards square d7 is occupied by a black queen, and on the other 53% d7 is empty, then the app will display on d7 a black queen icon surrounded by a colored ring that is 47% full. If it's got a black queen 47% of the time and a white pawn 10% of the time, the app will display both icons together in the space with appropriately colored rings. However, this is just a display gimmick! There is no sense in which "both pieces" occupy the space, and they cannot in any way "interact" with each other.
Feel free to leave comments (or even edit this answer directly), and I'll try to keep up.
1.c3 e5 2.d4 [Bb4], where [Bb4] indicates a "quantum move." Then the position before 3.c4 is:
There are two boards in the active set: one with the bishop on b4 and one with it still on f8. White normal-moves their pawn to c4. In the board where the bishop is still on f8, this is fine. In the board where the bishop is on b4, this move would leave White in check. Is the game state after this move "pawn 100% at c4," or "pawn 50% at c3 and 50% at c4"?
Can you castle through check? A slight variation on the former question.
Are there any equivalents of the 50-move or threefold repetition rules?
What happens if the chosen ordinary move is illegal on all active boards? For example, suppose the play is 1.[d3] e5 2.Nd2 f6 3.Nc4 d6 4.e3 b5, where [d3] indicates a "quantum move." Then the position before 5.Bxb5 is:
From the "flattened" display representation, you might think that the bishop stands a "25% chance" of making it to b5 for the capture; but in fact the knight and pawn are "entangled" (in Cantwell's terminology) — every active board contains either a knight blocking c4 or a pawn blocking d3. So, is Bxb5 simply not-allowed-as-a-move and White must play something else? or is Bxb5 permitted and considered tantamount to a "pass"?
Is the answer any different if White claims they're making [Bxb5] as a quantum move?
1.e3 e5 2.[Be2] Nf6 3.Qe2 Nh5. Then the position before 4.e2xh5 is:
Can White make an ordinary move like "I move my piece on e2 to capture on h5"? Or must White specify exactly which piece he's trying to move — e.g. "I move my queen from e2 to capture on h5" — and on boards where that exact piece isn't on e2, nothing happens?
What if the piece being moved is definitely a knight, but it's not clear which knight? What if the move is definitely a capture, but the identity of the captured piece is in question? What if the move is "Queen to h5" but it's not clear whether there's a (capturable) piece on h5 or not? (I.e., is Qxh5 considered a different move from Qh5?)
What if the move is an ordinary Qxh5, and there's definitely a black knight at h5, but on some boards there's also a black bishop at g4? On those latter boards, is it as if White said "pass" or is it as if White said Qxg4?
What if the move is e5xf6, always the same white pawn capturing the same black pawn, but on some boards it's en passant and on others it's not?
Its best to discuss using isolated pieces and see how probabilities will get associated. E.g. if a N is on D4. now if it is in Quantum state, it can be on D4 and any of the 8 allowed squares. So each of these squares gets probability of 1/9.
When you move a piece, already a measurement has been made.
Double moves don't make any sense. E.g. if a N is on b1 at original move, and in quantum next turn, it can atmost reach a3, c3, d2. it can't reach d5. That would be illegal in chess, its like making two moves at a time. Next turn, yes, it can be there. So original: N@b1 :-100% Prob 2nd turn :- N@b1,@c3,@a3,@d2 :- 25 % probability each. 3rd turn :- N@b1,a3,b5,c3,d5,e4,e2,b3,c4,f3,f1 with equal probability. one has to consider possibility that it went to c3 and returned to b1 also.
