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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.

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The board and pieces

  • Quantum chess is played on a regular 8x8 chess board with standard pieces.
  • Pieces have rings around them, filled in with colour. These rings show the probability that the piece is in that square.

Modifications to existing chess rules

  • A player is not required to move their king out of check and the game concludes when there is a 100% probability that one of the kings has been taken. As a result there is no checkmate.
  • All promotions result in a queen as of 10/8/17, since underpromoting has not been implemented into the game.

Movement

  • All pieces move as usual in normal circumstances.
  • On any given turn, the player can make one quantum move instead of a normal move.

Quantum moves

  • Quantum moves are notated with the ^ symbol. (e.g: N^d4)
  • When quantum moving a piece, it can make up to two normal moves.
  • The probability of the piece moving is equal to the probability that the piece has not moved, resulting in a superposition of multiple board states.
  • A quantum move cannot be used to take a piece.
  • Pawns cannot make quantum moves.

Quantum move of Knight

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.

Taking superposed pieces.

  • If a piece "takes" one of the copies of another piece in superposition and it is not certain that the piece was available to take it, a measurement occurs. A measured piece will collapse onto one square using a random selection. The probability of it collapsing onto a particular square is the same as the probability that is was moved there. All pieces that are entangled with the measured piece will also be measured and may have their outcome influenced by the measurement of the piece(s) that they are entangled with.

Measurement resulting from quantum capture.

  • A pawn can be put into superposition if it takes a piece which is in superposition. If the piece was there, the pawn could take it. If not, moving diagonally was an illegal move. The probability of the pawn being on the new square is the same as the probability that a piece was there to be taken.

Pawn superimposed by taking a piece in superposition.

  • A pawn can also be superimposed by moving two spaces forward, through a square which may contain a piece. This results in entanglement.

Pawn becoming entangled with a bishop.

Entanglement

  • Entanglement occurs when a piece attempts to move through an instance of a piece in superposition. The probability of the moved piece existing in its new square is the same as the probability that the obstruction was not there to block it.
  • If a piece is entangled with another piece and is measured, both pieces are affected by the results of the measurement accordingly.

Special circumstances

The following "rules" are combinations of rules from above, however they are worth mentioning as useful tactics:


Quantum Trading

  • It is possible to trade probabilities of pieces, leading to complex trading. For example, it is possible to trade knights in a way which leaves one player with a 2/3 chance of having a knight, and the other with a 1/3 chance of having a knight. The first player is more likely to have benefited from this trade, however it is uncertain until measured.

Schrödinger's King

  • If the king is in superposition and one of the instances is taken, the probability that that player has lost is equal to the probability that the king was in that square. Since it is not certain that they have lost, play continues.
  • This means that it is possible for a player to win a game with a probability that they have already lost by fully capturing the opposing king.

Using quantum moves to escape checkmate.

Ghost capture

  • This is a result of a pawn taking a piece in superposition. If there is one other state of the taken piece still on the board and a piece attacking it, ghost capture can take place. If the attacking piece must move through the square where the pawn would be if it hadn't moved in order to attack the superimposed piece, two outcomes can occur:
    1. The measurement confirms that the piece had not moved, so an attack is impossible and the opposing player gains "half a piece".
    2. The measurement shows that the piece has moved and has already been taken, so the opposing player loses "half a piece".

Attempted ghost capture of opposing queen using rook.

Quantum Castling

  • If there is a probability that a piece is between the king and rook, the probability of castling is equal to the probability that the piece was not there. A measurement may take place if two pieces end up on the same square.
  • If there is a probability that the rook may have moved, castling can occur with the probability that the rook is available to castle. The king and rook are now entangled since the castling move depends on whether the rook was on the correct square.

Double Castling

  • After a quantum castle, the king can castle the other way if the rook is available, with the probability of castling being the probability that the king is there.
    • In this example, both sides have castled both ways:

Double castled mayhem.

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  • With quantum trading, can you only trade the probabilities of the same piece? E.g can you only trade the probability of knight for the the probability of the opponent's knight, or can you trade probabilities between knights and pawns, too? – Geza Kerecsenyi Aug 11 '19 at 14:30
  • @GezaKerecsenyi it works with any pieces in superposition, including the king – Aric Aug 11 '19 at 16:46
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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.

Non-rules, and flattening for display purposes

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.

Some rules questions to which I personally don't know answers

Feel free to leave comments (or even edit this answer directly), and I'll try to keep up.

  • Can you move into check? Since Quantum Chess doesn't really do "check," does that mean that it's legal to move your king into check — or move another piece that discovers a check on your own king? Or are such moves considered "illegal, impossible, or nonsensical" (to use my own phrase) and therefore count as a "pass"? For example, suppose the play is 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?

  • What exactly is considered the "same" move on two different boards? For example, suppose the play is 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?

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