SCIENCE CHINA Information Sciences, Volume 64 , Issue 2 : 122501(2021) https://doi.org/10.1007/s11432-020-3038-x

## Board games for quantum computers

Biao WU 1,2,3,*,
• AcceptedAug 10, 2020
• PublishedNov 9, 2020
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### Acknowledgment

This work was supported by National Key RD Program of China (Grants Nos. 2017YFA0303302, 2018YFA0305602), National Natural Science Foundation of China (Grant No. 11921005), and Shanghai Municipal Science and Technology Major Project (Grant No. 2019SHZDZX01). We dedicate this work to Professor P. W. Anderson, who was a very good amateur Weiqi (or Go) player and would probably view a quantum game of Weiqi as a spin-1 system on a square lattice.

### References

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• Figure 1

Scalable board. The board is labelled horizontally by Roman letters and vertically by Arabic numbers so that each point (or intersection) is specified by coordinates. For example, the coordinates of the central point are $(E,5)$. The game is played by two players who take turn to place black and white stones on the points of the board. This board has a grid of 9$\times$9. Even though the usual boards used in FIR and Weiqi have grids of 15$\times$15 and 19$\times$19, respectively, both games can be played on boards of other sizes.

• Figure 2

Two examples of capture in the game of Weiqi. After one more black stone is placed on the board, (a) one white stone is captured and removed from the board; (b) three white stones are captured and removed from the board.

• Figure 3

Superposition move. With $S^{b+}_{G3,G4}$, Bob places one black stone on the points $(G,3)$ and $(G,4)$ simultaneously with equal probability.

• Figure 4

Entangled move: one white stone is placed on two points simultaneously to counter the superposition move of the black stone in Figure 3.

• Figure 7

Classical computer Alice plays a quantum board game with another classical computer Bob. In this case, the quantum state of board is stored classically on both two classical computers. Alice and Bob communicate classically between them about the moves.

• Figure 8

One possible way of capture a stone in the quantum game of Weiqi: the captured stone is removed from all the games playing in parallel. The numbers on the stones mark the order that the stones are placed on the board: the black stone with 1 is the first stone placed on the board, the white stone with 2 is the second placed, etc.

• Figure 9

Another possible way of capture a stone in the quantum game of Weiqi: the stone is removed from the game where it is captured and stays in the other game. The numbers on the stones mark the order that the stones are placed on the board: the black stone with 1 is the first stone placed on the board, the white stone with 2 is the second placed, etc.

• Figure 10

A legitimate way of capture a stone in the quantum game of Weiqi: the stone at a certain position is removed from the game when it is captured and a stone of the same color appears at the same position in the other game. The numbers on the stones mark the order that the stones are placed on the board: the black stone with 1 is the first stone placed on the board, the white stone with 2 is the second placed, etc.

• Figure 11

Ko rule for unending cycle of captures. According to this rule, after the capture of the marked white stone (upper left), Alice is not allowed to place a white stone on position B to capture the marked black stone. She has to wait for at least one move.

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