**QuhacMan: A Quantum Two-player PacMan Game**

by Tareq El Dandachi, Matthew Baldwin, Qichen Song, Haozhe Wang

**Highlights of the Quacman:**

- In our game, the result is quantum. No one knows the results until the measurement.
- Player can increase their probability of winning, but nothing is guaranteed.
- During the game, two players are building a quantum circuit together. They try interfering with the result of the entangled qubits to something they desire.
- In the game, we applied Quantum Random Number Generator to find the type and place of gates. The speeds of Quhacmans change with entangled two qubits circuit.
- The rule and strategy are quantum. We use Bloch sphere as a win indicator.

*Screenshot of the game*

**Introduction**

The development of quantum computing has offered revolutionary scope for many traditional areas. Game theory, a theory to study decision making in conflict situations, has been transferred into quantum version[1]. The quantum game theory is different from the traditional one in three different ways[2]:

- The initial status is entangled;
- The initial state is superposed;
- Player strategy is quantum.

These features indicate that in quantum scenario, the definition of win and lose in a game changes fundamentally.

In this project, we propose a quantum game, which is a modified version of the multi-player PacMan game. The game is a validation of quantum game theory and test quantum win strategies.

**Basic principles**

The two-qubit state is initialized at entangled states.

Each player will collect certain gates during the game (either acting on both qubits or a single qubit). Effectively, the prefactor alpha and beta change dynamically during the game. There are measurement gates available for the player to pick up as well. Upon one-shot measurement, the quantum state collapses to player one getting |0> and player two getting |1> or the opposite. Whoever gets |0> is the winner!

The evaluation of the quantum circuit generating from the gaming processes are conducted by qiskit package.

**Elements and rules**

Two players in the game are gambling the final output of a quantum circuit with two entangled qubits. One can gain an advantage to win by eating beans to modify the Bloch sphere of their qubit. After measurement, the entangled qubits collapse to classical states, determines the game result.

Below is the explanation of essential elements:

- Maze: The map is the same as the conventional PacMan game. Players can move at the route divided by walls.
- Pac-man: Two players are playing as PacMan in the game. The PacMan can move, eat beans/gates and avoid being caught by ghosts.
- Ghost: Ghost is automatically generated by the game to catch the PacMans in the game. The ghosts are moving gates. Once the ghost encountering one PacMan, Rx(-pi/9) gate is applied on one’s qubit, and Rx(pi/9) gate is applied on another’s qubit.
- Beans: Players will eat the beans in the route they move. In this game, one bean means a rotation gate of Rx(pi/36) in PacMan’s qubit.
- Gates: Gates are randomly added in the beans to change the final quantum circuit. We used a separate quantum circuit to generate random numbers to assign gates in the maze.
- S gate: a separate 2 qubit quantum circuit to randomly generate |00> and |11> states. If |00> is measured, the hit PacMan speeds up, the other PacMan slows down. If |11> is measured, the hit PacMan slows, the other PacMan speed up.
- Swap gate: The qubits of two players swap.
- H gate: adding Hadamard gate in the circuit.
- Measure gate: Once PacMan hits measure gate, the game ends.

Basic Rules:

- The game starts with initial maze and two players at the same place. Two players separately control two PacMan in the maze.
- PacMan can move, eat beans, eat gates, hit ghosts and eat measure gate.
- The two Bloch spheres on the side of maze represent the wave functions of two players.
- Player 1 uses WSAD and player 2 uses up down left right arrow key to control their PACMan

Advanced Strategies:

- Player should keep the pointer close to the win state to have the largest winning probability. Eating more beans after pointer reach the highest position will lower it down.
- Try to lower down the pointer of your opponent
- Finish/Measure the game when you have advantage (But moving there may eat more beans, worth it?)

**Demonstrations:**

The GitHub repository link is https://github.com/tareqdandachi/quhackman.

[1]Piotrowski, Edward W., and Jan Sładkowski. “An invitation to quantum game theory.” *International Journal of Theoretical Physics* 42.5 (2003): 1089–1099.

[2]Quantum Game Theory, Wikipedia. https://en.wikipedia.org/wiki/Quantum_game_theory