assignment1_quantum_tictactoe

Assignment $|01\rangle$ ⚛ Quantum Tic-Tac-Toe game ⚛

Picture Demo (Click Here)

YouTube Demo: https://www.youtube.com/watch?v=U6_wSh_-EQc

I. How the Game is Played

In this game, 2 players X and O are represented by 2 qubit classical states, $|1\rangle$ and $|0\rangle$ respectively. Therefore, the Pauli-X gate is used to flip $|0\rangle$ to $|1\rangle$ for player X, while the Identity gate represents player O's unchanged state $|0\rangle$. Players can choose between making a classical move or quantum move, leading to a range of possibilities upon measurement:

1. Classical Move: Each player alternates turns, placing their marker X or O on 1 unoccupied cell.
2. SWAP Move: Apply the SWAP gate swap the states of 2 occupied cells without adding new markers.
3. Superposition: Apply the Hadamard (H) gate to 1 unoccupied cell (qubit), resulting in the $|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}$ state, which does not commit to X or O until the board is measured and collapsed.
4. Entanglement: Link 2 or 3 unoccupied cells, affecting outcomes such that the final state of one cell will affect the other. Players must also carefully choose between 4 Risk Levels (see below) based on their current position and potential future board states. Lower levels allow more controlled outcomes, while higher ones can gamble for a win or potentially aid the opponent.

The X? and O? markers will be used to indicate the cells are superposed/entangled, meaning their final values as X or O are not yet determined until a measurement collapses their states and capture the state with the highest occurrence. This can happen when:

• The board is full. If there are still several X?/O? cells, an automatic collapse will be performed to resolve all superpositions.
• Reaching a set of superposed/entangled cells forming a potential winning line.
• Manually triggered by the Collapse button press.

[NOTE]:

• If the board is full with no cells in superposition and also no winning lines, the game is Draw.
• If you want to make a new choice for Entanglement Risk Levels, just click the Entanglement button again.

II. Ensure Fairness and Demonstrate a grasp of Quantum Concepts

In the beginning, this game only has 3 options: Classical move, Superposition, and Entanglement with the standard Bell state $|\Phi^+\rangle$. However, this Entanglement approach has 2 disadvantages.

• Firstly, when players select Entanglement and pick 2 cells on the board, this will be unfair for their opponents as they will lose 1 move.

• Secondly, the Bell state $|\Phi^+\rangle$ represents a maximally entangled state where measurement outcomes are perfectly correlated ($|00\rangle$ or $|11\rangle$). This strategy can be risky as it could lead to either a winning line quicker or accidentally help the opponent. For example, if X players have a plan of entangling 2 consecutive cells, they can have:

  • Multiple X in a row, increasing their possibility of winning.
  • Multiple O in a row, putting them at risk of losing as their O opponents now have more consecutive cells than them.

Therefore, to overcome these:

• I introduce the SWAP Move, which can disrupt existing lines or defenses in the opponent's strategy, limiting players from overusing Entanglement to dominate the game unfairly.
• Moreover, I introduce 4 Entanglement Risk Levels to add strategic depth and require players to think critically about the consequences of their quantum moves:
  • By adding a Pauli-X gate before the CNOT's target qubit, I turned the Bell state $|\Phi^+\rangle$ into $|\Psi^+\rangle$, which also represents a maximally entangled state but with opposing outcomes, ensuring a 50/50 chance and reducing the above risk of $|\Phi^+\rangle$.
  • Similarly, I further employed the same approach for Triple Entanglement and turn the Standard |GHZ⟩ state into |GHZ_Xs⟩ by applying 2 additional Pauli-X before the CNOT chains of 2 targeted qubit components.

III. Entanglement Risk Levels

There are 2 types of Entanglements in this game associated with 4 corresponding Risk Levels:

• Pairwise Entanglement: A player can entangle 2 empty cells, meaning the state of 1 cell depends on the state of the other, and their final states (either Xor O) aren't determined until a measurement is performed.
• Triple Entanglement: Advanced moves allow entangling 3 cells simultaneously.

Risk Level (Select via the "Dropdown")	Quantum Gates	Example Circuit	Effect on Measurement and Collapse
Lv1. Lowest Risk $|\Psi^+\rangle = \frac{|01\rangle + |10\rangle}{\sqrt{2}}$ (Pairwise Entangle)	Hadamard (H), Pauli-X (X), CNOT (CX)	┌───┐ q_0: ┤ H ├──■── ├───┤┌─┴─┐ q_1: ┤ X ├┤ X ├ └───┘└───┘	This Bell state results in anti-correlated and truly random outcomes upon collapse. When 1 qubit collapses to X, the other must collapse to O, and vice versa.
Lv2. Lower Risk $|GHZ_{Xs}\rangle = \frac{|010\rangle + |101\rangle}{\sqrt{2}}$ (Triple Entangle)	Hadamard (H), 2 Pauli-X (X), 2 CNOT (CX)	┌───┐ q_0: ┤ H ├──■─────── ├───┤┌─┴─┐ q_1: ┤ X ├┤ X ├──■── ├───┤└───┘┌─┴─┐ q_2: ┤ X ├─────┤ X ├ └───┘ └───┘	Similar randomness but for 3 squares, formed by modifying the Standard |GHZ⟩, leading to combinations where the outcome isn't uniformly all 3 Xs or all Os, but mixed.
Lv3. Moderate Risk $|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}$ (Pairwise Entangle)	Hadamard (H), CNOT (CX)	┌───┐ q_0: ┤ H ├──■── └───┘┌─┴─┐ q_1: ─────┤ X ├ └───┘	This Bell state yields the same outcome for both selected cells/qubits. Both qubits collapse to the same value upon measurement, either XX or OO.
Lv4. High Risk $|GHZ\rangle = \frac{|000\rangle + |111\rangle}{\sqrt{2}}$ (Triple Entangle)	Hadamard (H), 2 CNOT (CX)	┌───┐ q_0: ┤ H ├──■─────── └───┘┌─┴─┐ q_1: ─────┤ X ├──■── └───┘┌─┴─┐ q_2: ──────────┤ X ├ └───┘	This Standard |GHZ⟩ state can strategically benefit the player but also at the risk of benefiting the opponent just as much. All 3 qubits collapse to the same value, either all XXX or all OOO.

IV. Installation and Usage

1. Install Qiskit and PyLaTeXEnc

pip install qiskit --quiet
pip install qiskit-aer --quiet
pip install pylatexenc --quiet

2. Play the game

from qiskit_aer import AerSimulator
from gui import QuantumT3GUI
game = QuantumT3GUI(size=3, simulator=AerSimulator())

👉 Check this quantum_tic_tac_toe.ipynb for a demo. You should open it in Colab, the notebook viewer within GitHub cannot render the game's widgets.

V. Future Improvements

• Limit the Entanglement Risk Levels based on the board size. For example, 3x3 board can only use PAIRWISE entanglement (Level 1 & 3). Because if they use 2 or 4, they can win or lose the game in 1 move.
• Count the number of winning lines for each player as a score to demonstrate how confident the winner is or to determine the winner if the game is a draw.
• Apply phase shift gates like S or T gates before making a move. This could affect the probability amplitudes of the states, creating interference patterns in probabilities.
• Implement the Undo and Redo functionalities for the game. But this is a bit complex for cases like entanglement or after collapsing.
• Develop an AI opponent that adaptively uses quantum strategies, learning from the player's moves.