Quantum Minesweeper 💣

Game rules

Goal: Defuse all mines by bringing every cell to the |0> state. If you measure a cell and observe |1>, you lose.

Overview: Each cell is a qubit (quantum bit): it may be a mine |1>, empty |0>, or in a superposition. Use measurements to reveal the expected number of mines in neighbouring cells, or apply gates to change the quantum states of the unrevealed cells.

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Controls

• Measure: collapses a cell to |0> (no mine) or |1> (mine). If you observe |1>, you lose. If you observe |0>, the cell becomes revealed and displays the expected number of mines in the neighbouring cells.

• Apply gate: manipulates the quantum state of a cell without revealing it. (See the help pages for a description of the different gates you can use.)

Levels

• Classical (Level 0)

  • Start here to refresh your memory. Here, mines behave just like in the traditional Minesweeper game.

• Quantum (Level 1)

  • Cells can be in a superposition state, meaning that there is a 0.5 probability of being occupied or unoccupied by a mine. If you measure a cell in a superposition, you have a 0.5 probability of losing.
  • There are two kinds of actions: measurements and gates.
  • If you measure a cell, you either lose (if you find a mine), or reveal the expected number of mines in neighbouring cells.
  • Since you will not know the quantum state of the unrevealed cells, you will need to apply gates to see how it changes the numbers in the neighbouring revealed cells.

• Quantum (Level 2+)

  • At higher levels, mines can be entangled.
  • Entanglement means that the state of two (or more) cells can be correlated. For example, a mine can be in a superposition of being in either one cell or another cell. So knowing the state of one tells you about the other.
  • This makes clearing the board trickier, since measuring one cell can change the state of another!
  • In these levels, you unlock two-qubit gates, which you use on two cells at a time. You will need to apply these gates to disentangle the entangled cells before defusing the mines.
  • Higher levels of entanglement (Level 3 and 4) means that more than two cells can be entangled at the start of the round.

• Sandbox

  • A playground where you can freely try out gates and see what they do to the board, without worrying about winning or losing.

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The best way to learn is to start simple (classical mines), and then experiment step-by-step with different gates in Sandbox mode. You'll build intuition about how quantum mines behave and how to defuse them.

Game Rules

Quantum Minesweeper is a quantum twist on the classic game of Minesweeper. Instead of fixed mines, the board is prepared in quantum states:
The player interacts with the board by measuring qubits or applying quantum gates, and must either identify or clear the “quantum mines.”

Options in the Advanced Setup

1. Board size

Choose the number of rows and columns.
The board contains \(N = R \times C\) qubits.

2. Mines & Entanglement Level

Set the number of mines \(B\) and how they are prepared.

• Classical mines: qubits initialized in the computational basis state \(|1\rangle\), acting like ordinary mines in the classical game.
• Product stabilizers: qubits prepared in independent stabilizer states, e.g. \(|+\rangle\), \(|i\rangle\).
• Entangled stabilizers: groups of qubits prepared in random entangled stabilizer states (e.g. Bell pairs, GHZ-type states).

The expected number of mines during play can be estimated as
$$ \langle \text{Mines} \rangle = \sum_i p_i, $$
where \(p_i = \tfrac{1}{2}(1 + \langle Z_i \rangle)\) is the probability that qubit \(i\) contains a mine.

The entanglement level controls the size of stabilizer groups used to place mines:

• Level 0 — purely classical mines.
• Level 1 — mines are independent single-qubit stabilizers.
• Level \(k\) — mines are prepared as random stabilizer states over groups of \(k\) qubits. At Level 2 may produce Bell pairs, Level 3 GHZ-type states, etc.

3. Win condition

• Identify: you win by correctly identifying all mines by measuring all the empy sites.
• Clear: you win by applying gates and measurements to drive the system back to the trivial product state \(|0\ldots 0\rangle\).
• Sandbox: free exploration — no win/lose condition.

4. Move set

Determines which quantum operations are allowed:

• Classic : Only measurement (M) and pinning (P).

• One-qubit (core set): Add \(X\), \(Y\), \(Z\), \(H\), \(S\) enabling the player to defuse the mines.

• One-qubit (complete set):
  Covers the full single-qubit Clifford group by adding \(S^\dagger\), \(\sqrt{X}\), \(\sqrt{X}^\dagger\), \(\sqrt{Y}\), \(\sqrt{Y}^\dagger\).

• Two-qubit:
  Full Clifford set: \(CX\), \(CY\), \(CZ\), \(SWAP\).
  Enables creation and manipulation of entangled mines.