Bloch Sphere Representation

A single qubit state can be represented as |ψ⟩ = cos(θ/2)|0⟩ + e^iφsin(θ/2)|1⟩, where θ ∈ [0, π] and φ ∈ [0, 2π]. The Bloch sphere visualizes this: the north pole (+Z) corresponds to |0⟩, the south pole (−Z) to |1⟩, and any point on the surface represents a pure qubit state.

1-Qubit Gates

• X, Y, Z (Pauli): 180° rotations. X maps |0⟩ ↔ |1⟩.
• H (Hadamard): Creates superposition: |0⟩ → (|0⟩+|1⟩)/√2.
• S, T: Phase gates (90°, 45° about Z).
• R_x(γ), R_y(γ), R_z(γ): Parametric rotation gates; γ sets the rotation angle.

2-Qubit Gates

• CNOT: Q₀ control, Q₁ target. If control is |1⟩, applies X to target.
• SWAP: Exchanges the states of Q₀ and Q₁.
• CZ: Phase flip on Q₁ when Q₀ is |1⟩.

When the output is entangled, the Bloch spheres turn purple and show reduced density matrices (mixed-state approximation).

Usage

1. Input State: Use θ and φ sliders to set each qubit. For 1-qubit mode, only Q₁ is shown; for 2-qubit gates (CNOT, SWAP, CZ), both Q₀ and Q₁ appear.
2. Gates: 1-qubit gates apply to Q₁; 2-qubit gates act on both. For parametric gates (R_x, R_y, R_z), use the γ slider to set the rotation angle.
3. Measurement Probabilities: The histogram shows P(|00⟩), P(|01⟩), P(|10⟩), P(|11⟩) from the current 4D state. Purple bars indicate entanglement.
4. Measure System: Collapses the state to a single outcome (Born rule); output spheres jump to the measured basis state.
5. Reset Simulation: Clears the measurement and restores the full probability distribution.
6. Reset: Sets all inputs to |0⟩ and gate to H.