Bloch Sphere Representation

A single qubit state is parameterized by two angles θ ∈ [0, π] and φ ∈ [0, 2π]:

The Bloch sphere visualizes this: the north pole (+Z) is |0⟩, the south pole (−Z) is |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).

Simulation

The interactive simulator is below. Use the controls to explore the concepts described above.

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.

Limitations

• One or two qubits only. The state space is 1- or 2-qubit (up to a 4-D vector). Real quantum advantage needs many qubits, where the 2^N state space cannot be drawn or simulated classically at scale.
• Pure-state, noiseless. Gates are perfect unitaries with no decoherence, gate error, or readout error. Entangled outputs are shown via reduced density matrices as a mixed-state approximation on the Bloch spheres — a single sphere cannot fully represent an entangled qubit.
• Single gate at a time. The tool applies one gate to the input state; it is not a multi-gate circuit builder with sequential composition, ancillas, or mid-circuit measurement.
• Fixed gate set. Common Pauli/H/S/T/rotation and CNOT/SWAP/CZ gates only; arbitrary unitaries, Toffoli/multi-controlled gates, and parameterized 2-qubit gates are not included.
• Idealized measurement. "Measure" applies the Born rule and collapses instantly with no measurement back-action noise or repeated-shot statistics (it shows exact probabilities, not sampled counts).
• Visualization, not computation. This builds intuition for single-gate action on the Bloch sphere; it does not run quantum algorithms (see [[groveralgorithm]] for an algorithm-level demo).