Qubit Bloch Sphere Simulator

Quantum Gates

Phase gate S·φ 0°

Decoherence rate (1/T₂) 0.15

θ (polar)

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φ (azimuth)

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P(|0⟩) = |α|²

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P(|1⟩) = |β|²

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Purity |r|

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Gates applied

How it works

A single qubit's pure state |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩ maps onto a point on the Bloch sphere: θ is the polar angle from |0⟩ (north pole), φ the azimuthal phase. Gates act as rotations of this vector — Hadamard (H) swaps the Z-axis and X-axis, Pauli-X flips it through the equator (bit flip), Pauli-Z reflects it about the vertical axis (phase flip), and the phase gate spins it around the vertical axis by the chosen angle. Turning on decoherence continuously shrinks the vector's length r toward the center — a mixed state — modelling how a real qubit loses coherence to its environment over a timescale T₂.

|ψ⟩ = cos(θ/2)|0⟩ + e^(iφ)sin(θ/2)|1⟩
r(t) = r₀·exp(−t/T₂),  Γ = 1/T₂
P(|0⟩) = |α|² = cos²(θ/2)

H / X / Z — apply the corresponding single-qubit gate to the current state, instantly rotating the Bloch vector.
Phase gate — rotate the vector around the vertical (Z) axis by the chosen angle φ, changing the relative phase between |0⟩ and |1⟩ without altering measurement probabilities.
Decoherence rate — how fast the vector's length decays toward the center when noise is on; a shorter T₂ (higher rate) means faster loss of quantum coherence.
Noise toggle — switch environmental decoherence on or off to compare an idealized qubit against a realistic, noisy one.

Real-world relevance: this shrinking-vector picture is exactly what limits every physical qubit platform — superconducting circuits, trapped ions, photonic qubits — and is the reason quantum error correction and fault-tolerant architectures exist at all.

State vector |1⟩ (south pole) |0⟩ (north pole)

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