3D Qubit & Bloch Sphere Gates

Gates (2x2 complex matrix)

General rotation R_n(θ)

Axis polar θ_n 90°

Axis azimuth φ_n 0°

Rotation angle θ 90°

Set state directly

Polar θ 90°

Phase φ 0°

Measurement

P(|0⟩)

100%

P(|1⟩)

0 / 0

Trials run

Freq. of |0⟩

–

How it works

The qubit state |ψ⟩ = α|0⟩ + β|1⟩ is stored as two complex amplitudes. Every gate button and the general-rotation control funnel through the same 2×2 complex rotation matrix R_n(θ) = cos(θ/2)·I − i·sin(θ/2)·(n_xX + n_yY + n_zZ), applied to (α,β) by real complex matrix-vector multiplication — X, Y, Z, H, S and T are just that formula evaluated at fixed axes/angles.

|psi> = alpha|0> + beta|1>
R_n(t) = cos(t/2) I - i sin(t/2)(nx X + ny Y + nz Z)
[alpha', beta'] = R_n(t) . [alpha, beta]
P(0) = |alpha|^2   P(1) = |beta|^2

Gate buttons — X (bit flip, n=x, θ=180°), Y (n=y, θ=180°), Z (phase flip, n=z, θ=180°), H (n=(x+z)/√2, θ=180°), S (n=z, θ=90°), T (n=z, θ=45°).
General rotation — pick any axis n on the sphere via its own (θ_n,φ_n) and any signed angle, then apply the exact matrix — this is the operation every real single-qubit gate ultimately reduces to.
Measure once — a genuine projective measurement: draws a random outcome weighted by the live P(0)/P(1), then collapses the displayed state to that basis vector.
Run N trials — samples N independent copies of the current, uncollapsed state (a repeated-measurement / tomography experiment) without disturbing the interactive vector, and the running frequency converges to the true probabilities as N grows.

Drag to orbit the camera, scroll to zoom. The gold arrow is the Bloch vector r = (⟨X⟩, ⟨Y⟩, ⟨Z⟩) computed directly from α, β.

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