Grover's Algorithm in 3D: The Geometric Rotation

Search space

N basis states 8

Marked target |0⟩

Run

Iteration k

Optimal k*

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P(marked)

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Rotation angle

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How it works

The bar field around the ring is a real N-length amplitude array aᵢ, one bar per basis state, height = |aᵢ|². Every step applies the true Grover iteration to that array:

oracle:     a_target → −a_target
diffusion:  aᵢ → 2·mean(a) − aᵢ  (all i)

Grover's whole trajectory stays confined to the 2-D plane spanned by |target⟩ and the equal superposition of every other state, |other⟩. The arrow on the left is that exact 2-D state, embedded in 3-D so you can orbit around it: its |other⟩-component is Σ (unmarked aᵢ)/√(N−1) and its |target⟩-component is a_target, both read directly off the live array — nothing about the arrow's angle is hard-coded, it is the geometric image of the same numbers driving the bars. Each iteration turns that arrow by a fixed angle θ = 2·arcsin(1/√N) toward |target⟩. Keep stepping past k* = π/4·√N and the arrow keeps rotating — past |target⟩ and back down — which is exactly why the marked bar's height rises then falls if you over-rotate.

N basis states — size of the search space; more states means more bars around the ring and a smaller per-step rotation angle.
Marked target — which single basis state the oracle flags; changing it resets the run.
Orbit — drag to rotate the camera; the rotation plane the arrow sweeps is fixed by the algorithm, but you can view it from any angle, including edge-on where the sin² curve is easiest to see.

Real-world relevance: this is the same rotation picture used to derive Grover's O(√N) query bound — every quantum search/optimisation routine built on amplitude amplification (Grover-based counting, fixed-point search) reuses this exact 2-D geometric argument.

Unmarked amplitude bar Marked amplitude bar State-vector arrow

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