16 BASIS STATES · 3 ITERATIONS

Ordinary Grover

INITIAL |0000⟩

MARKED |1111⟩0.00%

Choose initial / marked state

Use the selectors or pick a square. Drag to orbit.

INPUT STATE

Quantum gate circuit

w = |1111⟩0 / 13 operations

Prepare once; repeat the four remaining gates 3 times.

H is the four-qubit Hadamard.

State diagnostics

Marked probability0.00%

At recommended stop96.13%

Total probability 100.00%

Mean & mixer shift

Current mean · Re(μ)

Three amplification rounds for 16 states.

Hover a column to inspect its amplitude.

Amplitude phase

−π−π/20π/2π

Reading signed amplitudes

The oracle changes the sign of the marked amplitude only. Hadamard interference can also make unmarked amplitudes negative inside the diffuser. Those signs are part of the circuit calculation.

Gate checkpoints show exact Grover states. Between checkpoints, a norm-preserving animation rotates amplitudes through the complex plane. This animation is a display choice; Grover specifies the gate endpoints, not a physical path between them.

The 2×2 and 4×4 gate endpoints are real. The odd grids can have complex amplitudes even at checkpoints. A zero real column can have nonzero probability: choose Imaginary amplitude to inspect its other component.

Gates and phase convention

H is the tensor-product qubit Hadamard: H† = H.

The three diffuser steps are H, Sᵢ, H, where Sᵢ = I − 2|i⟩⟨i|. The middle step flips only the initial-state amplitude. Every other amplitude remains fixed throughout this step.

For |sᵢ⟩ = H|i⟩, the completed diffuser is H Sᵢ H = I − 2|sᵢ⟩⟨sᵢ| = −D. This is Grover's inversion about the mean with an overall minus sign. It gives the same success probabilities, and the columns show that sign explicitly.

For the default 16-state example, the first Sᵢ step takes the initial column from +0.875 to −0.875. After the final H, the marked amplitude is −0.6875 and each unmarked amplitude is −0.1875. Nothing is replaced by an absolute value or silently rephased.