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ObsidianVault/SS2026/Quantum Computing/12. Grover's Algorithm/Grover's Algorithm.md
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Another well known quantum algorithm for searching. Takes a function f : \{0,1\}^n \rightarrow \{0,1\} where f(x) = 1 for exactly one x. The goal is to find x.

New gates: V_f and FLIP$_*$

\ket{*} denotes the superposition over al classical possibilities.

Oracle V_f

!Pasted image 20260805102238.png

We use the Unitary U_f as previously defined to construct V_f

!Pasted image 20260805102605.png

FLIP$_*$

We first need FLIP$_0$ defined as follows: !Pasted image 20260805102659.png

This is implemented via this circuit:

!Pasted image 20260805102717.png

Z is a Pauli matrix and the empty circles denote negative control wires. So Z is only applied if all other wires are \ket{0} Now we define the unitary FLIP$_*$:

!Pasted image 20260805103650.png

!Pasted image 20260805103706.png

The search algorithm: