quantum gate directory
Phase Oracle
Symbol
$O_f^\pm$
Description:
Encodes a Boolean function as phases on computational basis states.
Alternate notations:
- $U_f^\pm$
- $\mathcal{O}_f^\pm$
A phase oracle is a variant of the standard oracle that acts on a single register $|x\rangle$, encoding the value of a classical Boolean function $f: \{0, 1\}^n \to \{0, 1\}$ as a relative phase: $$ |x\rangle \xmapsto{O_f^\pm} (-1)^{f(x)}|x\rangle $$ This construction is particularly useful in algorithms like Grover's search, where interference between states allows the amplitudes of marked states to be amplified.
Properties
- Diagonal in the computational basis; unitary and Hermitian for Boolean $f$.
- Can be implemented from a standard oracle via phase kickback.
Usage
- Grover search and amplitude amplification.
- Encoding classical predicates into quantum phases.
References
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