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In quantum computation, there is a famous algorithm to search a marked item in an unstructured database called Grover's algorithm. It achieves a quadratic speedup over the best possible classical algorithm.

On the gate model of quantum computing, a phase oracle is used in Grover's algorithm. The phase oracle can be implemented by simulating the classical circuit that defines the function $f$, where $f$ indicates if an element (of the database for example) is marked $(f(x) = 1)$ or not $(f(x) = 0)$.

There exists a version of Grover's algorithm for the adiabatic quantum computer architecture as well.

However, here the final Hamiltonian has a very similar form to the phase oracle in the gate model.

The phase oracle had to be constructed with a quantum circuit simulating the classical circuit, that defines $f$.

But in the adiabatic case, how can the final Hamiltonian be constructed?

The problem is, that the Hamiltonian contains a projector onto the unknown state/the marked element.

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