Driving Hamiltonian in a Quantum Search Problem

Abstract

We examine the driving Hamiltonian in the analog analogue of Grover's algorithm by Farhi and Gutmann. For a quantum system with a given Hamiltonian $E|w><w|$, we explicitly show that while the driving Hamiltonian $E|s><s|$ optimally produces the state $|w>$ from an initial state $|s>$, the driving Hamiltonian $E^{\prime }|s><s|(E^{\prime }\ne E)$ does not provide any speedup compared even with a classical computation.

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