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.