Quantum search by partial adiabatic evolution

A quantum search algorithm based on the partial adiabatic evolution\cite{Tulsi2009} is provided. We calculate its time complexity by studying the Hamiltonian in a two-dimensional Hilbert space. It is found that the algorithm improves the time complexity, which is $O(\sqrt{N/M})$, of the local adiabatic search algorithm\cite{Roland2002}, to $O(\sqrt{N}/M)$.

💡 Research Summary

The paper introduces a quantum search algorithm that leverages the concept of partial adiabatic evolution, originally proposed by Tulsi (2009), to achieve a faster runtime than the conventional local adiabatic search algorithm (Roland and Cerf, 2002). The problem setting is the standard unstructured search: among N items, M of them are marked, and the goal is to find any marked item. The authors begin by defining the initial Hamiltonian (H_0 = I - |s\rangle\langle s|), where (|s\rangle) is the uniform superposition over all N basis states, and the final Hamiltonian (H_f = I - |w\rangle\langle w|), where (|w\rangle) is the uniform superposition over the M marked states. The time‑dependent Hamiltonian is taken as a linear interpolation (H(s) = (1-s)H_0 + s H_f) with the adiabatic parameter (s\in