Resonant Quantum Search with Monitor Qubits

We present an algorithm for the generalized search problem (searching k marked items among N items) based on a continuous Hamiltonian and exploiting resonance. This resonant algorithm has the same time complexity O(\sqrtN/k) as the Grover algorithm. A natural extension of the algorithm, incorporating auxiliary "monitor" qubits, can determine k precisely, if it is unknown. The time complexity of our counting algorithm is O(\sqrtN), similar to the best quantum approximate counting algorithm, or better, given appropriate physical resources.