Generalized Quantum Search Hamiltonian

There are hamiltonians that solve a search problem of finding one of N items in O( √ N) steps. They are hamiltonians to describe an oscillation between two states. In this paper we propose a generalized search hamiltonian, Hg. Then the known search hamiltonians become special cases of Hg. From the generalized search hamiltonian, we present remarkable results that searching with 100% is subject only to the phase factor in Hg and independent to the number of states or initialization. Grover proposed the quantum search algorithm that finds one of unsorted N items in O( √ N) steps.[1] Since it is known that classical search algorithms would take O(N) steps to solve a search problem, the quadratic speedup in Grover algorithm is remarkable. The reason of the speedup is effects of quantum mechanics. To apply quantum mechanics to a search problem, it needs to map each item to a state, respectively, in N dimensional Hilbert space. Then a search problem can be transformed to finding one of N states. We can use quantum mechanical characteristics, for instance, superposition and parallelism. Grover algorithm is in fact to find the state which corresponds to a target item. An iteration of Grover algorithm is composed of two operations: the first is to flip the target state about 0 and the next to invert all states about the average state.[2] An oracle, the heart of quadratic speedup, is applied in the first operation. Grover iterations amplify the amplitude of the target state up to nearly one in O( √ N) times, when an initial state is a uniform superposition of N states. Zalka showed Grover algorithm is optimal when unstructured items are considered.[3] On the other hand, there are a search algorithm based on hamiltonian evoution by Schrodinger equation. While Grover algorithm operates a state in discrete time, a search hamiltonian does a state in continuous time. They are hamiltonians to describe an oscillation between two states. Farhi et al. suggested harmonic-oscillation hamiltonian, exactly Hfa = E(|w〉〈w| + |s〉〈s|), where |w〉 is a target state and |s〉 is an initial state.[4] Fenner provided another hamiltonian, Hfe = 2iEx(|w〉〈ψ| − |ψ〉〈w|), where x is the overlap between an initial state and a target state, i.e., 〈w|s〉 = x(> 0).[5] A state under the hamiltonians evolves from an initial state to a target state in the half period of oscillation, Email address: [email protected] Email address: [email protected]