Quantum search by partial adiabatic evolution

Quantum adiabatic computation has attracted a lot of attention in the past decades, such as [1, 3–10], since it was proposed by Farhi et al.[11]. In [5], an adiabatic algorithm was proposed to solve the Deutsch-Jozsa problem. The algorithm took an exponential time which provided only a quadratic speed up over the best classical algorithm for the problem. A modified algorithm proposed by Wei and Ying [12] improved the performance to constant time. In [6], quantum adiabatic computation was proved to be polynomially equivalent to the quantum circuit model. The proof showed that adiabatic quantum computation using Hamiltonians with long-range fiveor three-body interactions, or nearest-neighbor two-body interactions with six-state particles, could efficiently simulate the circuit model. This results was soon modified to qubits with two-body interactions [13]. A simpler proof of the Equivalence was presented in [8]. In [14], quantum adiabatic computation was applied to solve random instances of NP-complete problems. A research outline of its application to solve NP-complete problems can also be found in the paper. A typical quantum adiabatic algorithm starts with the ground state of the initial Hamiltonian Hi, and evolves slowly to the ground state of the final Hamiltonian Hf . The system that implements the algorithm uses the timedependent Hamiltonian