1210.4647 (Avatar Tulsi)

Alternative to quantum adiabatic evolution using fixed-point quantum
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Avatar Tulsi

A quantum system can be evolved from the ground state of an initial Hamiltonian to that of a final Hamiltonian by adiabatically changing the Hamiltonian with respect to time. The system remains in the ground-state of time-changing Hamiltonian provided the change is slow enough. More precisely, if g is the minimum energy gap between the ground state and other eigenstates of time- changing Hamiltonian then the evolution time must scale as the inverse square of g for a successful evolution. Childs et al proposed an alternative, where the system is kept in the ground state of a time-changing Hamiltonian by doing measurements at suitably small enough time intervals. Their scheme is successful only if the time scales as the inverse cube of g, and thus the time-scaling is inferior to the adiabatic evolution. Here, we propose another alternative which is essentially similar to the Childs scheme but uses the concept of fixed-point quantum search (FPQS) algorithm to recover the inverse-square time-scaling behaviour of adiabatic evolution. Our algorithm uses selective transformations of the unknown ground states and phase-estimation algorithm (PEA) is the main tool to approximate such selective transformations. Though the approximation provided by PEA is not sufficient for our algorithm, we show that a combination of PEA and FPQS can significantly enhance the approximation making it sufficient for our algorithm. Thus we demonstrate interesting applications of fixed-point quantum search which achieves monotonic convergence towards the desired final state.

View original: http://arxiv.org/abs/1210.4647