Quantum search by local adiabatic evolution

Abstract

The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state which tends to the solution [1]. We apply this time-dependent Hamiltonian approach to the Grover's problem, i. e., searching a marked item in an unstructured database. We find that, by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order √ N , where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N . This is in contrast with the constant-rate adiabatic approach developed in [1], where the requirement of adiabaticity is expressed only globally, resulting in a time of order N .

Speeding up critical system dynamics through optimized evolution

“…II A. Furthermore, studying the QSL as a function of the system size, we show that the speed up obtained by the adiabatic GSA [33,38] can be reproduced and extended to other models with optimized, non-adiabatic evolutions. Finally, we introduce the action s = T ∆ as a parameter to characterize the evolution of a quantum system and we find that the QSL identifies a new dynamical regime, as discussed in Sec.…”

“…We study two paradigmatic critical systems, the adiabatic GSA [33] and the LMG model [39] and we compare them with the Landau-Zener (LZ) model to better understand the physics of the process. The GSA Hamiltonian is given by:…”

“…More recently with the developement of the quantum annealing [28] and adiabatic quantum computation [4], a renewed interest has been devoted to the subject in condensed matter and quantum information [29][30][31][32]. Here we investigate for the first time the QSL of the dynamics of a first order QPT in the adiabatic version of Grover's search algo-rithm (GSA) [33,34] and of a second order QPT [35] in the Lipkin-Meshkov-Glick (LMG) model. Specifically we consider the problem of converting the ground state on one side of the critical point into the ground state on the opposite side in the fastest and most accurate way by selecting an optimal time-dependence of the control field.…”

Robust coherent superposition of states using quasiadiabatic inverse engineering

“…[30], the quasiadiabaticity (constant adiabaticity) condition should provide a faster shortcut to adiabaticity. Compared with other local adiabaticity strategies [26,27,28,29,30], the current strategy provides the exact target state at short process times through inverse engineering. The STA strategies [20,21,24,25] can provide exact target state as well as robustness against errors by design, while the current approach offers robustness by driving the system as close to adiabaticity as possible, providing a versatile alternative to the aforementioned manipulation techniques.…”

“…A family of processes have been developed to optimize the efficiency and robustness of adiabatic passage using local adiabaticity constraints [26,27,28,29,30]. The fast quasiadiabatic dynamics (FAQUAD) approach reduces the time by homogeneously distributing the adiabaticity parameter along the process using a single control parameter [30].…”