On time slices, time complexity, and energy increase in implementing quantum adiabatic evolution by circuit model

Abstract

In this paper, motivated by previous related works on a kind of quantum local adiabatic evolution, we mainly study the circuit model of its corresponding quantum global adiabatic search algorithm, in which the evolving paths functions in the system Hamiltonian have a general form instead of the usual linear interpolating ones which are widely investigated in the literature. The main conclusion found here can be summarized as follows. Though this quantum adiabatic computation can demonstrate various algorithmic performances, when considering implementing it by a quantum circuit, the time slices thus produced always keep as an invariant which is proportional to inverse square of the overlap between the initial and final states in the problem of interest and also equal to that of multiplying the time complexity and the energy together, by noting that compared with the usual adiabatic evolutions, there is being a great increase of energy of the system. As far as we know, the result exposed here is new and is hoped to be useful for further understanding the kind of quantum adiabatic evolution itself and its connection with the quantum circuit model. Also, we hope the main conclusion here will be ubiquitous when considering applying this kind of quantum adiabatic evolution to solve the problems beyond quantum search.