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Science in China. Series G, Physics, mechanics & astronomy/Science in China. Series G, Physics, Mechanics & Astronomy

An upper bound for the adiabatic approximation error

Wang Yong Wen

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

In this paper,we derive an upper bound for the adiabatic approximation error,which is the distance between the exact solution to a Schr dinger equation and the adiabatic approximation solution. As an application,we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrdinger equation.

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In this paper,we derive an upper bound for the adiabatic approximation error,which is the distance between the exact solution to a Schr dinger equation and the adiabatic approximation solution. As an application,we obtain an upper bound for 1 minus the fidelity of the exact solution and the adiabatic approximation solution to a Schrdinger equation.
The discovery of adiabatic quantum computing has revealed a need for rigorously proven bounds for the error in the adiabatic approximation. We present in this paper, a rigorous and elementary derivation of upper and lower bounds on the error incurred from using the adiabatic approximation for quantum systems that can be used to assess the runtime of adiabatic algorithms. Our bounds are often asymptotically tight in the limit of slow evolution for fixed Hamiltonians, and are used to provide sufficient conditions for the application of the adiabatic approximation. We show that our sufficiency criteria exclude the Marzlin–Sanders counterexample from the class of Hamiltonians that obey the adiabatic approximation. Finally, we demonstrate the existence of classes of Hamiltonians that resemble the Marzlin–Sanders counterexample Hamiltonian, but also obey the adiabatic approximation.
We provide rigorous bounds for the error of the adiabatic approximation of quantum mechanics under four sources of experimental error: perturbations in the initial condition, systematic time-dependent perturbations in the Hamiltonian, coupling to low-energy quantum systems, and decoherent time-dependent perturbations in the Hamiltonian. For decoherent perturbations, we find both upper and lower bounds on the evolution time to guarantee that the adiabatic approximation performs within a prescribed tolerance. Our results include explicit definitions of constants, and we apply them to the spin-1/2 particle in a rotating magnetic field and to the superconducting flux qubit. We compare the theoretical bounds on the superconducting flux qubit to simulation results.