Hilbert-space average method and adiabatic quantum search

We discuss some aspects related to the so-called Hilbert-space average method, as an alternative to describe the dynamics of open quantum systems. First we present a derivation of the method which does not make use of the algebra satisfied by the operators involved in the dynamics, and extend the method to systems subject to a Hamiltonian that changes with time. Next we examine the performance of the adiabatic quantum search algorithm with a particular model for the environment. We relate our results to the criteria discussed in the literature for the validity of the above-mentioned method for similar environments.

Figure 1

(Color online) Numerical solution of the quantum search algorithm using the exact Schrödinger equation (blue, solid curves) or the derived master equation (41) (red, dashed curves). We plot the probability of overlapping with the searched-for state. Blue (solid) curves are obtained, from top to bottom, for $\lambda =0$, ${10}^{-4}$, and $5\times {10}^{-4}$. For the two last cases, the corresponding approximated solution is also shown, also from top to bottom. Also, for these two cases, green (dotted) curves show how decoherence manifests as time evolves, by plotting the magnitude defined in Eq. (43).