Quantum Adiabatic Markovian Master Equations

We develop from first principles Markovian master equations suited for studying the time evolution of a system evolving adiabatically while coupled weakly to a thermal bath. We derive two sets of equations in the adiabatic limit, one using the rotating wave (secular) approximation that results in a master equation in Lindblad form, the other without the rotating wave approximation but not in Lindblad form. The two equations make markedly different predictions depending on whether or not the Lamb shift is included. Our analysis keeps track of the various time- and energy-scales associated with the various approximations we make, and thus allows for a systematic inclusion of higher order corrections, in particular beyond the adiabatic limit. We use our formalism to study the evolution of an Ising spin chain in a transverse field and coupled to a thermal bosonic bath, for which we identify four distinct evolution phases. While we do not expect this to be a generic feature, in one of these phases dissipation acts to increase the fidelity of the system state relative to the adiabatic ground state.Comment: 31 pages, 9 figures. v2: Generalized Markov approximation bound. Included a section on thermal equilibration. v3: Added text that appears in NJP version. Generalized Lindblad ME to include degenerate subspaces. v3. Corrections made to Appendix E and F. We thank Kabuki Takada and Hidetoshi Nishimori for pointing out the errors. v4: Corrected a typo in Eqt. B

Quantum Simulations of Classical Annealing Processes

We describe a quantum algorithm that solves combinatorial optimization problems by quantum simulation of a classical simulated annealing process. Our algorithm exploits quantum walks and the quantum Zeno effect induced by evolution randomization. It requires order $1/\sqrt{\delta}$ steps to find an optimal solution with bounded error probability, where $\delta$ is the minimum spectral gap of the stochastic matrices used in the classical annealing process. This is a quadratic improvement over the order $1/\delta$ steps required by the latter.Comment: 4 pages - 1 figure. This work differs from arXiv:0712.1008 in that the quantum Zeno effect is implemented via randomization in the evolutio