Studies of Classical and Quantum Annealing

A summary of the results of recent applications of PIMC-QA on different optimization problems is given in Chapter 1. In order to gain understanding on these problems, we have moved one step back, and concentrated attention on the simplest textbook problems where the energy landscape is well under control: essentially, one-dimensional potentials, starting from a double-well potential, the simplest form of barrier. On these well controlled landscapes we have carried out a detailed and exhaustive comparison between quantum adiabatic Schrödinger evolution, both in real and in imaginary time, and its classical deterministic counterpart, i.e., Fokker-Planck evolution [17]. This work will be illustrated in Chapter 2. On the same double well-potential, we have also studied the performance of different stochastic annealing approaches, both classical Monte Carlo annealing and PIMCQA. The CA work is illustrated in Chapter 3, where we analyze the different annealing behaviors of three possible types of Monte Carlo moves (with Box, Gaussian, and Lorentzian distributions) in a numerical and analytical way. The PIMC-QA work is illustrated in Chapter 4, were we show the difficulties that a state-of-the-art PIMCQA algorithm can encounter in describing tunneling even in a simple landscape, and we also investigate the role of the kinetic energy choice, by comparing the standard non-relativistic dispersion, Hkin = Tau(t)p^2, with a relativistic one, Hkin = Tau(t)|p|, which turns out to be definitely more effective. In view of the difficulties encountered by PIMC-QA even in a simple double-well potential, we finally explored the capabilities of another well established QMC technique, the Green's Function Monte Carlo (GFMC), as a base for a QA algorithm. This time, we concentrated our attention on a very studied and challenging optimization problem, the random Ising model ground state search, for which both CA and PIMC-QA data are available [10, 11]. A more detailed summary of the results and achievements described in this Thesis, and a discussion of open issues, is contained in the final section `Conclusions and Perspectives'. Finally, in order to keep this Thesis as self-contained as possible, we include in the appendices a large amount of supplemental material.