Classical and Quantum Computation in Ground States and Beyond
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In this dissertation we study classical and quantum spin systems with applications to the theory of computation. In particular, we examine computational aspects of these systems beyond their ground states by considering the effects of non-zero temperatures and excited energy states. In the first part we show that universal classical computations can be encoded into equilibrium thermal states of classical spin systems, at sufficiently low temperatures which are independent of the system size. In the second part we explore different strategies for optimization with the quantum adiabatic algorithm, and we show that it is possible to increase the success probability for hard random instances by following the counterintuitive strategy of evolving along the Hamiltonian path more rapidly. In the third part we examine the performance of simulated quantum annealing in finding the minimum of an energy function which contains a high energy barrier, and we provide evidence that simulated quantum annealing inherits some of the advantages of quantum annealing which allow it to pass through the high barrier and minimize the energy function efficiently. In the last part we show that the path-integral quantum Monte Carlo method leads to a provably efficient algorithm for approximating the partition function of any 1D generalized transverse Ising spin chain (with position-dependent local fields and frustrated interactions), at temperatures which are independent of the system size.
- Physics