| dc.description.abstract | Spin glasses are experiencing a revival due to applications in quantum information
theory. In particular, they are the archetypal native benchmark problem for quantum annealing
machines. Furthermore, they find applications in fields as diverse as satisfiability,
neural networks, and general combinatorial optimization problems. As such, developing
and improving algorithms and methods to study these computationally complex systems
is of paramount importance to many disciplines. This body of work attempts to attack
the problem of solving combinatorial optimization problems by simulating spin glasses
from three sides: classical algorithm development, suggestions for quantum annealing
device design, and improving measurements in realistic physical systems with inherent
noise. I begin with the introduction of a cluster algorithm based on Houdayer’s cluster algorithm
for two-dimensional Ising spin-glasses that is applicable to any space dimension
and speeds up thermalization by several orders of magnitude at low temperatures where
previous algorithms have difficulty. I show improvement for the D-Wave chimera topology
and the three-dimensional cubic lattice that increases with the size of the problem.
One consequence of adding cluster moves is that for problems with degenerate solutions,
ground-state sampling is improved. I demonstrate an ergodic algorithm to sample ground
states through the use of simple Monte Carlo with parallel tempering and cluster moves.
In addition, I present a non-ergodic algorithm to generate new solutions from a bank of
known solutions. I compare these results against results from quantum annealing utilizing
the D-Wave Inc. quantum annealing device. Finally, I present an algorithm for improving
the recovery of ground-state solutions from problems with noise by using thermal fluctuations
to infer the correct solution at the Nishimori temperature. While this method has been
demonstrated analytically and numerically for trivial ferromagnetic and Gaussian distributions, a useful metric for more complex Gaussian distributions with added Gaussian noise
is unavailable. We show improved recovery of numerical solutions on the chimera graph
with a ferromagnetic distribution and added Gaussian noise. Next, I direct my focus to the
design of future generations of quantum annealers. The first design is the two-dimensional
square-lattice bimodal spin glass with next-nearest ferrromagnetic interactions proposed
by Lemke and Campbell claimed to exhibit a finite-temperature spin-glass state for a particular
relative strength of the next-nearest to nearest neighbor interactions. Our results
from finite-temperature simulations show the system is in a paramagnetic state in the thermodynamic
limit, thus not useful for quantum annealing device designs that would benefit
from a spin-glass phase transition. The second design is the diluted next-nearest neighbor
Ising spin-glass with Gaussian interactions in an attempt to improve the estimation
of critical parameter with smaller system sizes by implementing averaging of observables
over different graph dilutions. To date, this model has shown no improvement. Finally,
I make suggestions for the choice of distributions of interactions that are robust to noise
and present a method for using previously unaccessible continuous distributions. I begin
with showing the best-case performance of quantum annealing devices. I show results for
the resilience, the probability that the ground-state solution has changed due to inherent
analog noise in the device, and present strategies for developing robust instance classes.
The analog noise is also detrimental to interactions chosen from continuous distributions.
Using Gaussian quadratures, I present a method for discretizing continuous distributions
to reduce noise effects. Simulations on the D-Wave show that the average residual of
the ground-state energy with the true ground-state energy is calculated and shown to be
smaller in the case of the discrete distribution. | en |
