Ph.D. Thesis - Physics

Figure 4-3: Schematic diagram of spin frustration. The blue circles represent lattice sites,
while the arrows represent the spin state of the particle on a given site. Suppose the spins
are governed by an antiferromagnetic interaction, meaning that it is energetically favorable
for pairs of spins to anti-align. If two spins are in well-defined states, the third does not
“know” which way to align. Thus, the ground state of the system is a quantum-mechanical
superposition of spin states. As the number of particles grows, the system becomes very
difficult to simulate on a classical computer.
they are inclined to anti-align) exhibit a state in which the ground state wave function
is a superposition of many distinct spin states. This occurs because there is not a single
spin configuration which minimizes the energy of the system. A schematic diagram of spin
frustration is given in Fig. 4-3.
One of the reasons spin frustration is interesting is that it is believed to hold insight into
the phenomenon of high-temperature superconductivity [GP00, NGB92]. We pursue spin
frustration because it is hard to simulate classically, and offers an example of an “analog”
quantum simulation that may provide a great deal of insight into a perplexing physical
system. We consider such simulations to be analog because of the continuously varying
controls applied, even though the system Hilbert space is discrete.
In this section, we examine the proposal of Porras and Cirac for simulating spin models
with trapped ions [PC04b]. Their scheme is unique in its ability to use a limited set of
control techniques to simulate phase transitions in a large system of trapped ions. The
benefit of this analog approach is that even with a limited amount of control, one may
be able to observe quantum behavior that is not simulable on classical computers, such
as quantum phase transitions. Although the ultimate precision in computing quantities
such as transition points scales poorly with the number of measurements done, quantum
simulation might nevertheless be a powerful tool whenever classical approximations or exact
solutions are not known. Here we follow the approach of Porras and Cirac fairly closely.
The mathematical steps involved in the derivation are presented in more detail in the S.B.
thesis of Ziliang Lin [Lin08]. The challenge of simulating spin physics in two dimensions
motivates the work in the rest of this part of the thesis.
95