Ph.D. Thesis - Physics

study should be undertaken of the effects of control errors in scalable quantum computing
systems. Is it possible that the presence of control errors and the sensitivity of quantum
simulations to them is superior in one system relative to others? Is it even possible that
this system will be found not to be identical to the one that is best-suited for universal
quantum computation? We do not yet know the answers to these questions, but they are,
in our view, fascinating ones that must be pursued.
Moving to the subject of analog quantum simulation, we saw in Part II how an array
of trapped ions could be used for simulating physics such as Ising and Heisenberg spin
models, and specifically spin frustration in 2-D antiferromagnetic spin lattices. We also
saw, however, that building a scalable or even semi-scalable architecture for doing such
simulations in 2-D is actually quite difficult. This problem arises for arrays of microtraps
from the difficulty of maintaining both low secular frequencies and high trap depths at small
ion-ion distances. This is a fairly straightforward result of the physics of coupled oscillators
and the equations of motion of trapped ions. Although we have not proven that quantum
simulation in lattice traps is impossible, it does appear to be quite difficult.
We then looked for a different trap design for 2-D quantum simulation, and moved on
to studying elliptical ion traps, calculating and measuring such properties as the motional
frequencies and the structure of ion crystals in these traps. We also showed that the
unavoidable micromotion is not a fundamental detriment to quantum simulation, and that
interesting analog simulations, such as the search for quantum phase transitions, may be
possible in such an architecture. Although other trap designs exist that create 2-D lattices of
ions, elliptical traps have the advantage that they can be scaled down and microfabricated,
reducing overall micromotion amplitudes and also allowing for the integration of wires that
produce magnetic field gradients, which we showed can be used to provide a rich variety of
simulated Hamiltonians for quantum simulation.
To date, the only realization of the spin model simulation scheme has been with two
ions in a linear trap. This can and should be scaled to larger numbers of ions in a linear
chain, and efforts made to understand the sources of decoherence and control errors. At the
same time, prototype 2-D quantum simulators should also be built. We think that elliptical
traps are a good starting point. In the future, these simulations may enable us to solve
very difficult problems, such as calculating the phase diagrams of 2-D antiferromagnetic
lattices, that may, in turn, shed light on the causes of high-temperature superconductivity.
Other condensed-matter systems, such as the Bose-Hubbard model, may be simulable in
such a trap as well. Developing a good 2-D trap design is an important step toward tackling
problems that classical computers cannot efficiently solve.
Despite the possibility of exceeding the simulation power of classical computation even
with 40 ions in a single trap, the 2-D architectures we proposed are not truly scalable to
hundreds of ions or more. We mentioned a number of ideas for scaling ion trap quantum
simulators to such large numbers of qubits, including moving ions between separate micro-
214