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