Ph.D. Thesis - Physics

Chapter 4
Theory and history of quantum
simulation using trapped ions
In the previous part, we explored how NMR systems can be used to perform quantum
simulations. However, we also saw some drawbacks to the scalability of these systems, as
well as to the precision with which results can be calculated. In this chapter, we present
an alternative approach based on trapped ions. Trapped ions have a long coherence time
(potentially O(10 s) or longer), which exceeds that of nuclear spins in solution, and offer a
potentially scalable system in the sense that the number of ions that can be used is unlimited
if proper controls are available and decoherence is sufficiently small (or error correction is
used). Networking between ions may be accomplished by moving ions or by linking them
with photonic or electronic quantum communication, as discussed in Ch. 1.
We also noted in Ch. 1 that the analog approach to quantum simulation is promising for
quantum simulations of systems that do not require error correction, or systems which are
fundamentally not amenable to error correction, such as systems with continuous Hilbert
spaces. In particular, analog simulations of spin physics may be able to solve problems that
classical simulation cannot for as few as 36 interacting particles [RMR + 07]. Here, we turn
to such analog systems. Our main effort is to design and experimentally characterize an
ion trap in which such a simulation could be done, focusing on the problem of simulating a
2-D lattice of interacting spins using trapped ions.
In Sec. 4.1, we present the basic ion trap Hamiltonian and demonstrate how it permits
quantum control of the internal and external degrees of freedom of trapped ions using laser
radiation. Then, in Sec. 4.2, we outline the method proposed in Ref. [PC04b] for simulating
quantum spin models in an ion trap. Sec. 4.3 enumerates the challenges involved with
designing ion traps for such quantum simulations, framing the questions explored in the
remainder of Part II. We summarize in Sec. 4.4.
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