Ph.D. Thesis - Physics

digits of precision in the final result. We also note that the inefficiency increases if error
correction techniques need to be used, such that the time required scales as O(1/ǫ r ), where
r ≥ 2. Errors in the final answer are also induced by systematic control errors; we find that
evolution during single-qubit pulses due to the “always-on” NMR interaction Hamiltonian
is the main source of these errors. The time resources needed for error correction and
to compensate control errors may be compared to Table 1.2, which presents the required
resources accounting only for the error in the final measurement estimates.
Our results do not rule out the usefulness of digital quantum simulations. They merely
point out that quantum simulation outperforms classical only for specific problems and
numbers of qubits. It remains exponentially more efficient than classical simulation for
quantum systems for which no good approximate classical simulation is known, as long as
only fixed precision is required.
Part II
In the second part, we ask: how can one build a scalable two-dimensional array of trapped
ions that is suitable for analog quantum simulation? We will be focused, in particular,
on the goal of analog simulation of quantum spin models, particularly spin frustration. In
pursuing the answer to this question, we design, test, and evaluate two types of ion trap
that may be suitable for this purpose, a lattice ion trap and an elliptical surface-electrode
ion trap. It is essential not only to design, model, build, and test these traps, but also
to examine how analog quantum simulation might actually be performed in such traps. It
will be important to estimate the expected simulated coupling rates in each type of trap,
including all rf and dc fields that act on the ion. This requires us to ask, for example, how
the motional frequencies and simulated coupling rates scale with the size of the trap, how
the potentials that produce an effective Hamiltonian might be applied, and how rf-induced
micromotion might affect the fidelity of analog quantum simulations.
In Chapter 4, we begin with an introduction to quantum simulation with trapped
ions, including ion trap Hamiltonians, control techniques, and proposals for analog quantum
simulation. We particularly focus on the proposal of Porras and Cirac for the study of 2-D
spin models [PC04b]. The goal of building a 2-D simulator of antiferromagnetic spin lattices,
in which phenomena such as spin frustration could be observed, is the prime motivation for
the rest of the work in this part.
In Chapter 5, we examine the lattice ion trap architecture, in which ions are confined
in a 2-D array of individual potential wells, enabling the trapping of ions in virtually any
configuration (since sites can be loaded selectively) with a well-defined spacing between
ions. Such a trap could provide the stable array of trapped ions that is required for scalable
analog quantum simulation. We experimentally test certain predictions about the trapping
potentials, most importantly the motional frequencies of the ions, and find that the traps
agree well with our theoretical predictions. However, despite the apparent advantages of
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