Ph.D. Thesis - Physics

problems, the others have a strong effect upon our considerations. Estimates of decoher-
ence rates must be calculated for each trap design, and possible control errors must also
be considered. Furthermore, although we will focus on the simulation of spin frustration
as the ultimate goal, we note that the traps studied in this part may be useful for other
simulations as well, such as Bose-Hubbard physics.
The main considerations for a trap design are interaction rates, controls, and decoherence
rates. We discuss each of these below. The design must:
1. Provide a regular array of stationary qubits in at least two spatial dimensions.
The array of ions desired here is an extension to two dimensions of the linear ion
crystals used in most quantum information experiments. To do quantum simulations
of problems that are unique to configurations of spins in two or more dimensions,
such as spin frustration, we need a trap that will produce a two-dimensional array
of ions. The word regular means here that the ions are stored in some configuration
such that the ion-ion distance d between each pair of nearest neighbors is identical.
This is important because the simulated coupling rate J depends strongly on the
inter-ion distance d (varying as d −3 in the β ≪ 1 limit as discussed above). Quantum
simulation is also possible in an array for which this distance is not constant, but
this adds a constraint to the types of Hamiltonians that can be implemented (the
J-coupling will be site-dependent). When applying a global effective Hamiltonian,
this site dependence may translate into a control error, limiting the precision of the
simulation, but not necessarily rendering it useless. Furthermore, such systematic
control errors may be compensated if sufficient controls are available.
2. Enable sufficient control over each qubit to implement the desired simulation.
A trap design that fulfills the first condition does not guarantee that quantum sim-
ulation may be effectively done. Sufficient controls are required to implement the
desired effective Hamiltonian. These may include rotations of individual qubits, state-
dependent forces arising from an optical or magnetic force, and global or ion-specific
measurements. In addition, achieving some desired J-coupling rate requires a suf-
ficient interaction rate between individual ions. Recall that the parameter β is the
fractional transfer of the motional energy of one ion to another per secular period;
although the dipolar approximation holds for β ≪ 1, the resulting J-coupling rate
may prove too small to be observable within the decoherence time of the system.
Therefore, in considering the controls required for quantum simulation, one must take
into account the pertinent quantities β and J, and calculate each for each trap design.
3. Support a low enough decoherence rate to perform meaningful simulations given cer-
tain coupling rates.
Internal state decoherence depends on the choice of qubit states, ambient field fluctu-
ations, errors in the classical controls, and other factors that are not directly related
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