Ph.D. Thesis - Physics

2. Initialization to a simple fiducial state such as |000...0〉.
3. Decoherence times much longer than the gate times.
4. A universal set of quantum gates.
5. High efficiency, qubit-specific measurements.
We now briefly summarize how these criteria differ for quantum simulation. The first
criterion certainly holds, although the form of the large-scale quantum simulator may be
different for analog, as opposed to digital, quantum simulation (as discussed above). The
second and third also hold, but the initial state will depend greatly on the problem being
simulated. It may be necessary to create a state such as |000...0〉, which may then be
transformed into the correct initial state through single-qubit rotations. An example of this
is in Ref. [WBL02]. However, it is also possible that the relevant degrees of freedom may
be described by continuous variables. In the example of cold atoms in an optical lattice,
it is typically necessary for the translational degrees of freedom of the atoms to be cold;
however, the simulation itself may not depend on the internal states. This is one way in
which analog quantum simulation enables simulation with less control than digital; in the
digital case, the momenta of the atoms would be discretized and then programmed into a
set of qubits at the beginning of the simulation.
The fourth criterion must also be modified for quantum simulation, and again varies
between the digital and analog types. Depending on the problem being implemented, a
set of controls less powerful than those of a universal quantum computer may be required.
The optical lattices are again a good example. In this analog simulation, sufficient control
is present to simulate a Bose-Hubbard model. By contrast, a universal quantum simulator
is equally as powerful as a universal quantum computer, because for either any arbitrary
Hamiltonian (and corresponding unitary gate) may be approximated to arbitrary accuracy.
It is also true, however, that for certain digital quantum simulations, only a subset of a
universal gate set will be required. Finally, high measurement fidelity is important in both
analog and digital quantum simulation, but for the analog variety, it may be a measurement
of a continuous variable, and “qubit-specific” may not have meaning. Measuring the density
of atoms in space is an example of this.
Scaling up quantum simulators is a major research question no matter what physical
system forms the qubit. The two physical systems that are studied in this thesis are solution-
state NMR and trapped ions; however, solution-state NMR is intrinsically unscalable. Ion
traps, by contrast, do satisfy all the DiVincenzo criteria. The only caveat is that although
ion traps are scalable in principle, we do not yet know which method for scaling them up
will prove most effective. We therefore focus here on the scalability of ion traps.
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