Ph.D. Thesis - Physics

precision of a quantum simulation is then discussed, in light of these experiments, in Sec. 2.3.
In Sec. 2.4, we summarize the chapter and then enumerate some outstanding questions that
remain to be addressed.
2.1 Hamiltonian and control techniques
In this section we present a simple description of the NMR system, and with it the Hamilto-
nian that governs the nuclear spins. We then show how this Hamiltonian enables quantum
control of the nuclear spins. In so doing, we explain how a NMR system satisfies most of
the DiVincenzo criteria for quantum computation (Sec. 1.4.3).
We begin with a comment on the first criterion. Although suitable for small systems,
NMR is not a scalable architecture for quantum computation (or universal quantum sim-
ulation) because there is no way to do operations fault-tolerantly. It is not possible to
perform operations on the system conditioned on fast measurements, as would be required
in the diagnosis and correction of errors. This stems from the nature of measurement in
solution-state NMR, which is based on a slow recording of the bulk magnetization of the
system (see Sec. 2.1.6).
The other criteria are state initialization, a universal set of gates, high-efficiency mea-
surement, and a sufficiently long coherence time; the satisfaction of these by NMR will be
explained in the remainder of this section. Throughout we will use “nucleus,” “spin,” and
“qubit” interchangeably, since the exclusive topic of this section is spin-1/2 nuclei.
2.1.1 The static Hamiltonian
The NMR system consists of a sample of dissolved molecules, the nuclear spins of which,
in thermal equilibrium, are aligned (or antialigned) with a strong external magnetic field
of magnitude B0, which by convention points along the ˆz direction. Oscillating magnetic
fields of peak magnitude B1, and oriented along ˆx and ˆy, are applied to the nuclei to
rotate their spin states. Finally, the bulk magnetization of the sample may be read out
inductively, giving an ensemble measurement of the spin states. Details of this setup for a
real experimental system are given in Sec. 3.4. In this chapter, we focus on the Hamiltonian
and control techniques.
The static portion of the NMR Hamiltonian is composed of two terms: one describes
rotations of the nuclei about the ˆz axis due to the the static magnetic field, and the other
describes the spin-spin coupling between nuclei. Using i to index the individual nuclear
spins, the term due to B0 is written
H0 =
ω0Zi , (2.1)
i
where the Larmor frequency ω0 is given by ω0 = gNµnB0/, and gN is the nuclear g-factor,
48