Ph.D. Thesis - Physics

Chapter 2
Quantum simulation using nuclear
magnetic resonance
Nuclear magnetic resonance (NMR) is a well-known technique for manipulating and mea-
suring the spins of nuclei in molecules. It found its first widespread use in the identification
of chemical compounds and the elucidation of their chemical structures. Subsequently,
NMR came to form the basis of magnetic resonance imaging (MRI), a leading medical diag-
nostic technique. The success of NMR depends on the exquisite degree of control over the
quantum dynamics of the nuclei, combined with their long (O(seconds)) coherence times.
These same features allow NMR to serve as an ideal test-bed for quantum algorithms and
quantum simulation. In a 1997, two groups independently proposed implementing quantum
algorithms in bulk solution-state NMR: Gershenfeld and Chuang in Ref. [GC97], and Cory,
Havel, and Fahmy in Ref. [CFH97].
Following this stimulus, nuclear magnetic resonance (NMR) was the first system in
which a wide variety of quantum algorithms were implemented. NMR demonstrations of
the Deutsch-Josza [CVZ + 98], Grover [VSS + 99], and Shor [VSB + 01] algorithms were the
first to be realized in any technology. It offers an extremely convenient test-bed for quan-
tum algorithms: a customized off-the-shelf experimental system in which many quantum
computation protocols can be applied to a closed quantum system. In addition, implement-
ing quantum algorithms with NMR can lead to insights about quantum control that are
applicable to other systems as well, e.g. ion traps [GRL + 03]. Several detailed treatments
of solution-state NMR quantum computation have been written, including Ref. [VC05] and
the theses of Vandersypen [Van01] and Steffen [Ste03].
In this chapter, we show how an NMR system can be used as a quantum simulator. We
first discuss, in Sec. 2.1, the basic properties of an NMR system, including the Hamiltonian
under which atomic nuclei evolve and the techniques for controlling the nuclei. Then, in
Sec. 2.2, we examine prior experiments that have proven the capacity of NMR to act as
a quantum simulator for small numbers of qubits. The question of the limitations to the
47