Ph.D. Thesis - Physics

and thus requires invocation of the Trotter product formula, dramatically increasing the
number of necessary pulses. Second, we attempt to calculate a property of some physical
system, an eigenvalue of its Hamiltonian, and in doing so explore the limitations to the
precision with which such a quantity can in principle be calculated. In this sense, we test
the limits of digital quantum simulation. We not only calculate the bounds on the precision
that are inherent in the quantum protocol itself, but also those which are a result of the
control techniques used to implement it.
Specifically, we wish to use a digital quantum simulator to calculate ∆, the energy gap
between the ground state and first excited state of a pairing Hamiltonian. Algorithms such
as WBL, which require a polynomial number of gates in the problem size, are generally not
efficient with respect to the precision in the final answer. This was true in Lloyd’s proposals
[Llo96, AL97, AL99], and in the specific case we consider here.
We ask the following questions: What are the theoretical bounds on precision for the
algorithm studied here? Can an NMR implementation saturate the theoretical bounds on
precision? What control errors are most important, and what effect do they have on the
final result? Do simple methods of compensation for these errors have a predictable effect?
The chapter is organized as follows: in Sec. 3.1, we briefly review the BCS pairing theory;
in Sec. 3.2, we present the proposal of Wu et al. for the simulation of this Hamiltonian
using NMR; in Sec. 3.3, we discuss the bounds on precision in the expected result ; in
Sec. 3.4, we discuss our experimental setup; in Sec. 3.5, we present our implementation of
the algorithm; finally, in Sec. 3.6, we evaluate this work, including the inherent limitations
of digital quantum simulations in general and NMR quantum simulations in particular.
3.1 The BCS theory
The motivation for this work lies in the quantum simulation of superconducting systems.
Superconductivity, first discovered in 1911 by H. K. Onnes, is characterized by an abrupt
drop to zero of the electrical resistivity of a metal. Although this phenomenon defied theo-
retical explanation for some time, a phenomenological theory promulgated by Ginzburg and
Landau in 1950 explained most of the features of superconductivity, including the Meissner
effect (exclusion of magnetic flux) [GL50]. In 1957, Bardeen, Cooper, and Scrieffer (BCS)
published the first complete microscopic theory, realizing that the phenomenon of supercon-
ductivity was due to the superfluidity of pairs of electrons in the conductor [BCS57]. In the
BCS theory, electrons form the pairs through very weak interactions that are mediated by
phonons in the lattice (made up of the nuclei of atoms in the solid metal). The weakness of
these interactions is why superconductivity only occurs at very low temperatures; above the
superconducting transition temperature Tc for a given material, thermal energy is enough
to break the bonds that hold electron-electron pairs together.
Superconductors are grouped into two types based on their magnetic properties. Type I
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