Ph.D. Thesis - Physics

1996, right on the heels of Peter Shor’s 1994 factoring algorithm. Factoring and quantum
simulation were the first two practical (and classically intractable) problems that quantum
computers were shown to be able to efficiently solve. Since that time, numerous proposals
have been set forth to do quantum simulation of physical systems. And, slowly but surely,
experiments have started to catch up. Throughout this thesis, we will encounter several
small-scale implementations of quantum simulation.
There remain difficult problems to be solved, however, before a quantum simulator that
can outperform classical computation is realized. In this thesis we address some of these
problems. Our journey takes us across two different physical systems, nuclear spins and
trapped ions, as we explore two of the most important problems facing quantum simulation:
precision and scalability. The question of precision asks, for a given quantum simulation
algorithm, how many digits of precision can be obtained, in principle, and how does this
number scale with the space and time resources required? Also, in practice, how do the
real-world effects of decoherence and control errors impact this precision? The exploration
of this question in the context of solution-state nuclear magnetic resonance (NMR) forms
the first part of this thesis.
Quantum systems are generally hard for classical computers to simulate, although there
do exist classical techniques for solving some large quantum systems either exactly or ap-
proximately. To solve the exact dynamics of a general quantum system, however, requires
a quantum simulator of the same number of interacting subsystems as the model being
simulated. Moreover, each small quantum system is subject to the effects of decoherence
and control errors. The question of scalability is: how do you build a reliable large-scale
quantum simulator out of faulty small-scale quantum simulators? This is a highly nontrivial
question, but trying to answer it for the case of trapped ions occupies the second and third
parts of this work.
This chapter is organized as follows. In Sec. 1.1, we present a brief overview of the con-
cepts of quantum information processing, including motivations and experiments. Sec. 1.2
is an introduction to quantum simulation, including its history, most important literature,
and most enticing applications. In Sec. 1.3, we discuss two different approaches to quantum
simulation, termed digital and analog, both of which are of importance to this thesis. Some
of the outstanding challenges in quantum simulation, focusing on the issues of precision
and scaling mentioned above, are then discussed in Sec. 1.4, which in turn motivate the
main results and content of the thesis, presented in Sec. 1.5. The final section, Sec. 1.6,
enumerates the contributions of the author and coworkers to this thesis, as well as resulting
publications.
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