Ph.D. Thesis - Physics

Chapter 1
Introduction
The growth of computing technology over the past century has revolutionized how much
of the human race lives, works, and interacts with itself. Although the miniaturization of
transistors has enabled phenomenal computational feats, the same basic principles apply
to modern devices that applied to Babbage’s difference engine, and to the model system of
Turing. These principles are those of classical physics, in which any given physical system
exists in one and only one state at one time, evolves under the laws of Newton and Maxwell,
and can remain in the same state upon being measured. What this means is that classical
mechanics can be efficiently simulated on a computer: you need keep track only of the
degrees of freedom of each subsystem separately. For instance, in a system of 100 two-level
systems, you need only to keep track of the state (say, 0 or 1) of each individual system to
completely specify the state of the entire system.
The discovery of quantum mechanics has led to a more correct, but rather more unintu-
itive view of the workings of the universe. In the quantum world, systems can be thought
to exist in many possible configurations at once, evolve under a different dynamics, and are
irreversibly altered when measured. This quantum strangeness of the world has posed a
difficult problem for classical computers: given the laws of quantum mechanics, the number
of numbers needed to specify a given system scales not linearly, but exponentially with the
number of interacting subsystems. Our example of 100 systems is suddenly intractable,
because now not 100 but roughly 2 100 numbers (and complex ones at that) are required to
specify the state of the system. Thus, even printing the complete state of the system cannot
be done efficiently with respect to the number of particles. A host of problems that are
described by quantum mechanics are thus intractable, from chemistry to solid-state physics
and many others. Although the clever use of approximations, symmetries, and probabilistic
algorithms yields satisfactory solutions to many problems, the full quantum dynamics of
most systems cannot be simulated.
The first sign that this hopeless situation could be rectified came in 1982, when Richard
Feynman suggested that perhaps one quantum mechanical system could be used to calcu-
late the behavior of another. This visionary conjecture was confirmed by Seth Lloyd in
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