Quantum Query Complexity and Distributed Computing
Hein Roehrig
In complexity theory, the strengths and limitations of computers are
investigated on abstract models of computation. The choice of these models is
governed by three considerations: (1) how close is the model to existing
computers or computers that could be built in principle? (2) how well does it
lend itself to proving interesting properties of computers? (3) how elegant is
the model mathematically?
Quantum computation appeals to all three criteria. In functional analysis,
quantum mechanics has a beautiful mathematical underpinning, which benefits
quantum computing through new applications of linear algebra and matrix
analysis. Nowadays it is a widely-held belief that the physical theory of
quantum mechanics describes reality accurately at very small scales of length,
time, and energy. Where classical probabilistic Turing machines may be seen as
capturing the power of computers operating according to finite- precision
classical physics, the computational model of quantum circuits aims at modeling
what realistic computers in a quantum mechanical world can do. Query
complexity, a variant of time complexity, has a close analogue for quantum
computers; as in the classical case, our current mathematical tools are more
amenable to this restricted complexity measure than to general time complexity.
Sometimes, the implications of quantum query complexity shed new light even on
classical complexity theory.
This thesis investigates the properties and applications of quantum query
complexity and the related quantum communication complexity. It suggests new
cryptographic protocols and new experiments for probing the predictions of
quantum mechanics. Quantum states are very sensitive; this thesis examines ways
to deal with imperfections and errors in a number of different situations.