%Nr: DS-2001-06
%Author: Ronald de Wolf
%Title: Quantum Computing and Communication Complexity
Computers are physical objects and hence should follow the laws of
physics. Somewhat surprisingly, today's computers (theoretical Turing
machines as well as desk-top PCs) are developed on the model of
classical physics rather than on the model of its 20th century
successor quantum mechanics. The new field of quantum computing tries
to make up for this deficit by studying the properties of computers
that follow the laws of quantum mechanics. One of the striking
properties of a quantum computer is that it can be in a superposition
of many classical states at the same time, which can exhibit
interference patterns.
One of the main goals of the field of quantum computing is to find
quantum algorithms that solve certain problems much faster than the
best classical algorithms. Its two main successes so far are Shor's
1994 efficient quantum algorithm for finding the prime factors of
large integers (which can break most of modern cryptography) and
Grover's 1996 algorithm that can search an $n$-element space in about
$\sqrt{n}$ steps.
Part I: Query Complexity
------------------------
The starting point of part I of this thesis is the observation that
virtually all known quantum algorithms (including Shor's and Grover's)
can be described in terms of query complexity: they require far fewer
queries to input bits than classical algorithms do. It thus appears
that the model of query complexity captures a lot of the power of
quantum computing. Accordingly, in part I of the thesis we make a
detailed and general comparison of quantum query complexity versus
classical query complexity for various kinds of computational
problems.
Our main tool in this comparison is algebraic: we prove that the
quantum query complexity of a computational problem is lower bounded
by the degree of a certain polynomial that in some sense represents
that problem. This means that we can prove lower bounds on the
quantum query complexity of various problems by analyzing polynomials
for those problems. One of the main consequences of this technique is
the result that quantum query complexity can be at most polynomially
smaller than classical query complexity when we consider total
computational problems (which are defined on all possible inputs). In
other words, any exponential quantum speed-up in this model will have
to be based on some promise on the input, some property that the input
is known in advance to have. For example, for Shor's algorithm this
promise is the periodicity of a certain function to which the
factoring problem can be reduced.
Apart from these general results that hold for all total problems, we
also consider in more detail the quantum complexities of various
specific computational problems. For example, we prove that the error
probability in Grover's search algorithm can be reduced slightly
better if we do this in a quantum way than if we do it in the usual
classical way (which would just repeat Grover's algorithm many times).
We also derive an algorithm for the element distinctness problem
(which is: are the numbers on a list of $n$ elements all distinct?)
that takes about $n^{3/4}$ steps. This shows that for a quantum
computer the problem of element distinctness is significantly easier
than the problem of sorting, in contrast to the classical world, where
both problems require about $n\log n$ steps.
Finally, we show that the negative result for standard query
complexity (at most a polynomial quantum speed-up for all total
problems) does not hold in two other versions of query complexity:
average-case complexity and non-deterministic complexity. For both
models we exhibit total problems and quantum algorithms for solving
those problems that are exponentially better than the best classical
algorithms.
Part II: Communication and Complexity
-------------------------------------
It has been known since the early 1970s that quantum communication
cannot improve upon classical communication when it comes to
information transmission: if Alice wants to send Bob $k$ bits of
information, then she has to send him at least $k$ quantum bits
(Holevo's theorem). However, Cleve and Buhrman discovered that if the
goal of Alice and Bob is not to communicate information but to solve
some distributed computational problem (Alice gets $x$, Bob gets $y$,
and together they want to compute some function $f(x,y)$ with minimal
communication between them), then sometimes the amount of
communication can be reduced drastically by allowing quantum
communication. For example, a result of Buhrman, Cleve, and Wigderson
says that if Alice and Bob each have an $n$-slot agenda and they want
to find a slot where they are both free, then they can do this with
roughly $\sqrt{n}$ quantum bits of communication, whereas in the
classical world about $n$ bits of communication would be needed.
In part II of the thesis we look at this model of quantum
communication complexity from various angles. We first discuss the
main examples known where quantum communication complexity is
significantly less than classical communication complexity. Then we
consider the other side and develop techniques to show lower bounds on
quantum communication complexity, again using algebraic techniques.
These techniques imply, for example, that quantum communication cannot
improve significantly upon classical communication complexity for
almost all distributed problems. However, we also exhibit a new
example where quantum communication complexity does improve upon
classical complexity: in a specific 3-party model (Alice and Bob each
send a message to a referee, who should then compute $f(x,y)$), the
problem of testing equality between Alice and Bob's input can be
solved with exponentially less communication when we allow quantum
communication, using a new technique called quantum fingerprinting.
In the final chapter we address an issue of security. Suppose Alice
and Bob care not only about minimizing the amount of their
communication, but also about keeping it secret: if some third party
Eve is tapping the communication channel, then she should learn
nothing about the actual messages. Classically, it is known that a
shared secret $n$-bit key is necessary and sufficient to send a
classical $n$-bit message from Alice to Bob in a way that gives no
information to Eve (Shannon's theorem). We prove the quantum analogue
of this: $2n$ bits of shared secret key are necessary and sufficient
to securely send a message of $n$ quantum bits.