1
INTRODUCTION
AND BACKGROUND
1.1
Overview
A computer is a physical device that helps us process information by executing
algorithms. An algorithm is a welldefined procedure, with finite description,
for realizing an informationprocessing task. An informationprocessing task can
always be translated into a physical task.
When designing complex algorithms and protocols for various information
processing tasks, it is very helpful, perhaps essential, to work with some idealized
computing model. However, when studying the true limitations of a computing
device, especially for some practical reason, it is important not to forget the rela
tionship between computing and physics. Real computing devices are embodied
in a larger and often richer physical reality than is represented by the idealized
computing model.
Quantum information processing is the result of using the physical reality that
quantum theory tells us about for the purposes of performing tasks that were
previously thought impossible or infeasible. Devices that perform quantum in
formation processing are known as
quantum computers
. In this book we examine
how quantum computers can be used to solve certain problems more eﬃciently
than can be done with classical computers, and also how this can be done reliably
even when there is a possibility for errors to occur.
In this first chapter we present some fundamental notions of computation theory
and quantum physics that will form the basis for much of what follows. After
this brief introduction, we will review the necessary tools from linear algebra in
Chapter 2, and detail the framework of quantum mechanics, as relevant to our
model of quantum computation, in Chapter 3. In the remainder of the book we
examine quantum teleportation, quantum algorithms and quantum error correc
tion in detail.
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2
INTRODUCTION AND BACKGROUND
1.2
Computers and the Strong Church–Turing Thesis
We are often interested in the amount of
resources
used by a computer to solve a
problem, and we refer to this as the
complexity
of the computation. An important
resource for a computer is
time
. Another resource is
space
, which refers to the
amount of memory used by the computer in performing the computation. We
measure the amount of a resource used in a computation for solving a given
problem as a function of the length of the input of an instance of that problem.
For example, if the problem is to multiply two
n
bit numbers, a computer might
solve this problem using up to 2
n
2
+3 units of time (where the unit of time may be
seconds, or the length of time required for the computer to perform a basic step).
Of course, the exact amount of resources used by a computer executing an algo
rithm depends on the physical architecture of the computer. A different computer
multiplying the same numbers mentioned above might use up to time 4
n
3
+
n
+5
to execute the same basic algorithm. This fact seems to present a problem if we
are interested in studying the complexity of algorithms themselves, abstracted
from the details of the machines that might be used to execute them. To avoid
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 Fall '09
 Vandam
 Quantum Physics, Computational complexity theory, Quantum computing, Probabilistic Turing machine

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