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powerofquantumcomputation,aswellasthesourceoftheengineeringproblems
of actually building devices capable of enacting quantum algorithms.
1.4 Quantum computation
In1980,thephysicistRichardFeynmanpointedoutthatdigitalcomputerscan-
not simulate quantum systems without an exponential slowdown [9]. Feynman
wasn’t particularly interested in approximating quantum physics, what he dis-
cussedwasexactsimulation,thequestionofwhetherdigitalcomputerscoulddo
exactly the same as the quantum system would do. He came to the conclusion
that present day physics does not allow this, essentially due to a mismatch be-
tweenthe discretenatureofdigitalcomputersandthe exponentiallylargestate
space of quantum systems. Feynman’s interest in the physics of computers
sparked off the research into quantum computation in the 1980’s.
Other lines of research that contributed to the initial impetus of quantum
computation was the work of Bennet [10] and Fredkin and Toffoli [11] on re-
versible computation, as well as the previously cited work by Landauer [5].
In quantum computing we are interested in the computational strength of
physical systems that by their very nature must be analyzed or understood
according to quantum mechanics. Quantum computation relies on the exact
manipulation of individual quantum physical objects, in distinction from the
electronicscomputer.
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classicalcomputer, where averagedclassicalstatisticalproperties of the objects
suffices. This is the source of the strength of quantum computation as well the
difficulties in actually building quantum computing devices. There is to be no
transitiontotheclassicalregimeduringthecomputation,asthatwoulddestroy
the very features that lends the quantum computer its strength.
Some researchers in the field remark that, as classical physics is fundamen-
tally wrong, any correct theory of computability must be based on quantum
mechanics. This point ofview is veryclearlystatedin the paper by Deutsch[7]
where it is claimed that there is a physical assumption underlying the Church-
Turing thesis.
Deutsch argues that the Church-Turing thesis is actually at variance with
classical physics, but that it can be rephrased in agreement with quantum
physics. According to this point of view, quantum computers are not fun-
damentally more powerful than Turing Machines, though they might be faster.
Thebasic lineofthe argumentisthatthe continuousnatureofclassicalphysics
makesitinprinciple impossibletosimulate aclassicalphysicalsystembyadis-
crete computer. But quantum physics is fundamentally discrete, and therefore
the Church-Turingthesisconnectseffective methodsnottoclassicalcomputing
machines but to quantum computing machines. The argument is not entirely
convincing, and will be returned to in part II.
Andwiththisweplungeintothedetails! Butfirstadisappointingremarkis
perhapsinorder. Therearenoquantumcomputersasyetifonedoesnotcount
experimental setups working on the equivalent of a few bits. Because of this,
notmuchwillbesaidhereaboutrealizations. Anythingwrittenaboutpractical
implementations of quantum computing devices will surely soon be outdated
by new experimental developments, whereas the theoretical part of the topic is
likely to be more stable.
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Chapter 2
Classical computation
Quantum computation relies on the ability for quantum mechanical physical
systems to perform computations. In order to prepare some common ground
forthediscussion,wewillreviewthetheoryofclassicalcomputationintermsof
Turing machines,the Church-Turingthesis and the limitations ofcomputation.
There are two fundamental questions in the theory of computation; what
can be computed in principle, and what can be computed efficiently. The first
question is addressed by the theory of computability and second by the theory
of complexity. In order to discuss these questions in general without reference
toanyparticularcomputing machineorprogramminglanguage,onemustwork
withinsomeabstractmathematicalmodelofcomputation. Still,itmustbepos-
sibleunderstandpreciselytherelationshipbetweenthemodelandtheproperties
of actual physical computing machines.
The models of computation developed in the nineteen thirties were all at-
tempts to capture in precise mathematical terms what is meant by a compu-
tation. In order to answer the Entscheidungsproblem1, i.e. David Hilbert’s
questionofwhetherthereexistamechanicalmethodtodetermineifmathemat-
ical statements are true or not,2 it was necessary to have a precise definition of
amechanicalmethodinordertotreatthequestionwithmathematicaltools. In
this context, mechanical does not necessarily mean, and in fact did not mean,
a procedure performed by a machine. Mechanical means algorithmic.
As it turned out, the different models put forward; Church’s λ-calculus,
Herbrand-G¨odel’s recursive functions and Turing’s automatic computing
machines3,wereallshowntobeequivalentinthesensethattheyalldefinedthe
same set of computable functions (see several papers in [13]). All three models
were meant to be abstract mathematical models of computation. Turing, how-
ever, phrased his concepts in terms of machines reading and writing symbols
1Thiswasapparentlynottheonlyimpetustothiswork,see[6].
2ForHilbert,truthofastatementwasequivalenttoitbeingatheorem,otherwiseonewould
have to distinguish between truth and derivability when stating the Entscheidungsproblem.
JohnvonNeumanndiscussedtheproblemintermsofprovability.
3ThetermisTuring’sown.
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on a tape, and compared the process of computation to humans working with
paper and pencil. And it is also with the Turing model that the connection to
present day digital computers and to physics is most clearly seen.
Theformalmodelsofcomputationmustbeconnectedtotheintuitivenotion
of an algorithm. The models are meant to capture what it means to carry out
a mechanical, or algorithmic, procedure.
The Church-Turing thesis is often quoted in this context, but there seem
to be some confusion as to what it actually says. After 50 years of dramatic
evolutionofdigitalcomputers,thethesis hasperhapsnotsurprisingly,acquired
connotations or meanings not present in the original formulation [12]. We will
be concernedwith whatthe thesis did saywhenit was firstformulated,whatis
normally meant by it nowadays,and how it relates to quantum computation.
2.1 Some definitions
The following definitions are for the benefit of the reader not familiar with
computer science terminology, and for fixing the sense in which the terms will
be used in this text.
2.1.1 Algorithm