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5.6.1 Projective measurement . . . . . . . . . . . . . . . . . . . 118
2
5.6.2 General measurement . . . . . . . . . . . . . . . . . . . . 119
5.6.3 POVM measurement . . . . . . . . . . . . . . . . . . . . . 120
6 Abstract quantum computation: The circuit model 121
6.1 Quantum alphabets, strings and languages. . . . . . . . . . . . . 121
6.2 The circuit model of quantum computation . . . . . . . . . . . . 124
6.2.1 Gates and wires . . . . . . . . . . . . . . . . . . . . . . . 126
6.2.2 General notation . . . . . . . . . . . . . . . . . . . . . . . 126
6.2.3 Special discrete one-qubit gates . . . . . . . . . . . . . . . 128
6.2.4 One-qubit rotation operators . . . . . . . . . . . . . . . . 128
6.2.5 The rotation operators and the Bloch sphere . . . . . . . 129
6.2.6 Single qubit phase-shift operators. . . . . . . . . . . . . . 130
6.2.7 Some special controlled operations . . . . . . . . . . . . . 130
6.2.8 Some practical ”machinery” . . . . . . . . . . . . . . . . . 132
6.2.9 Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.10 Some important gate constructions . . . . . . . . . . . . . 137
6.2.11 Decomposing general two-level unitary operation on n-
qubit states . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.2.12 Universal sets of quantum gates. . . . . . . . . . . . . . . 143
6.2.13 Exact and approximate universality . . . . . . . . . . . . 144
6.2.14 Exact universality of two-level unitaries . . . . . . . . . . 145
6.2.15 Summary of universality results . . . . . . . . . . . . . . . 146
6.2.16 Discrete sets of gates . . . . . . . . . . . . . . . . . . . . . 147
6.2.17 General results . . . . . . . . . . . . . . . . . . . . . . . . 148
3
Foreword
Theintendedreadershipforthismaster’sthesisinComputerScienceisprimarily
the computer scientist wishing to get an idea of what quantum computing is
about. But I also have physicists in mind. Therefore, the physicist will find
material on physics that will appear to be obvious and the computer scientist
willfindmaterialoncomputersthatwilllikewiseappeartobetrivial. Soperhaps
the reader who will benefit the most from the text is the one who is unfamiliar
with both subjects. The point is that I’m actually not writing for the lucky
fewwhohaveexpertiseinbothfields butratherforthosewhocomefromeither
field, or from none of them. The text is thus basically introductory, but not
elementary.
There is also a further point. Since quantum computation straddles the
borderline between physics and computing science, it is interesting to spell out
the basic assumptions and facts of both fields in some detail.
Obviously, this text can be seen as a review article. But I have no intention
to treat every aspect of the subject which is simply to vast. The depth of the
treatment will also vary considerably. Some basic definitions and some, in my
opinion fundamental, results, will be spelt out in detail, whereas many topics
thatacomprehensivetextwouldtreat,willbepassedoverrapidly. Theprinciple
behind these choices is that I will attempt to be detailed on issues that has a
bearing on the connections between physics and computation. What has been
left out can be found in the textbook literature and original articles on the
subject as well as in other review articles.
The text is mostly written in theoretical physics style, introducing no more
formalism than needed to make the arguments clear. The degree of formal-
ization will vary. A high level of formalization throughout tends to make the
text unreadable, whereas a low level of formalization might leave the reader
unnecessarily confused. Definitions, derivations and results are presented and
proved in the running text, but occasionally, due to the nature of subject, a
more formal style will be adopted. I’ve chosen a level of formalization that I
found appropriate and in the end it reflects my own taste.
There are of course lots of review article on quantum computation. I have
thereforedecidednottorepeattomuchofthestandardcalculationsandderiva-
tions,insteadfocusingonwhatIfindinteresting,tryingtoputforwardaslightly
differentperspective,andinsteadbeingdetailedonpointsthatareoftenglossed
over. InthisrespectIhopethistextcanbeacomplementtothemanyexcellent
4
books and reviews already in circulation, a few of which are [1, 2, 3, 4, 39].
One seldom learns a subject by reading just one book or just one review
article. In writing chapter 4 on introduction to quantum mechanics, I realized
howmuchisleftimplicit,eventhoughyoutrytomakethetextselfcontained. If
youhaven’talreadymastereda subject, perhapsyoucannotgainsomuchfrom
just one review - you must read several articles and books to see the subject
treated in different ways.
Outline of contents
Chapter 1 is an introduction to text and a motivation for studying quantum
computation. Some fundamental questions on the connection between physics
andcomputationwillbe mentioned. Theywillbereturnedtoinaplannedpart
II of this work.
Chapter2isanoverviewofthecentralconceptsofclassicalcomputationsuch
as the notions of computational models, computability and complexity theory.
TogetherwithChapter5ongeneralquantumtheoryitservesasthe foundation
for a treatment of quantum computational models and quantum algorithms.
Chapter 3 is a brief introductionto quantum computation. It servesmainly
as motivating the subsequent two chapters on quantum mechanics.
Chapter 4 contains a quite extensive introduction to quantum mechanics
written in a physics style. Three important models are treated in some detail;
a particle trapped in a potential well, the harmonic oscillator and the theory
of angular momentum. Apart from being important in quantum physics, these
models are the standard ones employed when teaching introductory quantum
mechanics. Allconceptsofquantummechanicscanbeintroducedwhilestudying
these simple models.
Chapter 5 then sets up the formal theory of quantum mechanics in terms
of linear operators on Hilbert spaces. After that, the stage is set for treating
quantum computation.
Chapter 6 describes in an abstract way the quantum circuit model.
As this text is mainly on the abstract and theoretical aspects of classical
and quantum computational models, not very much will be said on practical
realizations of quantum computing devices, or quantum computers for short.
Presumably, the theoretical aspects of the subject matter will remain relevant,
whilethepractical,implementationaldetailsarelikelytoundergomoredramatic
change.
Onelastremark. MyinitialintentionswastotreatalsotheQuantumTuring
machine model and quantum algorithms. However, the scope of the project