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Inarealquantumcomputer,thisprocessoftransformingtheinputstateinto
theoutputstateisactuallyadynamicalprocessthatoccursintime,wespeakof
time evolution. Now it is a fundamental property of quantum mechanics that,
if no measurements aremade, the time evolutionis reversible. This means that
the input can be inferred from the output. This, and other requirements, puts
arestrictiononthe allowedcomputationalmatricesC. Derivingthisrestriction
demands tools that will be developed in the following two chapters. Here we
just state the result.
First, if we canfind aninverseC 1 to C with the propertyC 1C =1, then
− −
equation (3.19) can be inverted by multiplying through by C 1,
−
65
C−1 out =C−1C in = in ,
| i | i | i
so that
in =C 1 out .
−
| i | i
The requirementon the matrices C that makes it straight forwardto invert
them is that they must be unitary. This means that C is invertible and its
inverse is equal to its conjugate transpose (C )T =C , or
∗ †
C 1 =C .
− †
Taking the conjugate transpose is a computationally cheap process of rear-
ranging and complex conjugating the elements of the matrix.
66
Chapter 4
Introduction to quantum
mechanics
Quantummechanicsisnotaphysicaltheoryinitself,itisratheraframeworkin
which physical theories must be formulated. If one takes a more fundamental,
or philosophical point of view, quantum mechanics is a basic characteristic of
reality which transcends all descriptions or theories of physical systems. It sets
certainlimitsonwhatcanknowninprincipleaboutphysicalsystems. Thebare
bonesofquantummechanicscanbeformulatedasafewpostulateswhichevery
quantum mechanical description of a physical system must conform to.
In the community of physicists, opinions differs as to the proper philosophi-
calstatusofquantummechanics. Themajorityviewseemsto be totakeitasa
fact of life, and since physical theories based on quantum mechanics in general
agree very well with experiment, the only sensible thing to do is to go on and
use it. There are features to quantum mechanics (for example the uncertainty
principleandentanglement)thatareconsideredtobe counterintuitive froman
everyday or classical physics perspective, but there is not a single experimen-
tal fact contradicting quantum mechanics. Quite to the contrary, the theory is
verified every day in physics laboratories around the world. Quantum mechan-
ics has furthermore been corroborated during the last twenty years by special
experiments testing the very foundations of the theory [29].
There are, however, and has always been, a strand of physicists uncom-
fortable with quantum mechanics. For them the theory is, though in practice
successful, in principle tentative, and eventually due to be replaced by a more
satisfactorytheory. The discussiongoesbackto the verybeginning ofquantum
mechanics and in particular to the famous Bohr-Einstein debate.
Some physicists maintain that (as Niels Bohr is reported to have said) that
if you’re not confused by quantum mechanics, then you haven’t understood it,
while others, especially younger physicists, can’t understand what all the fuss
isabout. Clearly,thishasmoretodowithonesownphilosophicaloutlookthan
with the theory itself, and we will leave this discussion here. As this is not a
67
workonfundamentalprinciplesofnaturalphilosophy,Iwilladoptthe standard
view that quantum mechanics is the proper framework for describing and un-
derstandingphysicalsystems,andthatclassicaltheoriesoffersatbestverygood
approximations. It should be kept in mind though, that there are fundamental
problems having to do with the relation between quantum mechanics and rela-
tivity, especiallygeneralrelativityandthe theory ofgravitation. This is not,at
least not yet, of any importance to the theory of quantum computation.
Thetermquantum physics thusreferstoanyphysicalsystem,ortheory,for-
mulated according to the postulates of quantum mechanics. The term classical
physics on the other hand, refers to physics not formulated using quantum me-
chanics. Examples of classical theories are classical mechanics (or Newtonian
mechanics), relativity (both special and general) and classical electrodynam-
ics. These classical theories are, as already noted, excellent approximations to
physical phenomena that takes place on a macroscopic scale, and often even to
microscopic phenomena. But in principle, physics is ’quantum’.
Many physical theories come in both a classicaland a quantum versionand
therearewelldefinedprocedurestopassbetweenthem. Theprocedureofgoing
from classical to quantum is called quantization. Very often, quantum theories
are formulated by first writing down a classical theory which is then quantized
according to a set of heuristic rules.
Obviously,inordertoworkonquantumcomputationyouneedsomegraspon
quantummechanics. Itisinfactnotdifficulttorapidlygathertogetherthebasic
elementsofquantummechanicsonacoupleofpages,andthisseemstobewhat
mostreviewarticlesdoes. Suchabriefexposethroughthequantummechanical
toolboxtendshowevertoratherdull,andIthink,fairlyincomprehensibleifyou
haven’t already studied the subject.
Iwilladoptanotherstrategy. Quantummechanicswillbeintroducedthrough
asetofsimplephysicaltoymodels. Thesewillbethestandardmodelsthathave
traditionally proved their worth in physics education. In the course of working
throughthemodels,allrelevantquantummechanicalconceptscanbeabstracted
from these concrete models. We will recklessly assume that whats true in the
particular case is true in the general case unless otherwise stated. Of course,
suchanapproachisonlyusefulinafirstgeneralintroductiontoasubject,andis
not a substitute to proper study. I also think that this approachwill be helpful
whenimplementationsofquantumcomputationintermsofphysicaldevicesare
discussed briefly in chapter 9.
The simple model systems we will consider are:
Particle in a potential box
•
Harmonic oscillator