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stance by a so called witness. However, there need not be any such witnesses
for ”no” instances.
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A good example is factoring of integers. Suppose the language under con-
sideration is the set of composite natural numbers, i.e. non-prime numbers. A
”yes” instance, let’s say a number n, can always be ascertained by exhibiting a
factor, say wy. By simply dividing n by wy it can be verified (in polynomial
time) that n is indeed composite. Onthe other hand, supplying a ”no” witness
wn is quite useless, since even if wn does not divide n, there might very well
be another ”yes” witness not yet exhibited. So, without deeper insight into the
problemofdeterminingwhetheranumberisprimeorcomposite,21 alltentative
factors must be checked before the verdict prime can be passed.
A language L is in NP if there is a Turing machine such that
If x L, there exist a witness w such that when the machine is started
• ∈
withxandwasinputs,itshaltsinthe”yes”stateafteratimepolynomial
in the size x of x.
| |
Ifx / L,thenforallpurportedwitnessesw,themachinehaltsinthe”no”
• ∈
state after atime polynomialin the size x ofx, when startedwith xand
| |
w as inputs.
ItisnotknownwhetherPisastrictsubsetofNP.TheconjectureP=NP
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is one of the main unsolved problems in complexity theory.
The class PSPACE
The class PSPACE is the space analogue to P. It is defined as follows.
The space complexity class PSPACE is the collection of all languages that
are in SPACE(nk) for some constant k. That is, a language is in PSPACE if
itcanbe decidedby a deterministic Turing machineusing a number ofworking
bits polynomialin the input size. There is no limit to the amountoftime used.
It is clear that P is included in PSPACE simply because a machine that
haltsafterapolynomialnumberofstepscanonlytraverseapolynomialnumber
oftapesquares. Thus,P PSPACE,butitisnotknowwhethertheinclusion
⊆
is strict, i.e. if P=PSPACE or not.
6
The class BPP
Ifprobabilisticalgorithmsareconsidered,thencorrespondingprobabilisticcom-
plexity classes can be defined. The bounded error probabilistic class, BBP, is
definedto containalllanguagesLthatcanbedecidedbyaprobabilisticTuring
machine M, such that
If x L, then M accepts x with a probability at least 3/4.
• ∈
If x / L, then M rejects x with a probability at least 3/4.
• ∈
21Recently,suchinsighthasindeedbeengained,showingthatprimalitytestingisinPafter
all[24].
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The probability 3/4 is arbitrary, any probability strictly greater than 1/2
would suffice in the definition.
Therearemanymorecomplexityclasses,aswellaslotsofinclusionrelations
betweenthem. Thereaderisreferredtotheliteratureforathoroughdiscussion.
We will briefly return to the topic in chapter 6 on the complexity of quantum
computation.
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Chapter 3
Algebra of quantum bits
Thereareafewdifferentmodelsofquantumcomputationintheliterature. The
most popular, and most thoroughly worked out, is the quantum circuit model
[25]. Quantum circuits are the quantum analogue of classical circuits built out
of logic gates. Another model, the quantum Turing machine [7] is the quantum
analogue of the classical Turing machine. But just as general-purpose digital
computers are not really built as Turing machines, it does not seem practical
to build realquantumcomputers as quantumTuring machines. Inthis respect,
quantumcircuitsseemtobeclosertoactualimplementationasphysicaldevices.
This chapter is an introduction to the subject of quantum computation.
The circuit model of computation introduced in chapter 2 will be elaborated
andrealizedintermsofvectorsandmatrices,thusofferingwhatcouldbecalled
analgebraofbitsandquantumbits. Inthiswaywewillbeabletoseeprecisely
where the quantum paradigm of computation breaks away from the classical.
Precise and general definitions of concepts, as well as a detailed treatment will
follow in subsequent chapters.
3.1 Classical and quantum physical systems
In contrastto the case of classicaltheory of computation, the physical substra-
tum of the computer is more focused in the researchon quantum computation.
Inpartthisisduetotheveryrealproblemsofactuallybuildingdevicescapable
of performing quantum computations. It is appropriatetherefore to begin with
a short discussion of the concept of a physical system.
A simple example of a classical system is a gas in container. Pressure,
temperature, and volume give the macroscopic state of the gas. In classical
physics,these variablescanrange overa continuousset of values corresponding
to a continuous state space. The microscopic state is given by the values of all
positions and all momenta of all the particles in the gas. This forms a huge
continuous state space.
Inquantumphysics,statespacescanbediscreteorcontinuousorboth. Con-
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tinuousstatespacesoccurforfreeparticlesorparticlesscatteredoffapotential,
whereas discrete state spaces occur for bound state systems, notably particles
bound by a potential field. The standard example of a bound state system is
the hydrogen atom which can be in a set of discrete states, each state being
characterized by values of energy and a couple of other variables. In this case
the energy can only range over a discrete set of values. If the atom absorbs
enough energy, it will become ionized and the electron will no longer be bound
by the potential of the nucleus. This corresponds to the continuous part of the
state space.
A gas of quantum particles in a container will have a huge discrete micro-
scopic state space. There is no practical way to distinguish different states of
sucha systemandit is uselessfor computationalpurposes. In orderfor a phys-