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putation
The paradigm of computation is capable of encompassing widely different sys-
tems. Onacertainlevelofabstraction,thedescriptionofaphysicalsystemand
a computing machine is very similar, not to say identical. Almost everything
conceivable can be described by an input function output model. A
→ →
physical system is defined as some well defined portion of space and time with
a well defined interface and interaction with the environment. Then specifying
theinputusingsomelabeling,theoutputcan,inprincipleatleast,becomputed
using the dynamical laws.
In computer science, we always know the dynamics of the system, because
this is the program, and we set up the system in order for it to compute previ-
ously unknown outputs from given inputs. Furthermore, essentially due to the
discrete finite nature of input and output, there is a well agreed on paradigm
for this input-processing-output model. As soon as the labeling (the alphabet)
of the input/output states are defined, the computation is just a syntactically
ruled shuffling of the labels.
Inphysicsthefocusisdifferent. First,wesometimesdon’tknowthedynam-
icallaws. Theveryobjectoffundamentalphysicsistoinvestigatethedynamics
through theory, experiments and observations.
Is there a difference in the computational strength of different physical sys-
tems? What algorithms can be performed with what physical systems? These
arequestionsnotnormallyposedintheoreticalcomputersciencewherethe dis-
cussionisfromthe outsetframedwithin theclassicalcomputationalmodels,all
of which are basically notational systems.
However, it is generally believed that any physical system constrained to
work in a discrete stepwise fashion, working to precisely and finitely stated
rules according to the logical description of a Turing Machine or an electronic
computer, or even a human computer as envisioned by Turing, are equivalent.
TheChurch-Turingthesisidentifiesthesetofeffectivelyorintuitivelycalcu-
lable functions with the set of functions computable within any of the classical
computational models. In a historical context, effective computability meant
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computability by an abstract human being working to precise rules.
Thethesishashoweveracquiredconnotationsconnectingittomachinecom-
putation, in particular electronic digital computing machines. In this latter
sense the thesis is true; what can be computed by a general purpose digital
computer can be computed by a Turing machine. This is due to the fact that
a digital computer can be modeled as an abstract RAM machine. Whether
any conceivable physical computing machine is constrained by the thesis is not
known(butsee[52]). Thisisaquestionaboutallofphysics,andwedon’tknow
all of physics yet.
Ontheotherhand,theprecisemappingbetweenmodelsofcomputationand
humananddigitalmachinecomputationandtheconsequentpossibilitytostudy
computationintheabstracthasleadtotheviewthatthelimitsofcomputation
are set by mathematics and logic. The development of quantum computation
has, to a degree, challenged this view of computation. Since computations are
basically physical processes when actually carried out, it can be argued that
what can be computed is a question of physics, not a question of mathematics
or logic [7].
1.3 Classical physics and the computer
The CPU of a digital electronic computer as well as the main memory used for
intermediate storage consist of huge amounts of transistors and other semicon-
ducting devices working in an on/off fashion corresponding to distinct voltage
levels. These voltage levels constitutes a concrete realization of the abstract
bit of information. The semiconductors in their turn are arranged into circuits
implementinglogicalgates. Theprecisemappingbetweenabstractlogicalgates
working on bits and circuits working with voltage levels are the basis for the
success of the electronic digital computer. But without the extreme fastness
with which the switching between on and off can be performed (on the order
of nanoseconds) the computer would not be so powerful. There is also an en-
gineering aspect of this. The transistors must work in parallel, and in practice
the CPU clock controls the working of the computer so that at each time tick,
bit flips are performed in parallel.
The electronic digital computer can therefore be seen as a special physical
systemconstrainedorengineeredto workin a discrete way. All operationsper-
formed by the computer are discrete, but the underlying physical processes are
continuous,or at leastthe descriptionof these physicalprocessesis continuous.
The actually bit flips between zero and one, when studied at the physical level
take a certain amount of time and the transition between voltage levels can
actually be studied by employing a good enough oscilloscope. But once one
has abstracted away from these physical considerations, the operations of the
electronic computer could as well be performed by other physical systems, for
exampleelectriccircuitsworkingwithmagneticrelays. The performancewould
be much slower and other engineering problems would ensue.2
2Actuallysuchcomputers,andcomputersbasedonvacuumtubes,precededthesolidstate
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The workings of a transistor, as far as electronics goes, can be described
by classical electrodynamics. But electronics is not far from quantum physics.
Transistors are quantum mechanical devices that could not possibly be under-
stood or built without a knowledge of quantum physics. However these tran-
sistors are wired to work in a discrete fashion as switches. Logically there is
no difference between a transistor switch, a mechanicalswitch or an electrome-
chanical switch. The only difference lies in performance measures like speed,
reliability and energy consumption. The underlying physics of the transistor
must be understood in terms of quantum mechanics, but once that is done,
the transistor as a circuit element can be understood in classical terms. And
furthermore, as pointed out above, these circuit elements can be abstractly
modeled and reliably worked with without at every step consulting the under-
lying physics. When implementing the circuit, design issues like power supply,
switching times, delays et cetera must of course be faced, but this does not, in
principle, influence the logical design of the circuit.
Intheenditallcomesdowntothefactthatwecanbuildfastelectroniccom-
puters that can effectively carryout algorithms. These computers are basedon
quantummechanicalphysicalsystemsconstrainedtoworkinadiscrete,classical
fashion. Furthermore, miniaturization notwithstanding, these systems involve
the collective behavior of large numbers of particles (electrons), thus relying on
statistical properties of the systems.
Incontrast,inthewouldbequantumcomputers,itistheindividualbehavior
of the particles we have to rely on. This is at the same time the source of the