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Quantum algorithmic information theory
K. Svozil
Institutfu¨r Theoretische Physik
University of Technology Vienna
WiednerHauptstraße 8-10/136
A-1040 Vienna, Austria
5991 e-mail: svozil@tph.tuwien.ac.at
www: http://tph.tuwien.ac.at/svozil
August 15, 2018
e
tcO
5 qait.tex
Abstract
1v5000159/hp-tnauq:viXra
The agenda of quantum algorithmic information theory, ordered ‘top-down,’ is the quan-
tumhaltingamplitude,followedbythequantumalgorithmicinformationcontent,whichinturn
requires the theory of quantum computation. The fundamental atoms processed by quantum
computation are the quantum bits which are dealt with in quantum information theory. The
theory of quantum computation will be based upon a model of universal quantum computer
whose elementary unit is a two-port interferometer capable of arbitrary U(2) transformations.
Basic to all these considerations is quantum theory, in particular Hilbert space quantum me-
chanics.
1 Information is physical, so is computation
The reasoning in constructive mathematics [17, 18, 19] and recursion theory, at least insofar as
their applicability to worldly things is concerned, makes implicit assumptions about the opera-
tionalizability of the entities of discourse. It is this postulated correspondence between practical
and theoretical objects, subsumed by the Church-Turing thesis, which confers power to the for-
mal methods. Therefore, any finding in physics concerns the formal sciences; at least insofar as
they claim to be applicable in the physical universe. In this sense one might quite justifyably
say that the Church-Turing thesis is under permanent physical attack.1 Converseley, any fea-
ture ofthe (constructive ornon-constructive[90]) formalismshould correspondto some physically
operationalizable [22] property.
Hence, any theory of information, if applicable, has to deal with entities which are operational
[22, 60, 57, 58, 62]. In Bridgman’s words (cf. [23], p. V),
“the meaning of one’s terms are to be found by an analysis of the operations which
one performs in applying the term in concrete situations or in verifying the truth of
statements or in finding the answers to questions.”
1 Foranearlydiscussionofthistopic,seeDavis(cf. [29],p. 11):
“ ... how can we ever exclude the possibility of our presented, some day (perhaps by some extrater-
restrial visitors), with a(perhaps extremely complex) device or“oracle” that “computes” a noncom-
putable function?”
AmainthemeofLandauer’sworkhasbeentheconnectionsbetweenphysicsandcomputation;see,forexample,his
1967 article [57] “Wanted: a physically possible theory of physics,” or his more recent survey [60] “Information is
physical.” SeealsoRosen[81]. AsDeutschputsitmorerecently(cf. [31],p. 101),
“The reason why we find it possible to construct, say, electronic calculators, and indeed why we can
perform mental arithmetic, cannot be found in mathematics or logic. The reason is that the laws
of physics ‘happen to’ permit the existence of physical models for the operations of arithmetic such
as addition, subtraction and multiplication. If they did not, these familiar operations would be non-
computable functions. We might still know of them and invoke them in mathematical proofs (which
would presumably be called ‘non constructive’)but we could not perform them.”
1
In particular, the fundamental atom of information, the bit, must be represented by whatever
physicaltheoriesareavailableandmustbeexperimentallyproducibleandmanipulablebywhatever
physical operations are available.
The classical digital computer, at least up to finite resources,seems to be a canonical example
for physical information representation and processing. Classical digital computers, however, are
designed to behave classically. That is, if functioning correctly, certain of their physical states
can be mapped one-to-one onto the set of classical bit states. (This is achieved by appropriately
filtering out noise.) The set of instructions implement the classical propositional calculus and so
on.
In miniaturizing components, however, one encounters limits to the quasi-classical domain.
The alternative is either to stop miniaturization before quantum effects become dominant, or to
take the quantum domain seriously. The latter alternative (at least to the author) seems the only
progressive one, but it results in a head-on collision with long-held classical properties. Several
long-heldassumptionsonthecharacterofinformationhavetobeadapted. Furthermore,theformal
computational techniques in manipulating information have to be revised.
This can be rather negatively perceived as a failure of the old models; but I think that we are
justified to think of it in very posive terms: Physics, in particular quantum physics, stimulates us
tore-considerourconceptions. Wecouldhopethattheoutcomewillbenewtoolsandtechnologies
in computing.
Indeed, right now, we are experiencing an attack on the “Cook-Karp thesis,” putting into
question the robustness of the notion of tractability or polynomial time complexity class with
respect to variations of “reasonable” models of computation. In particular, factoring may require
polynomialtimeonquantumcomputerswithin“reasonablestatistics”[82]. Iwouldsuspectthatit
is wise of mathematiciens and computer scientists to keep aneye onnew developments in physics,
just as we physicists are required to be open for the great advances in the formal sciences.
2 Hilbert space quantum mechanics
“Quantization”has been introduced by Max Planck in 1900 [76]. Planck assumeda discretization
of the total energy U of N linear oscillators (“Resonatoren”), U = Pǫ 0,ǫ,2ǫ,3ǫ,4ǫ,... ,
N N
where P N is zero or a positive integer and ǫ stands for the smallest quan∈ tu{ m of energy. ǫ is} a
0
∈
linear function of frequency ν and proportional to Planck’s fundamental constant h; i.e., ǫ=hν.