1. Quantum Computer
on a Turing Machine
Infinite but Converging Computation

2. Vasil Penchev
vasildinev@gmail.com, vaspench@abv.bg
http://www.scribd.com/vasil7penchev
http://www.wprdpress.com/vasil7penchev
CV: http://old-philosophy.issk-bas.org/CV/cv-
pdf/V.Penchev-CV-eng.pdf

3. Quantum computer: mathematical
model or technical realization?
The term of “quantum computer” means both:
1. A mathematical model like a Turing machine,
which is the general model of any usual
computer we use, and:
2. Any concrete technical realization involving
the laws of quantum mechanics to implement
computations

4. Mathematical models: quantum
computer and Turing machine
• Only the mathematical model is meant here and
in comparison with that of a standard computer,
namely a Turing machine (Turing 1937)
• That mathematical model raises a series of
philosophical questions about model and
quantum model, quantum model and reality,
infinity and even actual infinity as a physical
entity, computational and physical process,
information and quantum information,
information and its carrier, etc.

5. Quantum Turing Machine
• The quantum Turing machine (Deutsch 1985)
is an abstract model computationally
equivalent (Yao 1993) to the quantum circuit
(Deutsch 1989) and can represent all features
of quantum computer without entanglement
• Deitsch (1985) did not use the notion of
‘qubit’ to define ‘quantum Turing machine’

6. Quantum computer in terms of
‘Turing machine’
• Another way to generalize the Turing machine to
the quantum computer is by replacing all bits or
cells of a Turing tape with “quantum bits” or
“qubits”
• Then all admissible operations on a cell of the
quantum tape are generalized to those two:
“write/ read a value of a qubit” just as “write/ read
a value of a bit” on the tape of a classical Turing
machine
• There are not other generalizations from a Turing
machine to a quantum one in that model: All the
rest is the same

7. A “classical” Turing machine
A quantum Turing machine
1 ... n n+1 ...
The
last
cell
A classical Turing
tape of bits:
A quantum Turing
tape of qubits:
1 ... n n+1 ...
The
/No
last
cell
The list of all
operations on a cell:
1. Write!
2. Read!
3. Next!
4. Stop!

8. A possible objection about reversibility
• All quantum computations are reversible
unlike the classical ones
• However the input/ output of a value in a
qubit is irreversible
• Thus a quantum Turing machine is not
reversible just as a classical one
• Quantum reversibility is “bracketed” and
“hidden” by the non-constructiveness of the
choice of a value for the axiom of choice

9. For what can and for what cannot
that model serve?
That model is intended:
- For elucidating the most general mathematical
and philosophical properties of quantum computer
or computation
- For their comparison with those of a classical
computer or computation
That model cannot serve to design any technical
realization of quantum computer just as the true
machine of Turing cannot as to a standard
computer

10. The qubit as a 3D ball

12. Hilbert space as a “tape” of qubits

13. Components
“Axes”Hilbert space
Quantum Turing tape
1 ... n n+1 ...
The/No
last
cell
... ...
......
..

14. Bit vs. qubit
• Then if any bit is an elementary binary choice
between two disjunctive options usually
designated by “0” and “1”, any qubit is a choice
between a continuum of disjunctive options as
many as the points of the surface of the unit ball:
• Thus the concept of choice is the core of
computation and information. It is what can unify
the classical and quantum case, and the
demarcation between them is the bound between
a finite vs. infinite number of the alternatives of
the corresponding choice

15. 0
1
0
1
One bit (a finite choice)
One qubit (an infinite choice)
Choice Well-ordering

16. Qubit & the axiom of choice
• That visualization allows of highlighting the
fundamental difference between the Turing
machine and quantum computer: the choice
of an element of an uncountable set
necessarily requiring the axiom of choice
• The axiom of choice being non-constructive is
the relevant reference frame to the concept of
quantum algorithm to involve a constructive
process of solving or computation having an
infinite and even uncountable number of
steps

17. Choice and information
• The concept of information can be interpreted as
the quantity of the number of primary choices
• Furthermore the Turing machine either classical
or quantum as a model links computation to
information directly:
• The quantity of information can be thought as the
sum of the change bit by bit or qubit by qubit, i.e.
as the change of number written by two or
infinitely many digits
• Thus: a cell of a (quantum) Turing tape =
a choice of (quantum) information = a “digit”

18. Much Many
Information A choice
Finite
(binary)
Infinite
A cell Values
0
1
...
... ...
Turing tapes = well orderings:

19. Algorithm and information
• Furthermore the fundamental concept of
choice connects the algorithm to the
information:
• Any algorithm either classical or quantum is a
well-ordered series of choices:
• The quantity of information either classical or
quantum is the quantity of those choices in
units of primary choices: either bits or qubits
• In general the quantity of information does
not require the set of choices to be well-
ordered

20. Information and quantum information
• The generalization from information to
quantum information can be interpreted as
the corresponding generalization of ‘choice’:
from the choice between two (or any finite
number of) disjunctive alternatives to
infinitely many alternatives
• Thus the distinction between the classical and
quantum case can be limited within any cell of
an algorithm or (qu)bit of information

21. Quantum algorithm and quantum
information
• Obviously the concept of quantum algorithm
should involve infinity unlike the classical one
• Furthermore that infinity should be actual since
quantum algorithm can process an infinite
number of alternatives per a finite period of time
unlike a classical one needing an infinite time for
that aim
• Nevertheless the quantity of quantum
information in a quantum algorithm can have a
finite value being measured in qubits, i.e. in
“units of infinity” (figuratively said)

22. Turing machine and information
• The Turing machine as a general model of
calculation postulates the processing of
information bit by bit serially
• The processing is restricted to a few, exactly
defined operations stereotyped on any cell
(bit)
• Thus the Turing machine is designed to
represent any algorithm as the serial
processing of the primary units of
information: Information underlies algorithm
by that model

23. Quantum Turing machine and
quantum information
• The quantum Turing machine processes
quantum information correspondingly qubit
by qubit serially but in parallel within any
qubit, and the axiom of choice formalizes that
parallel processing as the choice of the result
• Even the operations on a qubit can be the
same as on a bit. The only difference is for
“write/ read”: to be a value of either a binary
(finite) or an infinite set

24. Information and information carrier
What is the relation between information and
its carrier, e.g. between an empty cell of the
tape and the written on it?
The classical notion of information or algorithm
separates them disjunctively from their
corresponding carriers.
The Turing machine model represents that
distinction by an empty cell, on the one
hand, and the set of values, which can be
written on it, or a given written value, on the
other hand

25. The “material” The “ideal”
The carrier of
information
The information
as a given and
conventional form
of that carrier
0
1
An empty cell

26. The classical disjunction of
information from information carrier
The classical concept of information divides
unconditionally information from its carrier and
excludes information without some energetic or
material carrier:
Information obeys the carrier: no information
without its carrier: Information needs something
with nonzero energy, on which is written or from
which is read. Otherwise it cannot exist
OK, but all this refers to the classical
information, not to the quantum one. One can call
the latter emancipated information

27. The classical disjunction of potential
and actual choice
• Furthermore it separates disjunctively the option
of choice (the set of possible values) from the
chosen alternative of choice (e.g. either “0” or
“1”) and thus the possible or potential from
the real or actual
• The act of choice is the demarcation between
“virtuality” and reality. That act is irreversible.
Thus it creates a well-ordering of successive
choices just because of irreveresibility

28. That disjunction also in the definition
of information

29. The coincidence of quantum information
and quantum-information carrier
All those classical demarcations are removed in
quantum information:
It coincides with its carrier
Potential and actual choice merge
The empty cells and the written on them are
interchangeable (as a basis and as a vector in an
orthonormal vector space like Hilbert space)
However all this contradicts our prejudices
borrowed from “common sense”: so much the
worse for the prejudices ...

30. The quantum case The classical case
The particle “carries”
the information of all its
properties and quantities:
That is: the set of them
is ‘particle’ or the ‘carrier
of information’
Space
Time A trajectory
‘Particle’= ‘Carrier’
The ‘particle’ is split into
two complementary sets
of properties, each of
which can be as if the
carrier of the other. Their
interchange is identical
...
...
...
...
Energy-
momentum
Position

31. That coincidence and the definition of
the quantity of quantum information

32. That invariance and the definition
of quantity in quantum mechanics

33. Quantity in quantum mechanics and quan-
tum computation: a process and a result
• Thus any quantity in quantum mechanics can be
interpreted as a quantity of quantum information
and as quantum computation, and its value as the
result of that computation
• Indeed (in more detail, see Slide 10), any point in
Hilbert space (= a wave function) is equivalent to a
quantum Turing state, and the selfadjoint operator
is what conserves the sequence of qubits changing
their values. Thus the action of a selfadjoint
operator is equivalent to the change of the
quantum Turing state, i.e. to a quantum
computation

34. The “tape” of a quantum Turing machine
• As an illustration, the tape of quantum Turing
machine coincides with the written on it: Any
quantum Turing machine calculating should create
itself in a sense
• More exactly, if one transforms one qubit dually
(i.e. one empty cell from the basis and its value
interchange their positions), it will coincide with
the initial one: Any quantum Turing cell and the
written on it are one and the same in this sense of
invariance to interchange

35. Two dual, complementary qubits
Each one can be considered as the “carrier” of the
other: The “carrier” and information are identical

36. The concept of quantum invariance
• The term of “quantum invariance” can be coined
to outline the important role assigned to the
axiom of choice in the theory of quantum
computer and inherited from quantum
mechanics:
• Quantum invariance means the following principle
as to quantum computation:
The result chosen by the axiom of choice is the
same as the result of the corresponding quantum
algorithm. Or: the non-constructive choice and
the quantum-constructive choice coincide and can
be accepted as one and same

37. The justification of quantum invariance
That principle of quantum invariance is quite not
obvious and even contradicts “common sense”: It can
obtain relevant foundation from quantum mechanics
and quantum measurement:
Quantum measure underlies quantum measurement:
It is a fundamentally new kind of measure, which
transfers Skolem’s “relativity of ‘set’” (1922 *1970+)
into the theory of measure as that measure, to which
a “much” and a “many” are relative and can share it
and thus measured jointly
The justification of quantum invariance is as follows:

38. Quantum measurement and well-
ordering
• The theorems about the absence of hidden
variables in quantum mechanics (Neumann
1932; Kochen, Specker 1968) exclude any well-
ordering before measurement
• However the results of the measurements are
always well-ordered and thus any quantum
model implies the well-ordering theorem
equivalent to the axiom of choice

39. Quantum reality vs. orderablity
• Furthermore quantum reality according to the
cited theorems is not well-orderable in principle
• So if one measures the unorderable quantum
reality, one needs quantum measure to be able
to unify the measured and the results of
measurement:
• Quantum reality is always a “much” versus the
“many” of the measured results: Quantum
measure is only what can unify them and
underlies quantum invariance about all
measurable by it

40. Quantum model vs. quantum reality:
the axiom of choice
• Thus the relation between quantum model
and quantum reality requires correspondingly
the axiom of choice and its absence, or the
coined quantum invariance, to designate that
extraordinary relation between model and
reality specific to quantum mechanics and
trough it, to the theory of quantum computer:
• Quantum computation coincides with physical
process and thus with reality

41. Quantum invariance and Skolem’s
“paradox”
• That quantum invariance is well known in
mathematics in the form of Skolem’s paradox
(Skolem 1922 [1970]: ), who has introduced the
notion of “relativity” as to set theory discussing
infinity
• He even spoke that the notions of finite and infinite
set are relative and interchangeable (ibid.: [143-
144]) and the so-called “paradox” of Skolem can
comprise finite sets, too. Thus he is the immediate
predecessor of the concept of quantum measure

42. Quantum invariance: quantum
computer on a Turing machine
• Quantum invariance as to quantum computer can
be exhaustedly described by the mapping of
quantum computer on a Turing machine having an
infinite tape in general
• That mapping is always possible to be one-to-one
just because of the axiom of choice
• Quantum invariance means for that mapping to be
one-to-one
• Furthermore the unit of quantum measure can be
defined as that “one-to-one” of two heterogeneous
quantities like a “much” and a “many”

43. Quantum computer on a tape of qubits

44. A single qubit by a Turing machine
• Any qubit of it being a choice of one between
a continuum of disjunctive options can be
replaced by a Turing machine (possibly with a
tape consisting of infinitely many cells)
utilizing the axiom of choice for replacing
• However the qubit itself as the unit of
quantum measure can be considered as any
one-to-one mapping of anything into a bit of
information
• Thus quantum information can mean the
equivalent mapping of anything into classical
information

45. Quantum computation: infinite but
convergent
• Given all that, any quantum computational
process can by defined in terms of a standard
one on a Turing machine as infinite but
convergent
• Consequently ‘quantum computer’ is that
extension of ‘Turing machine’, which
comprises infinite computational
processes, which are only infinite “loops” for a
Turing machine without any result

46. The result of quantum computation
The limit, to which it converges, is the result of
this quantum computation
That definition raises two questions:
• Does any series representing a quantum
computation converge and thus: Is the
existence of a limit point always guaranteed?
• Is that generalization of computation to
comprise infinite ones is only possible? Or in
other words: Is quantum and infinite
computation one and the same and does they
map to each other one-to-one?

47. Quantum computation and actual infinity
Quantum computation involves the notion of actual
infinity since the computational series is both
infinite and considered as a completed whole by
dint of its limit
Furthermore quantum computation unifies both
definitions of ‘function”:
• That as a constructive and thus computational
process
• That as a mapping of a set into another under
condition of a single image in the latter
That unifying cannot be obtained without involving
actual infinity

48. Quantum algorithm & quantum result
• As the model of a Turing machine unifies the
utilized algorithm with the result obtained by
it, quantum computer can be interpreted both as
a convergently advanced algorithm and a
convergently improved result for the former
• Quantum computer extends that equivalence of
algorithm and calculation to the
interchangeability of an “atom” of data (a qubit)
and the “atomic” operations on it:
• This is due to the interchangeability of quantum
information and its carrier as well as that of
computational and physical process

49. The coincidence of reality and
quantum computation
• If its objectivity is to model a concrete reality by
the computed ultimate result, it coincides with
reality unlike any standard Turing machine which
has to be finite and thus there is always a finite
difference between the computed reality and any
completed result of a Turing computation
• Quantum epistemology should be defined as
studying the discrete or computational hypostasis
of reality rather than the relation of cognition and
reality after cognition and reality have coincide

50. The coincidence of quantum model
and reality
• One can state that quantum computer
calculates reality or that quantum model and
reality coincide
• All classical epistemology assumes that there
is an irremovable essential difference between
any model and reality: No model can coincide
with reality and epistemology is that science,
which studies that difference. Consequently
that mismatch is the subject of classical
epistemology enabling it

51. The most general case of infinitely
many limit points
The offered model of quantum computer on a Turing
machine as a convergent and infinite process
comprises the more general case where that infinite
process does not converge and even has infinitely
many limit points
This is due to quantum invariance, which allows of two
equivalent “hypostases” of quantum computation:
The one is expanded, without the axiom of choice
being unorderable in principle
The other is compacted, well-ordered by the axiom
of choice and thus converging

52. The axiom of choice and the limit points
One can use the granted above axiom of choice to
order the limit points even being infinitely many as
a monotonic series, which necessarily converges if
it is a subset of any finite interval, and to accept
this last limit as the ultimate result of the quantum
computer
Consequently quantum invariance underlain by all
quantum mechanics is what guarantees that any
quantum computation has a single result, and thus
it unlike a Turing machines in general is complete

53. The physical and philosophical meaning
of Hilbert space by the axiom of choice
• The axiom of choice can be used in another
way to give the same result thus elucidating
the physical and even philosophical meaning
of Hilbert space, the basic mathematical
structure of quantum mechanics:
• Hilbert space is that common space where all
measured by quantum measure can be in one
place together co-existing: It allows of any
unorderable quantum “much” and its image
of a “many” to be seen as one and the same

54. Qubit as a limit point of a Turing machine
Any qubit represents equivalently a limit point
of the “tape” of the Turing machine, on which
the quantum computer is modeled
That qubit or that limit point can be expanded
into a series of qubits (i.e. a subspace of Hilbert
space) or to a series, which converges to this
limit point
The axiom of choice implies that “reverse
action” as above: Indeed, given the set of all
series converging to a limit point, it enables a
series to be chosen from it

55. The “axes” of Hilbert space as qubits
If those limit points are even infinitely many, they
can be represented equivalently by a point in
Hilbert space where any “axis” of it corresponds
one-to-one to a qubit ant thus to a limit point of the
quantum computational process (see Slide 10)
So any limit point corresponds one-to-one to a
subspace of Hilbert space, and any that one can be
compacted into a single qubit by the axiom of
choice. The same compacting as to a series means
to be chosen its limit point to represent all series

56. ... ... ... ...
Limit point m
Qubit m
Limit point n
Qubit n
Limit point p
Qubit p
A series with infinitely many limit points
... ... ......
The ultimate result of any quantum
computation exists always!

57. Wave function as quantum
computation
• Then obviously any change of the state of any
quantum system being a wave function and a
point in Hilbert space can be interpreted as a
quantum calculative process, and the physical
world as a whole as an immense quantum
computer
• The concept of computation and physical
reality converge to each other at the point
visible from quantum mechanics

58. The axiom of choice on a bounded
set of limit points
Using the axiom of choice, one can always reorder
monotonically a bounded set of limit points to
converge or represent a point in Hilbert space as a
single qubit by the Banach-Tarski paradox (Banach,
Tarski 1924):
Both are only different images of one and the same
quantum computation:
The one is compacted into a qubit or reordered as a
converging series
The other is expanded as Hilbert space (a converging
vector in it) or as an arbitrary series non-converging,
non-reordered, but reorderable in principle

59. Quantum vs. standard computer
• The model of quantum computer on a Turing
machine allows of clarifying the sense and
meaning of a quantum computation in terms
of a usual computer equivalent to some finite
Turing machine:
• It generalizes the notion from finite to infinite
and even to actual infinite computation.
Furthermore it allows of comparing between a
standard and a quantum computer on the
distinction of the finite/ infinite

60. Quantum vs. standard computer:
tendency & image vs. result as a value
• While the standard computer gives a result,
the quantum computer offers a tendency
comprising a potentially infinite sequence of
converging algorithms and results as well as
the limit of this tendency both as an ultimate
algorithm-result coinciding with reality and as
an image (“Gestalt”) of the tendency as a
completed whole
• Thus quantum computation generalizes the
finite calculation in a way close to human
understanding and interpretation

61. Quantum computer and human
understanding and interpretation
• The transition from the result of a usual
computer to the ultimate result of a quantum
computer is a leap comparable with human
understanding and interpretation to restore the
true reality on the base of a finite set of sensual
or experimental data
• One can rise the question whether that
comparison is only a metaphor or it reveals a
deeper link between quantum computation and
the human understanding and interpretation of
reality

62. End
Thank You
for
Your kind attention!

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mechanics,” Journal of Mathematics and Mechanics. 17 (1): 59-87.
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