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### Quantum Computer on a Turing Machine

Infinite but Converging Computation

VasilPenchev

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

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

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.

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’

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

A classical Turing

tape of bits:

1. Write!

2. Read!

3. Next!

4. Stop!

The list of all

operations on a cell:

A quantum Turing

tape of qubits:

A quantum Turing machine

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

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

The qubit as a 3D ball

- ‘Qubit’ is: where are two complex numbers such that +, and are any two orthonormal vectors (e.g. the orthonormal bases of any two subspaces) in any vector space (e.g. Hilbert space, Euclidean space, etc.)
- A qubit is equivalently representable as a unit ball in Euclidean space and two points, the one chosen within the ball, and the other being the orthogonal projection on its surface, i.e. as a mapping of a unit ball onto its surface (or any other unit sphere)

are two complex numbers:

1

α

+

β

1

, are two orthonormal

vectors or a basis such as two orthogonal

great circles of the unit ball

defines a point of the unit ball

and define a point of the unit sphere

Hilbert space as a “tape” of qubits

Given any point in (complex) Hilbert space as a vector one can replace any successive couple of its components such as (, with a single corresponding qubitsuch that:

;

if are not both 0. However if both are 0 one needs to add conventionally the center ( to conserve the mapping of Hilbert space and an infinite qubit tape to be one-to-one (*) (**)

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

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

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”

Information

0

Finite

(binary)

1

Many

Much

A cell

Values

Infinite

Turing tapes = well orderings:

...

...

...

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

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

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)

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

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

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

as a given and

conventional form

of that carrier

The carrier of

information

The pure

sign

A sign

Read!

An empty cell

WRITE!

0

1

The “material”

The “ideal”

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

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

That disjunction also in the definition of information

The definition (e.g. of Shannon) of classical information delimits the quantity of some information from the number of cells on its carrier: that definition is not invariant to the transformation between the quantity of information and the number of cell for it.

Indeed:

Thus:

And:

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 ...

...

The particle “carries”

the information of all its

properties and quantities:

That is: the set of them

is ‘particle’ or the ‘carrier

of information’

Position

Energy-

momentum

A trajectory

Time

...

...

Energy-

momentum

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

Position

‘Particle’= ‘Carrier’

Space

The quantum case

The classical case

That coincidence and the definition of the quantity of quantum information

- For that coincidence one can suppose that quantum information (unlike classical one) is invariant to the transformation between the quantity of quantum information and the number of cell for it
- That is: One searches for a suitable operation generalizing that of logarithm in a sense so that:

X(B) should mean the number of qubits necessary to store the quantity of information A, and X(A), that of B

That invariance and the definition of quantity in quantum mechanics

- That invariance is a fact in the definition of quantity (observable) in quantum mechanics by a selfadjoint operator (for “Operation X”) in Hilbert space: the interchange between a wave function (for “A”) and its conjugate (for “B”) conserves the value of the quantity:

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

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

Each one can be considered as the “carrier” of the

other: The “carrier” and information are identical

Two dual, complementary qubits

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

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:

Quantum measurement and well-ordering

- The theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen, Specker1968) 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

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

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

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

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 machinehaving 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”

Quantum computer on a tape of qubits

- The offered above visualization of quantum computer as a tape of qubits is about to be used
- Any qubit can be thought as a mapping of any “much” or even of anything, which cannot be counted, into a unit of counting such as a bit of classical information
- Indeed a bit can be interpreted also as a unit of counting being due to:

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

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

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?

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

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

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

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

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

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

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

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

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

A series with infinitely many limit points

...

...

...

...

Limit point m

Qubit m

Limit point n

Qubit n

Limit point p

Qubit p

...

...

...

...

The ultimate result of any quantum

computation exists always!

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

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

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

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

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

References:

Banach, Stefan, Alfred Tarski 1924. “Sur la decomposition des ensembles de points en parties respectivementcongruentes.” FundamentaMathematicae. 6, (1): 244-277.

Deutsch, David 1985. “Quantum theory, the Church-Turing principle and the universal quantum computer,” Proceedings of the Royal Society of London A. 400: 97-117.

Deutsch, David 1989. “Quantum computational networks,” Proceedings of the Royal Society of London. Volume A 425 73-90

Kochen, Simon and Ernst Specker1968. “The problem of hidden variables in quantum mechanics,” Journal of Mathematics and Mechanics. 17 (1): 59-87.

Neumann, Johan von 1932. MathematischeGrundlagen der Quantenmechanik, Berlin: Verlag von Julius Springer.

Skolem, Thoralf 1922. “EinigeBemerkungenzuraxiomatischenBegründung der Mengenlehre. ‒ In: T. Skolem,” in Selected works in logic (ed. E. Fenstad), Oslo: Univforlaget (1970).

Turing, Allen 1937. “On computable numbers, with an application to the Entscheidungsproblem,” Proceedings of London Mathematical Society, series 2. 42 (1): 230-265

Andrew Yao (1993). "Quantum circuit complexity". Proceedings of the 34th Annual Symposium on Foundations of Computer Science. Los Alamos: IEEE Computer Society Press pp. 352–361

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