Quantum Computer on a Turing Machine

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

2. 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

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 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

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 • ‘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)

11. The qubit as a unit ball 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

12. 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 (*) (**)

13. Components “Axes” ........ Hilbert space Quantum Turing tape

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

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. A choice Information 0 Finite (binary) 1 Many Much A cell Values Infinite 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 information 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”

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 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:

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 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

31. 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

32. 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:

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

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, 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

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 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”

43. 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:

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