Quantum Computer
on a Turing Machine
Infinite but Converging Computation
Vasil Penchev
vasildinev@gmail.com, vaspench@abv.bg
http://www.scribd.com/vasil7penchev
http://www.wprdpress.com/vasil7pen...
Quantum computer: mathematical
model or technical realization?
The term of “quantum computer” means both:
1. A mathematica...
Mathematical models: quantum
computer and Turing machine
• Only the mathematical model is meant here and
in comparison wit...
Quantum Turing Machine
• The quantum Turing machine (Deutsch 1985)
is an abstract model computationally
equivalent (Yao 19...
Quantum computer in terms of
‘Turing machine’
• Another way to generalize the Turing machine to
the quantum computer is by...
A “classical” Turing machine
A quantum Turing machine
1 ... n n+1 ...
The
last
cell
A classical Turing
tape of bits:
A qua...
A possible objection about reversibility
• All quantum computations are reversible
unlike the classical ones
• However the...
For what can and for what cannot
that model serve?
That model is intended:
- For elucidating the most general mathematical...
The qubit as a 3D ball
Hilbert space as a “tape” of qubits
Components
“Axes”Hilbert space
Quantum Turing tape
1 ... n n+1 ...
The/No
last
cell
... ...
......
..
Bit vs. qubit
• Then if any bit is an elementary binary choice
between two disjunctive options usually
designated by “0” a...
0
1
0
1
One bit (a finite choice)
One qubit (an infinite choice)
Choice Well-ordering
Qubit & the axiom of choice
• That visualization allows of highlighting the
fundamental difference between the Turing
mach...
Choice and information
• The concept of information can be interpreted as
the quantity of the number of primary choices
• ...
Much Many
Information A choice
Finite
(binary)
Infinite
A cell Values
0
1
...
... ...
Turing tapes = well orderings:
Algorithm and information
• Furthermore the fundamental concept of
choice connects the algorithm to the
information:
• Any...
Information and quantum information
• The generalization from information to
quantum information can be interpreted as
the...
Quantum algorithm and quantum
information
• Obviously the concept of quantum algorithm
should involve infinity unlike the ...
Turing machine and information
• The Turing machine as a general model of
calculation postulates the processing of
informa...
Quantum Turing machine and
quantum information
• The quantum Turing machine processes
quantum information correspondingly ...
Information and information carrier
What is the relation between information and
its carrier, e.g. between an empty cell o...
The “material” The “ideal”
The carrier of
information
The information
as a given and
conventional form
of that carrier
0
1...
The classical disjunction of
information from information carrier
The classical concept of information divides
uncondition...
The classical disjunction of potential
and actual choice
• Furthermore it separates disjunctively the option
of choice (th...
That disjunction also in the definition
of information
The coincidence of quantum information
and quantum-information carrier
All those classical demarcations are removed in
qua...
The quantum case The classical case
The particle “carries”
the information of all its
properties and quantities:
That is: ...
That coincidence and the definition of
the quantity of quantum information
That invariance and the definition
of quantity in quantum mechanics
Quantity in quantum mechanics and quan-
tum computation: a process and a result
• Thus any quantity in quantum mechanics c...
The “tape” of a quantum Turing machine
• As an illustration, the tape of quantum Turing
machine coincides with the written...
Two dual, complementary qubits
Each one can be considered as the “carrier” of the
other: The “carrier” and information are...
The concept of quantum invariance
• The term of “quantum invariance” can be coined
to outline the important role assigned ...
The justification of quantum invariance
That principle of quantum invariance is quite not
obvious and even contradicts “co...
Quantum measurement and well-
ordering
• The theorems about the absence of hidden
variables in quantum mechanics (Neumann
...
Quantum reality vs. orderablity
• Furthermore quantum reality according to the
cited theorems is not well-orderable in pri...
Quantum model vs. quantum reality:
the axiom of choice
• Thus the relation between quantum model
and quantum reality requi...
Quantum invariance and Skolem’s
“paradox”
• That quantum invariance is well known in
mathematics in the form of Skolem’s p...
Quantum invariance: quantum
computer on a Turing machine
• Quantum invariance as to quantum computer can
be exhaustedly de...
Quantum computer on a tape of qubits
A single qubit by a Turing machine
• Any qubit of it being a choice of one between
a continuum of disjunctive options can ...
Quantum computation: infinite but
convergent
• Given all that, any quantum computational
process can by defined in terms o...
The result of quantum computation
The limit, to which it converges, is the result of
this quantum computation
That definit...
Quantum computation and actual infinity
Quantum computation involves the notion of actual
infinity since the computational...
Quantum algorithm & quantum result
• As the model of a Turing machine unifies the
utilized algorithm with the result obtai...
The coincidence of reality and
quantum computation
• If its objectivity is to model a concrete reality by
the computed ult...
The coincidence of quantum model
and reality
• One can state that quantum computer
calculates reality or that quantum mode...
The most general case of infinitely
many limit points
The offered model of quantum computer on a Turing
machine as a conve...
The axiom of choice and the limit points
One can use the granted above axiom of choice to
order the limit points even bein...
The physical and philosophical meaning
of Hilbert space by the axiom of choice
• The axiom of choice can be used in anothe...
Qubit as a limit point of a Turing machine
Any qubit represents equivalently a limit point
of the “tape” of the Turing mac...
The “axes” of Hilbert space as qubits
If those limit points are even infinitely many, they
can be represented equivalently...
... ... ... ...
Limit point m
Qubit m
Limit point n
Qubit n
Limit point p
Qubit p
A series with infinitely many limit poin...
Wave function as quantum
computation
• Then obviously any change of the state of any
quantum system being a wave function ...
The axiom of choice on a bounded
set of limit points
Using the axiom of choice, one can always reorder
monotonically a bou...
Quantum vs. standard computer
• The model of quantum computer on a Turing
machine allows of clarifying the sense and
meani...
Quantum vs. standard computer:
tendency & image vs. result as a value
• While the standard computer gives a result,
the qu...
Quantum computer and human
understanding and interpretation
• The transition from the result of a usual
computer to the ul...
End
Thank You
for
Your kind attention!
References:
Banach, Stefan, Alfred Tarski 1924. “Sur la decomposition des ensembles de points en
parties respectivement co...
Quantum Computer on a Turing Machine
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Quantum Computer on a Turing Machine

  1. 1. Quantum Computer on a Turing Machine Infinite but Converging Computation
  2. 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. 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. 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. 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. 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. 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. 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. 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. 10. The qubit as a 3D ball
  11. 11. Hilbert space as a “tape” of qubits
  12. 12. Components “Axes”Hilbert space Quantum Turing tape 1 ... n n+1 ... The/No last cell ... ... ...... ..
  13. 13. 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
  14. 14. 0 1 0 1 One bit (a finite choice) One qubit (an infinite choice) Choice Well-ordering
  15. 15. 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
  16. 16. 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”
  17. 17. Much Many Information A choice Finite (binary) Infinite A cell Values 0 1 ... ... ... Turing tapes = well orderings:
  18. 18. 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
  19. 19. 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
  20. 20. 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)
  21. 21. 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
  22. 22. 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
  23. 23. 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
  24. 24. The “material” The “ideal” The carrier of information The information as a given and conventional form of that carrier 0 1 An empty cell
  25. 25. 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
  26. 26. 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
  27. 27. That disjunction also in the definition of information
  28. 28. 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 ...
  29. 29. 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
  30. 30. That coincidence and the definition of the quantity of quantum information
  31. 31. That invariance and the definition of quantity in quantum mechanics
  32. 32. 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
  33. 33. 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
  34. 34. Two dual, complementary qubits Each one can be considered as the “carrier” of the other: The “carrier” and information are identical
  35. 35. 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
  36. 36. 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:
  37. 37. 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
  38. 38. 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
  39. 39. 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
  40. 40. 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
  41. 41. 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”
  42. 42. Quantum computer on a tape of qubits
  43. 43. 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
  44. 44. 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
  45. 45. 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?
  46. 46. 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
  47. 47. 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
  48. 48. 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
  49. 49. 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
  50. 50. 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
  51. 51. 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
  52. 52. 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
  53. 53. 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
  54. 54. 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
  55. 55. ... ... ... ... 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!
  56. 56. 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
  57. 57. 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
  58. 58. 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
  59. 59. 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
  60. 60. 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
  61. 61. End Thank You for Your kind attention!
  62. 62. References: Banach, Stefan, Alfred Tarski 1924. “Sur la decomposition des ensembles de points en parties respectivement congruentes.” Fundamenta Mathematicae. 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 Specker 1968. “The problem of hidden variables in quantum mechanics,” Journal of Mathematics and Mechanics. 17 (1): 59-87. Neumann, Johan von 1932. Mathematische Grundlagen der Quantenmechanik, Berlin: Verlag von Julius Springer. Skolem, Thoralf 1922. “Einige Bemerkungen zur axiomatischen Begründung 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. pp. 352–361

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