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Infinite but Converging Computation
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
A “classical” Turing machine realization?
A classical Turing
tape of bits:
The list of all
operations on a cell:
A quantum Turing
tape of qubits:
A quantum Turing machine
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 realization? qubit as a unit ball
are two complex numbers:
, 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
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 (*) (**)
Quantum Turing tape
One bit realization? (a finite choice)
One qubit(an infinite choice)
A choice realization?
Turing tapes = well orderings:
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
The information realization?
as a given and
of that carrier
The carrier of
An empty cell
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 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.
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 ...
... quantum-information carrier
The particle “carries”
the information of all its
properties and quantities:
That is: the set of them
is ‘particle’ or the ‘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
The quantum case
The classical case
X(B) should mean the number of qubits necessary to store the quantity of information A, and X(A), that of B
Each one can be considered as the “carrier” of the quantum information
other: The “carrier” and information are identical
Two dual, complementary qubits
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 ) 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:
The limit, to which it converges, is the result of this quantum computation
That definition raises two questions:
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 unifying cannot be obtained without involving actual infinity
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:
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
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
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 the axiom of choice
Limit point m
Limit point n
Limit point p
The ultimate result of any quantum
computation exists always!
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
Your kind attention!
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