Quantum Computer: Quantum Model and Reality

There are a few most essential questions about the philosophical interpretation of quantum computer:

1. Can a quantum model unlike a classical model coincide with reality?

2. Is reality interpretable as a quantum computer?

3. Can physical processes be understood better and more generally as computations of quantum computer?

4. Is quantum information the real fundament of the world?

5. Does the conception of quantum computer unify physics and mathematics and thus the material and the ideal world?

6. Is quantum computer a non-Turing machine in principle?

7. Can a quantum computation be interpreted as an infinite classical computational process of a Turing machine?

8. Does quantum computer introduce the notion of “actually infinite computational process”?  

Any computer can create a model of reality. The hypothesis that quantum computer can generate such a model designated as quantum, which coincides with the modeled reality, is discussed. Its reasons are the theorems about the absence of “hidden variables” in quantum mechanics. The quantum modeling requires the axiom of choice. The following conclusions are deduced from the hypothesis:

A quantum model unlike a classical model can coincide with reality. Reality can be interpreted as a quantum computer. The physical processes represent computations of the quantum computer. Quantum information is the real fundament of the world. The conception of quantum computer unifies physics and mathematics and thus the material and the ideal world. Quantum computer is a non-Turing machine in principle. Any quantum computing can be interpreted as an infinite classical computational process of a Turing machine. Quantum computer introduces the notion of “actually infinite computational process”.

The hypothesis is consistent with quantum mechanics. The conclusions address a form of neo-Pythagoreanism. Unifying the mathematical and physical, quantum computer is situated in an intermediate domain of their mutual transformations.

References:

1. Kochen, Simon and Ernst Specker {1968) “The problem of hidden variables in quantum mechanics,” Journal of Mathematics and Mechanics 17 (1):   59-87.

2. Neumann, Johan von (1932) Mathematische Grundlagen der Quantenmechanik, Berlin: Verlag von Julius Springer.

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Appendix:

The Metaphysics of Quantum Computing

There are a few closely connected and most essential questions about the philosophical interpretation of quantum computing: 1. Can quantum model unlike a classical model coincide with reality and thus: 2. Can reality be interpreted as a quantum computer? 3. Can physical processes be understood better and more generally as computations of quantum computer? 4. Is quantum information the real fundament of the world? 5. Does the conception of quantum computer unify physics and mathematics and thus the material and the ideal world? 6. Is quantum computer a non-Turing machine in principle? 7. Can quantum computing be interpreted as an infinite classical computational process of a Turing machine? 8. Does quantum computer introduce the notion of “actually infinite computational process”?  

The answers of these questions could be searched in the following directions correspondingly:

1. The theorems about the absence of hidden variables in quantum mechanics (Neumann 1932; Kochen, Specker 1967) can be interpreted as that coincidence. Those alleged hidden variables would be situated in the complement of the quantum model to reality, but that complement has to be an empty set, which means the coincidence of quantum model and reality.

Quantum model requires the axiom of choice in the following sense: It maps any coherent quantum state before measurement with a well-ordered set of measured values. This means that the well-ordering theorem has to have been utilized, and it in turn is equivalent to the axiom of choice. 

2. The answer depends on whether the computation of quantum computer is a quantum model. If the former is a computation in Hilbert space and the latter is a model in Hilbert space, the answer would be positive. The computation in Hilbert space is a generalized computation where the integers are represented as qubits. This can be visualized geometrically thus: any integer is like a point, which can be thought as a degenerate unit ball (with a point on its surface), which is equivalent to a qubit.   

3. The fundamental duality of quantum mechanics or its complementarity in Bohr’s sense generates an analogical and derivative interrelation between any physical process and its computational counterpart of a quantum computer. This admits an intermediate domain between physics and mathematics. Any pair of wave functions corresponds both a physical process and to a quantum computation representable by replacing all bits in a Turing machine with qubits. Wave function represents a completed whole of an infinite number of qubits while quantum computation is the same set of qubits as a successive process: Thus actual and potential infinity turn out to be mapped between each other one-to-one.

4. If quantum information can be defined as a relation or even ratio corresponding to quantities of any physical process and of its quantum-computational counterpart, it can offer a better and more general viewpoint. Its physical sense is to be a quantity for the transformation of energy into some probability distribution like entropy. Energy and probability distribution should refer to one and the same quantum system. So the quantum system can be seen as a “Janus” with a physical and a mathematical “face”, which are complementary. Then quantum information as a quantity unifies both “faces” and their mutual mapping into each other. The physical “face” is depicted as a successive process point after point while the mathematical one is given immediately as a completed whole though different from the former in general.

5. The conception of quantum computer can unify physics and mathematics since it adds a computational and thus mathematical counterpart of any physical process and allows of discussing the energetic value of information or the informational value of energy. It creates a bridge cherished long ago between the material as the physical and the ideal as the mathematical. The mathematical represents the global result, which directs locally the physical process as a quantum computation.

6. Since any result obtained by a Turing machine keeps the fundamental difference between that result and the reality modeled by it, quantum computer under the above conditions should be a non-Turing machine in principle. If a Turing machine can choose between finitely many of alternatives, the quantum computer can do it from an infinite set.  The latter choice directs the former one to be able to reach it as a limit.

7. If the quantum-computational process be projected in a Turing machine, it has to be represented by an infinite Turing computation. The kind of that projection is the same as measurement in quantum mechanics: A coherent whole has to be represented as a finite time series. If that projection is on a finite Turing computation, it will turn out randomly chosen. Nevertheless the result of any quantum computation is representable as that limit, to which an infinite computational process of a Turing machine converges. If that process does not converge, it has at least one quantum counterpart, which converges.

8. Quantum computer introduces necessarily the notion of actual infinity after being projected on a Turing machine as it requires an infinite computational process to be reckoned as completed and as a whole. The actual infinity is representable as a single number, to which the infinite computational process converges. This number can be embedded in the process to direct it to converge.

All these extraordinary features of quantum computing reveal its importance in philosophy at all rather than only in philosophy of mathematics, information, computation. They can be summarized as a form of Pythagoreanism. Furthermore the quantum computer can be interpreted as an infinite series of Turing machines and thus to be investigated the conditions, under which it converges. Consequently that circle of fundamental mathematical and philosophical problems outlined in the beginning allows of being tested in physical or quantum-computational experiments at least in principle.

References:

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.