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Journal of Artificial Intelligence & Cloud Computing

The Foundations and Results of Parallel Computing in Terms of Gaisi Takeutis’ Quantum Set Theory

Miklós Banai

Published Apr 2, 2026• 0 citations
Publisher

Abstract

The foundations and results of parallel computing in terms of Gaisi Takeutis’ quantum set theory are presented in this talk. This approach is based on • The extension of the von Neumann’s uniqueness theorem for the solutions of the canonical commutations relations in quantum mechanics of finitely many degrees of freedom to quantum local field theories of infinitely many degrees of freedom; • The results of the mathematician Gaisi Takeuti about quantum set theory in extension theory; and • The results of the physicist Lev Davidovics Landau about the behavior of the quantum liquids He3 and He4 resulting in a more general approach than the one based on the notion of the quantum bit.

DOI
10.47363/jaicc/icadccs2026/2026(5)19

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The foundations and results of parallel computing in terms of Gaisi Takeutis’ quantum set theory are presented in this talk. This approach is based on • The extension of the von Neumann’s uniqueness theorem for the solutions of the canonical commutations relations in quantum mechanics of finitely many degrees of freedom to quantum local field theories of infinitely many degrees of freedom; • The results of the mathematician Gaisi Takeuti about quantum set theory in extension theory; and • The results of the physicist Lev Davidovics Landau about the behavior of the quantum liquids He3 and He4 resulting in a more general approach than the one based on the notion of the quantum bit.
The foundations and results of parallel computing in terms of Gaisi Takeutis’ quantum set theory are presented in this review paper. This approach is based on 1) the extension of the von Neumann’s uniqueness theorem for the solutions of the canonical commutations relations in quantum mechanics of finitely many degrees of freedom to quantum local field theories of infinitely many degrees of freedom [2]; 2) the results of the mathematician Gaisi Takeuti about quantum set theory in extension theory; and 3) the results of the physicist Lev Davidovics Landau about the behavior of the quantum liquids He3 and He4resulting in a more general approach than the one based on the notion of the quantum bit [1-5].
In this letter we discuss computing in terms of Gaisi Takeuti's quantum set theory using an explicit example of a rigid body of cuboid form. The universe V(L) of Takeuti will be determined. A set of real numbers in this Universe are also explicitly described including a set of binary numbers. Thus we arrive at the foundations of von Neumann' theory of computing in terms of ordinary binary numbers. Then we see that this extension of computing to the Universe V(L) provides a sound, mathematically well defined theory of quantum computing.
The term quantum parallelism is commonly used to refer to a property of quantum computations where an algorithm can act simultaneously on a superposition of states. However, this is not the only aspect of parallelism in quantum computing. Analogously to the classical computing model, every algorithm consists of elementary quantum operations and the application of them could be parallelised itself. This kind of parallelism is explored in this thesis in the one way quantum computing (1WQC) and the quantum circuit model. In the quantum circuit model we explore arithmetic circuits and circuit complexity theory. Two new arithmetic circuits for quantum computers are introduced in this work: an adder and a multiply-adder. The latter is especially interesting because its depth (i.e. the number of parallel steps required to finish the computation) is smaller than for any known classical circuit when applied sequentially. From the complexity theoretical perspective we concentrate on the classes QAC0 and QAC0[2], the quantum counterparts of AC0 and AC0[2]. The class AC0 comprises of constant depth circuits with unbounded fan-in AND and OR gates and AC0[2] is obtained when unbounded fan-in parity gates are added to AC0 circuits. We prove that QAC0 circuits with two layers of multi-qubit gates cannot compute parity exactly. This is a step towards proving QAC0 6= QAC0[2], a relation known to hold for AC0 and AC0[2]. In 1WQC, computation is done through measurements on an entangled state called the resource state. Two well known parallelisation methods exist in this model: signal shifting and finding the maximally delayed general flow. The first one uses the measurement calculus formalism to rewrite the dependencies of an existing computation, whereas the second technique exploits the geometry of the resource state to find the optimal ordering of measurements. We prove that the aforementioned methods result in same depth computations when the input and output sizes are equal. Through showing this equivalence we reveal new properties of 1WQC computations and design a new algorithm for the above mentioned parallelisations.
We discuss the q deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in Quantum Mechanics. We show that the q-deformation parameter labels the Weyl systems in Quantum Mechanics and the unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. The von Neumann theorem in Quantum Mechanics (QM) states that for systems with finite number of degrees of freedom the representations of the canonical commutation relations (ccr), also called the Weyl representations, are each other unitarily equivalent. In the limit of infinite number of degrees of freedom, namely in Quantum Field Theory