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Soft Computing

Fuzzy Tsetlin automata and their application in analog computing

Eugene Kagan, Alexander Rybalov

Published Apr 29, 2026• 0 citations
Publisher

Abstract

Abstract In the paper we suggest a system of fuzzy Tsetlin automata for analog computations. The system deals with fuzzy analogs of qubits and models operations of quantum gates used for quantum computations. In contrast to the existing models, the suggested system directly emulates reversible operators and the “square root of not” operator which is a key operator in quantum computations. Fuzzy information units called f-bits are defined by matrix with fuzzy normalized rows and emulate qubits with complex elements. Operations with these units are conducted by fuzzy Tsetlin automata based on the uninorm and absorbing norm. It is demonstrated that the suggested system emulates basic quantum computations; hence, it can implement analog computations and can support analog emulations of quantum algorithms.

DOI
10.1007/s00500-026-11303-0

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Abstract In the paper we suggest a system of fuzzy Tsetlin automata for analog computations. The system deals with fuzzy analogs of qubits and models operations of quantum gates used for quantum computations. In contrast to the existing models, the suggested system directly emulates reversible operators and the “square root of not” operator which is a key operator in quantum computations. Fuzzy information units called f-bits are defined by matrix with fuzzy normalized rows and emulate qubits with complex elements. Operations with these units are conducted by fuzzy Tsetlin automata based on the uninorm and absorbing norm. It is demonstrated that the suggested system emulates basic quantum computations; hence, it can implement analog computations and can support analog emulations of quantum algorithms.
An approach for emulating quantum circuits using conventional analog hardware is presented based on the intuitive similarity between fuzzy logic and quantum superposition, as well as some geometrical analogies. This approach has the advantage of being easy to implement on dedicated hardware for parallel processing of membership functions and fuzzy inference, compared to conventional quantum computing, which requires quantum mechanical systems which are extremely sensitive to noise and difficult to extend to large scale systems. Using geometrical analogies and a suitable transformation, qubits are modeled as pairs of fuzzy membership functions evolving on the unit square and basic one qubit gates are modeled as transformations on this unit square. A fuzzy implementation of the one bit and two bit Deutsch-Jozsa algorithm is proposed. Physical implementation and advantages, such as the possibility of implementing nonlinear or non-unitary gates, as well as drawbacks of the proposed model compared to conventional quantum computing are shown.
The concept of fuzzy qubit is proposed as an approach to emulating quantum interference using fuzzy logic. This makes it possible to implement certain quantum computing algorithms, especially quantum associative memory, on hardware dedicated to fuzzy logic. The fuzzy qubit model is briefly reviewed and the fuzzy implementation of basic 1 and 2 qubit gates, such as Hadamard or C-not gates, is presented. The possibility of emulating quantum algorithms is discussed. Then quantum associative memory is presented and its implementation using the fuzzy qubit model is discussed. The advantages over the previous associative memory models, both quantum and classical, are discussed.
The theory of logical gates in quantum computation has suggested new forms of quantum logic, called quantum computational logics. The basic semantic idea is the following: the meaning of a sentence is identified with a quregister (a system of qubits in a pure state) or, more generally, with a mixture of quregisters (called qumix). Following an approach proposed by Domenech and Freytes, we apply residuated structures associated with fuzzy logic to develop certain aspects of information processing in quantum computing from a logical perspective. For this purpose, we introduce an axiomatic system whose natural interpretation is the irreversible quantum Poincaré algebra. Such a system allows to establish a completeness theorem for the treatment of quantum information.
Abstract In this paper, a novel method for the emulation of qutrit and ququart based quantum computers is proposed. The Hilbert space and Bloch sphere representations of qutrits are initially discussed, preceding the emulation of a single qutrit using a classical system. The signal representing the state of the quantum system is defined as the sum of quadrature and in-phase 2-dimensional space-varying signals, which represent the basis states of the quantum system. Focus is mainly conferred on the frequency-domain representation of the signal, due to an innate bijective mapping found between the dimensions in the Hilbert space and the frequencies of the signal. An m -qutrit addressing scheme is then proposed which is used in the application of quantum ternary gates and measurement operations. The proposed theory is then used to simulate a ternary version of the Deutsch-Jozsa algorithm on MATLAB. This paper also includes a section on the extension of the theory to the case of ququarts. It concludes with a brief introduction to a potential system for emulation, and lists algorithms that can be implemented on the proposed system.
We construct a completely analog framework that emulates universal quantum gates and quantum algorithms. It is based on electronic circuits made of operational amplifiers, resistors and capacitors. In these circuits, input and output lines represent the computational basis states (CBSs) and thus 2n2^n2n lines are required to represent nnn qubits. An operation of the circuits is based on classical evolution and interference of complex amplitudes associated with each CBS. The framework can emulate entangled states and is free from decoherence, measurements are classical and do not collapse states. Similar to physical quantum computers, emulated quantum algorithms can be constructed as a sequence of the gates belonging to a universal set (phase shift, Hadamard, controlled-NOT), as a unitary matrix and as combinations of the two. Circuits representing the universal gates have been made and tested. We also have made a matrix-based emulator of a 3-qubit Grover search algorithm. We tested it by searching for one and two particular states with one and two iterations and found that its outputs accurately match predicted values. We anticipate that the emulators can work as sub-components of physical quantum computers. On their own, the emulators can be used for operations that require a few qubits or operations that can be split into independent (and perhaps weakly entangled) blocks of qubits.