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International Multi-Conference on Industrial Engineering and Modern Technologies (FarEastCon)

Fuzzy Quantum Control

D. V. Chernyshev

Published Oct 1, 2018• 1 citation

Abstract

The article presents a method of fuzzy quantum control and the structure of a fuzzy quantum controller. The authors suggest a method to represent the membership function of fuzzy variables as quantum values. A fuzzy quantum controller calculates the parameters for the optimum control of a quantum system on the basis of dynamic programming. The control task is the transfer of the system from its initial state into a required final state with a set merit functional. At the setting of the fuzzy quantum controller the authors use expert data. The input variable is transformed into fuzzy values and the membership functions are created. The membership functions are represented as quantum values. Further, the controlled system equation is derived. The authors set up a merit functional for optimal regulation and apply the optimal control method. The authors present a circuit to connect the fuzzy quantum controller to a control system.

DOI: 10.1109/fareastcon.2018.8602606

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In this work we consider a navigation of quantum-controlled mobile robot by the use of fuzzy methods. A fuzzy model of the robot behavior is obtained by transformation of the robot's states (that appear in a form of qubits) into the pairs of membership functions. For a full description of the robot's turns, we define a reverse fuzzy Hadamard operator that completes the known Hannachi, Hatakeyama and Hirota model. The suggested transformations and methods were comparatively studied by field trials using small mobile robots and by numerical simulations that model the robot's behavior in the presence of noise.

The goal of this study is to integrate fuzzy mathematics and quantum mechanics to seek deeper cosmic principles. The concept of fuzzy wave function is proposed to represent the fuzzy quantum probability by utilizing fuzzy mathematics to blur the probability (wave) of quantum mechanics. By applying the non-fuzzy formula of fuzzy quantity and the Schrödinger wave equation of quantum mechanics, the membership function equation is constructed to describe the evolution of the fuzzy wave function. The concept of membership degree amplitude is introduced to calculate fuzzy probability amplitude, and the membership degree amplitude equation is derived. Some important concepts in fuzzy mathematics are also illustrated.

Computer systems based on fuzzy logic are capable of generating a reliable output even when handling inaccurate input data by applying a rule based system. The main contribution of this paper is to show that quantum computing can be used to extend the class of fuzzy sets. The central idea associates the states of a quantum register to membership functions (mFs) of fuzzy subsets, and the rules for the processes of fuzzyfication are performed by unitary quantum transformations. Thus, this paper describes multi-dimensional quantum registers, associated to mFs on the unitary interval U, in order to introduce a novel interpretation of aggregations found in fuzzy set theory. In particular, t-norms and t-conorms based on quantum gates allows the modeling and interpretation of union, intersection, difference and implication among fuzzy sets, also including an expression for the class of fuzzy S-implications. Furthermore, an interpretation of the symmetric sum was achieved by considering the quantum register sum operator.

This paper is concerned with finite-level quantum memory systems for retaining initial dynamic variables in the presence of external quantum noise. The system variables have an algebraic structure, similar to that of the Pauli matrices, and their Heisenberg picture evolution is governed by a quasilinear quantum stochastic differential equation. The latter involves a Hamiltonian whose parameters depend affinely on a classical control signal in the form of a deterministic function of time. The memory performance is quantified by a mean-square deviation of quantum system variables of interest from their initial conditions. We relate this functional to a matrix-valued state of an auxiliary classical control-affine dynamical system. This leads to a pointwise control design where the control signal minimises the time-derivative of the mean-square deviation with an additional quadratic penalty on the control. In an alternative finite-horizon setting with a terminal-integral cost functional, we apply dynamic programming and obtain a quadratically nonlinear Hamilton-Jacobi-Bellman equation, for which a solution is outlined in the form of a recursively computed asymptotic expansion.