Eigenlogic Quantum Operators for Quantum Computing and Robotics (Invited Conference)

Zeno Toffano

Zeno Toffano, François Dubois

This work proposes a formulation of propositional logic, named Eigenlogic, using quantum observables as propositions. The eigenvalues of these operators are the truth-values and the associated eigenvectors the interpretations of the propositional system. Fuzzy logic arises naturally when considering vectors outside the eigensystem, the fuzzy membership function is obtained by the Born rule of the logical observable.This approach is then applied in the context of quantum robots using simple behavioral agents represented by Braitenberg vehicles. Processing with non-classical logic such as multivalued logic, fuzzy logic and the quantum Eigenlogic permits to enlarge the behavior possibilities and the associated decisions of these simple agents.

Giuliano Benenti, Giulio Casati, Simone Montangero

Giuliano Benenti, Giulio Casati, Simone Montangero

Zeno Toffano, François Dubois

This chapter proposes a formulation of propositional logic, named Eigenlogic, using quantum observables as propositions. The eigenvalues of these operators are the truth-values and the associated eigenvectors the interpretations of the propositional system. This approach is then applied in the context of quantum robots using simple behavioral agents represented by Braitenberg vehicles. Processing with non-classical logic such as multivalued logic, fuzzy logic and the quantum Eigenlogic permit to enlarge the behavior possibilities and the associated decisions of these simple agents. Valentino Braitenberg was a Cyberneticist and former director at the Max Planck Institute for Biological Cybernetics in Tubingen. A Braitenberg vehicle is an agent that can autonomously move around based on its light sensor inputs. Depending on the sensor-motor wiring, it appears to achieve certain situations and to avoid others, changing course when situation changes. The internal emotional state could be described now by a quantum circuit built from counterparts of fuzzy operators.

Zeno Toffano, François Dubois

Purpose The purpose of this paper is to apply the quantum “eigenlogic” formulation to behavioural analysis. Agents, represented by Braitenberg vehicles, are investigated in the context of the quantum robot paradigm. The agents are processed through quantum logical gates with fuzzy and multivalued inputs; this permits to enlarge the behavioural possibilities and the associated decisions for these simple vehicles. Design/methodology/approach In eigenlogic, the eigenvalues of the observables are the truth values and the associated eigenvectors are the logical interpretations of the propositional system. Logical observables belong to families of commuting observables for binary logic and many-valued logic. By extension, a fuzzy logic interpretation is proposed by using vectors outside the eigensystem of the logical connective observables. The fuzzy membership function is calculated by the quantum mean value (Born rule) of the logical projection operators and is associated to a quantum probability. The methodology of this paper is based on quantum measurement theory. Findings Fuzziness arises naturally when considering systems described by state vectors not in the considered logical eigensystem. These states correspond to incompatible and complementary systems outside the realm of classical logic. Considering these states allows the detection of new Braitenberg vehicle behaviours related to identified emotions; these are linked to quantum-like effects. Research limitations/implications The method does not deal at this stage with first-order logic and is limited to different families of commuting logical observables. An extension to families of logical non-commuting operators associated to predicate quantifiers could profit of the “quantum advantage” due to effects such as superposition, parallelism, non-commutativity and entanglement. This direction of research has a variety of applications, including robotics. Practical implications The goal of this research is to show the multiplicity of behaviours obtained by using fuzzy logic along with quantum logical gates in the control of simple Braitenberg vehicle agents. By changing and combining different quantum control gates, one can tune small changes in the vehicle’s behaviour and hence get specific features around the main basic robot’s emotions. Originality/value New mathematical formulation for propositional logic based on linear algebra. This methodology demonstrates the potentiality of this formalism for behavioural agent models (quantum robots).