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Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Robert Nowotniak, Jacek Kucharski Computer Engineering Department, Technical University of Łód´ z SŁOK, June 23-25, 2010 Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs propagation in SGA Situation for Simple Genetic Algorithm Holland’s schema theorem Short, low order, above average schemata receive exponentially increasing trials in subsequent generations of the classical genetic algorithm  — binary gene 1 1 1 1 1 0 0      1 1 1 1 1 1 0 — chromosome    population 0 1 0 1 0 1 0  1 1 0 0 0 1 0      0 1 1 0 0 0 1   � � � � � � � � � � � � � � � � schema 1 * 0 * * * * Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 1 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs propagation in SGA Situation for Simple Genetic Algorithm Holland’s schema theorem Short, low order, above average schemata receive exponentially increasing trials in subsequent generations of the classical genetic algorithm  — binary gene 1 1 1 1 1 0 0      1 1 1 1 1 1 0 — chromosome    population 0 1 0 1 0 1 0  1 1 0 0 0 1 0      0 1 1 0 0 0 1   � � � � � � � � � � � � � � � � schema 1 match 1 * 0 * * * * Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 1 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs propagation in SGA Situation for Simple Genetic Algorithm Holland’s schema theorem Short, low order, above average schemata receive exponentially increasing trials in subsequent generations of the classical genetic algorithm  — binary gene 0 0 0 0 1 0 0      1 0 0 1 0 1 1 — chromosome    population 1 1 1 0 0 1 0  1 1 0 0 0 1 0      0 0 1 0 1 1 0   � � � � � � � � � � � � � � � � schema 2 matches 1 * 0 * * * * Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 1 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs propagation in SGA Situation for Simple Genetic Algorithm Holland’s schema theorem Short, low order, above average schemata receive exponentially increasing trials in subsequent generations of the classical genetic algorithm  — binary gene 1 1 0 1 0 1 0      1 0 0 1 0 0 0 — chromosome    population 0 0 1 0 1 1 0  1 0 0 0 1 0 1      0 0 1 0 0 0 1   � � � � � � � � � � � � � � � � schema 3 matches 1 * 0 * * * * Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 1 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Qubits and Binary Quantum Genes Qubits and Binary Quantum Genes qubit (quantum bit): | ψ � = α | 0 � + β | 1 � where: α, β ∈ C , | α | 2 + | β | 2 = 1 Pr | ψ � : F { 0 , 1 } �→ [ 0 , 1 ] Pr | ψ � ( { 0 } ) = | α | 2 Pr | ψ � ( { 1 } ) = | β | 2 Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 2 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Qubits and Binary Quantum Genes Qubits and Binary Quantum Genes qubit (quantum bit): | ψ � = α | 0 � + β | 1 � where: α, β ∈ C , | α | 2 + | β | 2 = 1 Pr | ψ � : F { 0 , 1 } �→ [ 0 , 1 ] | 1 � Pr | ψ � ( { 0 } ) = | α | 2 Pr | ψ � ( { 1 } ) = | β | 2 √ | ψ � 2 | 0 � + 1 3 | ψ � = 2 | 1 � β | 0 � α Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 2 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Qubits and Binary Quantum Genes Qubits and Binary Quantum Genes qubit (quantum bit): | ψ � = α | 0 � + β | 1 � where: α, β ∈ C , | α | 2 + | β | 2 = 1 Pr | ψ � : F { 0 , 1 } �→ [ 0 , 1 ] | 1 � Pr | ψ � ( { 0 } ) = | α | 2 Pr | ψ � ( { 1 } ) = | β | 2 | ψ � √ √ 2 2 | ψ � = 2 | 0 � + 2 | 1 � β | 0 � α Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 2 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Qubits and Binary Quantum Genes Qubits and Binary Quantum Genes qubit (quantum bit): | ψ � = α | 0 � + β | 1 � where: α, β ∈ C , | α | 2 + | β | 2 = 1 Pr | ψ � : F { 0 , 1 } �→ [ 0 , 1 ] | 1 � Pr | ψ � ( { 0 } ) = | α | 2 | ψ � Pr | ψ � ( { 1 } ) = | β | 2 √ β | ψ � = 1 3 | 0 � + 2 2 3 | 1 � | 0 � α Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 2 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs Schemata for Quantum Genetic Algorithm  — quantum gene          — quantum chromosome   quantum   population              � � � � � � � � � � � � � � � � � � � � � schema 1 * 0 * * * * Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 3 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs Schemata for Quantum Genetic Algorithm  — quantum gene          — quantum chromosome   quantum   population              � � � � � � � � � � � � � � � � � � � � � schema 1 * 0 * * * * Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 3 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs Schemata for Quantum Genetic Algorithm  — quantum gene          — quantum chromosome   quantum   population              � � � � � � � � � � � � � � � � � � � � � schema 1 * 0 * * * * Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 3 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs Schemata for Quantum Genetic Algorithm  — quantum gene          — quantum chromosome   quantum   population              � � � � � � � � � � � � � � � � � � � � � schema 1 * 0 * * * * Problem: How many chromosomes match the schema? Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 3 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs Quantum Chromosomes Matching The Schema Proposal L – random variable corresponding to the number of binary quantum chromosomes matching the schema Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 4 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs Quantum Chromosomes Matching The Schema Expected number:     N � �  � � E ( L ) =  w · M ( q i , S ) (1 − M ( q k , S ))     w =0 C ∈ { X ∈ 2 { 1 ,...,N } : | X | = w } j ∈ C k ∈{ 1 ,...,N }\ C Variation:     N �  w 2 �  � � V ( L ) = M ( q j , S ) (1 − M ( q k , S ))     w =0 C ∈ { X ∈ 2 { 1 ,...,N } : | X | = w } j ∈ C k ∈{ 1 ,...,N }\ C 2       N � �  � � − M ( q j , S ) (1 − M ( q k , S ))  w         w =0 C ∈ { X ∈ 2 { 1 ,...,N } : | X | = w } j ∈ C k ∈{ 1 ,...,N }\ C where: N – population size, M ( q , S ) = � m i = 1 Pr gi ( { S [ i ] } ) – probability that q matches S m – length of chromosomes Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 5 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs Building Blocks Propagation Comparison 10 number of chromosomes matching the schema 8 6 4 2 Expected Propagation in QIGA Actual Propagation in QIGA Actual Propagation in SGA 0 0 20 40 60 80 100 120 140 160 generation number BB: 01001***************, popsize = 10, chromlen = 20 Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 6 / 6

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Thank you for your attention Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010