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Building Blocks Propagation in Quantum-Inspired Genetic Algorithm - - PowerPoint PPT Presentation

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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 d z SOK,

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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

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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 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1

                

  • population

— chromosome — binary gene 1 * 0 * * * * schema

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 1 / 6

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SLIDE 3

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 1 1 1 1 1 0 0 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1

                

  • population

— chromosome — binary gene 1 * 0 * * * * schema 1 match

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 1 / 6

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SLIDE 4

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 0 0 0 0 1 0 0 1 0 0 1 0 1 1 1 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1 1 0

                

  • population

— chromosome — binary gene 1 * 0 * * * * schema 2 matches

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 1 / 6

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SLIDE 5

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 1 1 0 1 0 1 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 1 0 0 0 1

                

  • population

— chromosome — binary gene 1 * 0 * * * * schema 3 matches

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 1 / 6

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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

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Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Qubits and Binary Quantum Genes

Qubits and Binary Quantum Genes

|ψ =

√ 3 2 |0 + 1 2|1

|0 |1 |ψ α β

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

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Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Qubits and Binary Quantum Genes

Qubits and Binary Quantum Genes

|ψ =

√ 2 2 |0 + √ 2 2 |1

|0 |1 |ψ α β

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

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Building Blocks Propagation in Quantum-Inspired Genetic Algorithm Qubits and Binary Quantum Genes

Qubits and Binary Quantum Genes

|ψ = 1

3|0 + 2 √ 2 3 |1

|0 |1 |ψ α β

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

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Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs

Schemata for Quantum Genetic Algorithm

                          

  • quantum

population — quantum chromosome — quantum gene 1 * * * * * schema

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 3 / 6

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SLIDE 11

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs

Schemata for Quantum Genetic Algorithm

                          

  • quantum

population — quantum chromosome — quantum gene 1 * * * * * schema

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 3 / 6

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SLIDE 12

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs

Schemata for Quantum Genetic Algorithm

                          

  • quantum

population — quantum chromosome — quantum gene 1 * * * * * schema

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 3 / 6

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SLIDE 13

Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs

Schemata for Quantum Genetic Algorithm

                          

  • quantum

population — quantum chromosome — quantum gene 1 * * * * * schema

Problem: How many chromosomes match the schema?

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 3 / 6

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SLIDE 14

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

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Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs

Quantum Chromosomes Matching The Schema

Expected number:

E(L) =

N

  • w=0

  w ·

  • C∈{X∈2{1,...,N}:|X|=w}

 

j∈C

M(qi, S)

  • k∈{1,...,N}\C

(1 − M(qk, S))

    

Variation:

V (L) =

N

  • w=0

  w2

  • C∈{X∈2{1,...,N}:|X|=w}

 

j∈C

M(qj, S)

  • k∈{1,...,N}\C

(1 − M(qk, S))

    

−

  

N

  • w=0

  w

  • C∈{X∈2{1,...,N}:|X|=w}

 

j∈C

M(qj, S)

  • k∈{1,...,N}\C

(1 − M(qk, S))

       

2

where: N – population size, M(q, S) = m

i=1 Prgi ({S[i]}) – probability that q matches S

m – length of chromosomes

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 5 / 6

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Building Blocks Propagation in Quantum-Inspired Genetic Algorithm BBs Propagation in QGAs

Building Blocks Propagation Comparison

20 40 60 80 100 120 140 160 generation number 2 4 6 8 10 number of chromosomes matching the schema

Expected Propagation in QIGA Actual Propagation in QIGA Actual Propagation in SGA

BB: 01001***************, popsize = 10, chromlen = 20

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010 6 / 6

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Building Blocks Propagation in Quantum-Inspired Genetic Algorithm

Thank you for your attention

Robert Nowotniak, Jacek Kucharski SŁOK, June 23-25, 2010