prezentacja

State of the art New Theory Order-2 QIGA Experiments Conclusions

Higher-Order Quantum-Inspired

Genetic Algorithms

Robert Nowotniak, Jacek Kucharski

Institute of Applied Computer Science

Lodz University of Technology

Federated Conference on Computer Science

and Information Systems

September 7, 2014

Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms


Presentation outline

1 Background, state of the art

2 The new theory fundamental notions

3 Order-2 Quantum-Inspired Genetic Algorithm (QIGA2)

4 Numerical experiments and results

5 Conclusions

Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 1 / 20


Quantum Computing + Artificial Intelligence

← Classical Computing, Computational Intelligence

← Quantum-Inspired Computational IntelligenceQuantum Computer NOT REQUIREDA few hundred papers (total) since late 90's:

1 Quantum-Inspired Neural Networks2 Quantum-Inspired Fuzzy Systems3 Quantum-Inspired Genetic Algorithms4 Quantum-Inspired Immune Systems5 ...

← Quantum Computational Intelligence?Quantum Computer REQUIRED

State of the art: Science ction?



Qubits and Binary Quantum Genes

|ψ〉 =√32︸︷︷︸α

|0〉+ 12︸︷︷︸β

|1〉

|0〉

|1〉

|ψ〉

α

β

qubit (quantum bit): |ψ〉 = α|0〉+ β|1〉where: α, β ∈ C, |α|2 + |β|2 = 1

Pr(0) = |α|2Pr(1) = |β|2




|ψ〉 =√22︸︷︷︸α

|0〉+√22︸︷︷︸β

|1〉

|0〉

|1〉

|ψ〉

α

β


Pr(0) = |α|2Pr(1) = |β|2




|ψ〉 = 13︸︷︷︸α

|0〉+ 2√23︸︷︷︸β

|1〉

|0〉

|1〉|ψ〉

α

β


Pr(0) = |α|2Pr(1) = |β|2




|ψ〉 = 0︸︷︷︸α

|0〉+ 1︸︷︷︸β

|1〉

|0〉

|1〉|ψ〉

α

β


Pr(0) = |α|2Pr(1) = |β|2



Simple 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

populationof solutions

— chromosome

— binary gene




1 1 0 1 0 1 0

1 0 0 1 0 0 0

0 0 1 0 1 1 0

0 0 1 0 0 0 1

1 0 0 0 1 0 1


— chromosome

— binary gene




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


— chromosome

— binary gene




0 0 0 0 1 0 1

0 1 1 0 1 1 0

1 0 0 1 1 1 0

1 1 0 1 0 1 1

1 1 0 1 0 1 1


— chromosome

— binary gene




0 1 0 0 1 0 0

0 0 0 0 1 1 0

0 1 0 1 0 0 1

1 0 0 1 0 0 1

0 1 0 0 0 1 0


— chromosome

— binary gene



Quantum-Inspired Genetic Algorithm [1]

0 = 1 = 0101110 =

[1] Han, K.-H., Kim, J.-H., Quantum-inspired evolutionary algorithm for a class of combinatorialoptimization. Evolutionary Computation, IEEE Transactions on, 2002, 6(6), pp. 580-593.



Quantum-Inspired Genetic Algorithm [1]

0 = 1 = 0101110 =

quantumpopulation

— quantum chromosome

— quantum gene

[1] Han, K.-H.; Kim, J.-H., Quantum-inspired evolutionary algorithm for a class of combinatorialoptimization. Evolutionary Computation, IEEE Transactions on, 2002, 6(6), pp. 580-593.




Evolutionary Algorithms – The New Theory

Fundamental notions of the theory:

Quantum order r ∈N+ of Quantum-Inspired algorithm:

the size of the biggest quantum register in the algorithm

(e.g. separate qubits-based algorithms are Order-1)

Quantum factor λ ∈ [0, 1] of Quantum-Inspired algorithm:

the ratio of the algorithm space dimension to dimension of

quantum register state space.

λ =2r · N

r

2N

where:N ∈N+ the problem size

r ∈ 1, . . . ,N quantum order

λ ∈ [0, 1] quantum factor




Evolutionary Algorithms – The New Theory

Fundamental notions of the theory:

Quantum order r ∈N+ of Quantum-Inspired algorithm:

the size of the biggest quantum register in the algorithm

(e.g. separate qubits-based algorithms are Order-1)

Quantum factor λ ∈ [0, 1] of Quantum-Inspired algorithm:

the ratio of the algorithm space dimension to dimension of

quantum register state space.

λ =2r · N

r

2N

where:N ∈N+ the problem size

r ∈ 1, . . . ,N quantum order

λ ∈ [0, 1] quantum factor


5 6 7 8 9 10 11

Problem size N

0.0

0.2

0.4

0.6

0.8

1.0

Qua

ntum

fact

orλ

Quantum factor λ for different r and N

r = 1, r = 2

r = 3

r = 4

r = 5


The Algorithms Spaces

Space Properties Meaning λ, r

binary strings Xnite, discrete set

X = 0, 1N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

λ = 0

schemata ΩHnite, discrete set

ΩH = 0, 1, ∗N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

H=*1*0***

quantum-inspired

chromosomes ΩQI

linear space,

dim(ΩQI ) = N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

λ ≈ 10−6

r = 1

Higher-dimensional spacesλ→ 1

r → N

quantum register

state space Hcomplex Hilbert space,

dim(H) = 2N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

λ = 1

r = N



Binary strings

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

000100111101011



Binary strings

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

010011110101110



Binary strings

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

101100110011001



Binary strings

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

111000111101011



Schemata

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

H = ∗1 ∗ 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗



Schemata

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

H = 01 ∗ 01 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗



Schemata

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

H = 01001 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗



Schemata

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

H = 01110 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗



Schemata

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

H = 0 ∗ 110 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗



Schemata

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

H = 01 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗



Schemata

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

H = 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗



Schemata

000...0 010...0 100...0 110...0 111...1x

0

20

40

60

80

100

f(x

)

H = ∗ ∗ ∗ ∗ 0



Binary quantum chromosomes




Space Properties Situation λ, r


X = 0, 1N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

λ = 0


ΩH = 0, 1, ∗N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

H=*1*0***

quantum-inspired

chromosomes ΩQI

linear space,

dim(ΩQI ) = N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

λ ≈ 10−6

r = 1


r → N

quantum register


dim(H) = 2N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

λ = 1

r = N




Space Properties Situation λ, r


X = 0, 1N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

λ = 0


ΩH = 0, 1, ∗N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

H=*1*0***

quantum-inspired

chromosomes ΩQI

linear space,

dim(ΩQI ) = N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

λ ≈ 10−6

r = 1


r → N

quantum register


dim(H) = 2N000...0 010...0 100...0 110...0 111...1

x

0

20

40

60

80

100

f(x

)

λ = 1

r = N


Higher-Order Quantum-Inspired Algorithms


Order-2 Quantum-Inspired Genetic Algorithm

for Combinatorial Optimization

QIGA2

1: t ← 0

2: Initialize quantum population Q(0)3: while t ≤ tmax do4: t ← t + 1

5: Generate P(t) by observing quantum pop. Q(t − 1)6: Evaluate classical population P(t)7: Update Q(t)8: Save best classical individual to b

9: end while





Quantum-Inspired Genetic Algorithm:





Order-2 Quantum-Inspired Genetic Algorithm:(quantum modelling of interactions in pairs of genes)





Observation of pair of genes in QIGA2 algorithm:

Require: qij = [α0 α1 α2 α3]T quantum register of 2 qubits

1: r ← uniformly random number from [0,1]2: if r < |α0|2 then3: p ← 00

4: else if r < |α0|2 + |α1|2 then5: p ← 01

6: else if r < |α0|2 + |α1|2 + |α2|2 then7: p ← 10

8: else9: p ← 11

10: end if



Chromosomes in QIGA2

000...0 010...0 100...0 110...0 111...1x

20

40

60

80

100

f(x

)

[0.0001.0000.0000.500|

0.8000.4000.8000.200|

0.5000.6000.7000.800

]




000...0 010...0 100...0 110...0 111...1x

20

40

60

80

100

f(x

)

[0.5001.0002.0000.500|

0.8000.4000.8000.200|

0.5000.8000.7000.500

]




000...0 010...0 100...0 110...0 111...1x

20

40

60

80

100

f(x

)

[0.5001.0000.0000.500|

0.8000.4000.8000.200|

0.5000.8000.7000.500

]



Numerical experiments

Performance of four algorithms has been compared:

SGA, QIGA1, GPU-tuned QIGA1, QIGA2

Recognizable benchmark of 20 deceptive combinatorialoptimization problems has been used has been used(knapsack + SATLIB benchmark)

1 Knapsack problem, problem size N = 100, . . . , 10002 SAT (NP-complete),

coding various combinatorial optimization problems,problem size N = 11, . . . , 1000

Objective:nd the binary strings that have maximum tness value

Stopping criterion:Maximum number of tness evaluations: MaxFE = 50, 000.

Average of 50 runs of each algorithm has been compared.



Numerical experiments – results




0 1000 2000 3000 4000 5000

Fitness evaluation count (FE)

1300

1320

1340

1360

1380

1400

1420

1440

1460

1480

Ave

rage

fitne

ssof

the

best

indi

vidu

alAlgorithms performance comparison

Problem: knapsack250, size N = 250

QIGA-2QIGA-1 tunedQIGA-1SGA




0 1000 2000 3000 4000 5000


620

640

660

680

700

720

740

760

Ave

rage

fitne

ssof

the

best

indi

vidu

alAlgorithms performance comparisonProblem: bejing-252, size N = 252





0 1000 2000 3000 4000 5000


5300

5350

5400

5450

5500

5550

5600

5650

5700

5750

Ave

rage

fitne

ssof

the

best

indi

vidu

alAlgorithms performance comparison

Problem: knapsack1000, size N = 1000




Problem N SGA QIGA-1 QIGA-1 tuned QIGA-2

anomaly 48 251.4 252.55 254.65 255.25

sat 90 284.9 289.2 293.2 293.7

jnh 100 826.15 831.05 839.05 836.05

knapsack 100 577.709 578.812 592.819 596.476

sat 100 408.6 413.6 418.6 419.7

bejing 125 297.35 302.1 305.35 306.2

sat-uuf 225 886.75 898.25 921.65 921.5

knapsack 250 1387.916 1406.528 1449.905 1467.407

sat1 250 981.45 995.15 1021.2 1023.1

sat2 250 982.95 994.6 1019.1 1020.6

sat3 250 984.2 994.3 1021.3 1019.7

bejing 252 709.85 731.0 724.4 745.75

parity 317 1141.65 1158.2 1179.35 1180.75

knapsack 400 2209.925 2222.160 2284.969 2334.494

knapsack 500 2803.266 2812.740 2869.774 2929.469

bejing 590 1263.8 1343.15 1284.0 1353.2

lran 600 2310.9 2330.35 2386.8 2398.95

bejing 708 1510.65 1605.9 1523.15 1611.55

knapsack 1000 5451.656 5462.718 5568.234 5709.116

lran 1000 3819.65 3848.4 3918.5 3937.3



Conclusions

In 17 out of 20 test problems (85%), the authors'QIGA2 algorithm presented on average a better solutionthan both the original and tuned QIGA12 algorithm.

Quantum order r = 2 allows to improve eciency of QIGA

algorithm in combinatorial optimization problems.

QIGA2 running time is about 15-30% faster than QIGA1

(due to simplications in comparison to the previous algorithm)



Our Recent Papers on QIEA algorithms

1 Nowotniak, R., Kucharski, J., GPU-based Tuning of Quantum-Inspired Genetic Algorithm for aCombinatorial Optimization Problem, Bulletin of The Polish Academy of Sciences:TechnicalSciences, Vol. 60, No. 2, 2012, ISSN 0239-7528

2 Nowotniak, R., Kucharski, J., Meta-optimization of Quantum-Inspired Evolutionary Algorithms inThe Polish Grid Infrastructure, 2nd Scientic Session of TUL PhD Students, ISBN978-83-7283-490-4

3 Nowotniak, R., Kucharski, J., Meta-optimization of Quantum-Inspired Evolutionary Algorithm,2010, Proceedings of the XVII International Conference on Information Technology Systems,ISBN 978-83-7283-378-5

4 Nowotniak, R., Kucharski, J., Building Blocks Propagation in Quantum-Inspired GeneticAlgorithm, 2010, Scientic Bulletin of Academy of Science and Technology, Automatics, 2010,ISSN 1429-3447

5 Nowotniak, R., Survey of Quantum-Inspired Evolutionary Algorithms, 2010, Proceedings of theFIMB PhD students conference, ISSN 2082-4831

6 Nowotniak, R., Kucharski J., GPU-based Tuning of Quantum-Inspired Genetic Algorithm fora Combinatorial Optimization Problem", XIV International Conference System Modelling andControl, 2011, ISBN 978-83-927875-1-8

7 Nowotniak, R., Quantum-Inspired Evolutionary Algorithms in Search and Optimization, IWyjazdowa Sesja Naukowa Doktorantów P, Rogów, ISBN 978-83-7283-411-9

8 Nowotniak, R., Kucharski J., GPU-based massively parallel implementation of metaheuristicalgorithms, Przetwarzanie i analiza sygnaªów w systemach wizji i sterowania, Sªok, 2011

9 Nowotniak, R., Draus C., Nowak M., Rybak G., "Modelling Reality In Visual Python", INotice2011, ISBN 978-83-7283-407-2

10 Je»ewski, S., aski, M., Nowotniak, R., Comparison of Algorithms for Simultaneous Localizationand Mapping Problem for Mobile Robot, 2010, Scientic Bulletin of Academy of Science andTechnology, Automatics, ISSN 1429-3447

11 Jopek, ., Nowotniak, R., Postolski, M., Babout, L., Janaszewski, M., Application of QuantumGenetic Algorithms in Feature Selection Problem, 2009, Scientic Bulletin of Academy ofScience and Technology, Automatics, ISSN 1429-3447



Thank you for your attention

Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms

prezentacja

Documents

qubit quantum

art new theory order

binary quantum genes

conclusions robert nowotniak

numerical experiments

genetic algorithm qiga2

new theoryfundamental

science ction