This paper presents a theory and an empirical evaluation of Higher-Order Quantum-Inspired Genetic Algorithms. Fundamental notions of the theory have been introduced, and a novel Order-2 Quantum-Inspired Genetic Algorithm (QIGA2) has been presented. Contrary to all QIGA algorithms which represent quantum genes as independent qubits, in higher-order QIGAs quantum registers are used to represent genes strings which allows modelling of genes relations using quantum phenomena. Performance comparison has been conducted on a benchmark of 20 deceptive combinatorial optimization problems. It has been presented that using higher quantum orders is beneficial for genetic algorithm efficiency, and the new QIGA2 algorithm outperforms the old QIGA algorithm which was tuned in highly compute intensive metaoptimization process.

1. 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

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

3. State of the art New Theory Order-2 QIGA Experiments Conclusions
Quantum Computing + Artificial Intelligence
← Classical Computing, Computational Intelligence
← Quantum Computational Intelligence?
Quantum Computer REQUIRED
State of the art: Science ction?
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 2 / 20

4. State of the art New Theory Order-2 QIGA Experiments Conclusions
Quantum Computing + Artificial Intelligence
← Classical Computing, Computational Intelligence
← Quantum-Inspired Computational Intelligence
Quantum Computer NOT REQUIRED
A few hundred papers (total) since late 90's:
1 Quantum-Inspired Neural Networks
2 Quantum-Inspired Fuzzy Systems
3 Quantum-Inspired Genetic Algorithms
4 Quantum-Inspired Immune Systems
5 ...
← Quantum Computational Intelligence?
Quantum Computer REQUIRED
State of the art: Science ction?
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 2 / 20

5. State of the art New Theory Order-2 QIGA Experiments Conclusions
Qubits and Binary Quantum Genes
|ψ =
√
3
2
α
|0 +
1
2
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|2
Pr({1}) = |β|2
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 3 / 20

6. State of the art New Theory Order-2 QIGA Experiments Conclusions
Qubits and Binary Quantum Genes
|ψ =
√
2
2
α
|0 +
√
2
2
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|2
Pr({1}) = |β|2
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 3 / 20

7. State of the art New Theory Order-2 QIGA Experiments Conclusions
Qubits and Binary Quantum Genes
|ψ =
1
3
α
|0 +
2
√
2
3
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|2
Pr({1}) = |β|2
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 3 / 20

8. State of the art New Theory Order-2 QIGA Experiments Conclusions
Qubits and Binary Quantum Genes
|ψ = 0
α
|0 + 1
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|2
Pr({1}) = |β|2
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 3 / 20

9. State of the art New Theory Order-2 QIGA Experiments Conclusions
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



population
of solutions
— chromosome
— binary gene
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 4 / 20

10. State of the art New Theory Order-2 QIGA Experiments Conclusions
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



population
of solutions
— chromosome
— binary gene
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 4 / 20

11. State of the art New Theory Order-2 QIGA Experiments Conclusions
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



population
of solutions
— chromosome
— binary gene
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 4 / 20

12. State of the art New Theory Order-2 QIGA Experiments Conclusions
Simple Genetic Algorithm
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



population
of solutions
— chromosome
— binary gene
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 4 / 20

13. State of the art New Theory Order-2 QIGA Experiments Conclusions
Simple 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
of solutions
— chromosome
— binary gene
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 4 / 20

14. State of the art New Theory Order-2 QIGA Experiments Conclusions
Simple Genetic Algorithm
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



population
of solutions
— chromosome
— binary gene
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 4 / 20

15. State of the art New Theory Order-2 QIGA Experiments Conclusions
Simple Genetic Algorithm
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



population
of solutions
— chromosome
— binary gene
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 4 / 20

16. State of the art New Theory Order-2 QIGA Experiments Conclusions
Quantum-Inspired Genetic Algorithm [1]
0 = 1 = 0101110 =
[1] Han, K.-H., Kim, J.-H., Quantum-inspired evolutionary algorithm for a class of combinatorial
optimization. Evolutionary Computation, IEEE Transactions on, 2002, 6(6), pp. 580-593.
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 5 / 20

17. State of the art New Theory Order-2 QIGA Experiments Conclusions
Quantum-Inspired Genetic Algorithm [1]
0 = 1 = 0101110 =



quantum
population
— quantum chromosome
— quantum gene
[1] Han, K.-H.; Kim, J.-H., Quantum-inspired evolutionary algorithm for a class of combinatorial
optimization. Evolutionary Computation, IEEE Transactions on, 2002, 6(6), pp. 580-593.
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 5 / 20

18. State of the art New Theory Order-2 QIGA Experiments Conclusions
Quantum-Inspired Genetic Algorithm [1]
0 = 1 = 0101110 =



quantum
population
— quantum chromosome
— quantum gene
[1] Han, K.-H.; Kim, J.-H., Quantum-inspired evolutionary algorithm for a class of combinatorial
optimization. Evolutionary Computation, IEEE Transactions on, 2002, 6(6), pp. 580-593.
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 5 / 20

19. State of the art New Theory Order-2 QIGA Experiments Conclusions
Quantum-Inspired Genetic Algorithm [1]
0 = 1 = 0101110 =



quantum
population
— quantum chromosome
— quantum gene
[1] Han, K.-H.; Kim, J.-H., Quantum-inspired evolutionary algorithm for a class of combinatorial
optimization. Evolutionary Computation, IEEE Transactions on, 2002, 6(6), pp. 580-593.
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 5 / 20

20. State of the art New Theory Order-2 QIGA Experiments Conclusions
Quantum-Inspired Genetic Algorithm [1]
0 = 1 = 0101110 =



quantum
population
— quantum chromosome
— quantum gene
[1] Han, K.-H.; Kim, J.-H., Quantum-inspired evolutionary algorithm for a class of combinatorial
optimization. Evolutionary Computation, IEEE Transactions on, 2002, 6(6), pp. 580-593.
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 5 / 20

21. State of the art New Theory Order-2 QIGA Experiments Conclusions
Higher-Order Quantum-Inspired
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 =
2
r · N
r
2N
where:
N ∈ N+ the problem size
r ∈ {1, . . . , N} quantum order
λ ∈ [0, 1] quantum factor
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 6 / 20

22. State of the art New Theory Order-2 QIGA Experiments Conclusions
Higher-Order Quantum-Inspired
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 =
2
r · N
r
2N
where:
N ∈ N+ the problem size
r ∈ {1, . . . , N} quantum order
λ ∈ [0, 1] quantum factor
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 6 / 20
5 6 7 8 9 10 11
Problem size N
0.0
0.2
0.4
0.6
0.8
1.0
Quantumfactorλ
Quantum factor λ for different r and N
r = 1, r = 2
r = 3
r = 4
r = 5

23. State of the art New Theory Order-2 QIGA Experiments Conclusions
The Algorithms Spaces
Space Properties Meaning λ, r
binary strings X
nite, discrete set
X = {0, 1}N
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
λ = 0
schemata ΩH
nite, discrete set
ΩH = {0, 1, ∗}N
000...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) = N 000...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 spaces
λ → 1
r → N
quantum register
state space H
complex Hilbert space,
dim(H) = 2
N
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
λ = 1
r = N
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 7 / 20

24. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary strings
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
000100111101011
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 8 / 20

25. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary strings
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
010011110101110
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 8 / 20

26. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary strings
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
101100110011001
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 8 / 20

27. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary strings
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
111000111101011
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 8 / 20

28. State of the art New Theory Order-2 QIGA Experiments Conclusions
Schemata
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
H = ∗1 ∗ 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 9 / 20

29. State of the art New Theory Order-2 QIGA Experiments Conclusions
Schemata
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
H = 01 ∗ 01 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 9 / 20

30. State of the art New Theory Order-2 QIGA Experiments Conclusions
Schemata
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
H = 01001 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 9 / 20

31. State of the art New Theory Order-2 QIGA Experiments Conclusions
Schemata
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
H = 01110 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 9 / 20

32. State of the art New Theory Order-2 QIGA Experiments Conclusions
Schemata
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
H = 0 ∗ 110 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 9 / 20

33. State of the art New Theory Order-2 QIGA Experiments Conclusions
Schemata
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
H = 01 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 9 / 20

34. State of the art New Theory Order-2 QIGA Experiments Conclusions
Schemata
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
H = 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 9 / 20

35. State of the art New Theory Order-2 QIGA Experiments Conclusions
Schemata
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
H = ∗ ∗ ∗ ∗ 0
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 9 / 20

36. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary quantum chromosomes
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 10 / 20

37. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary quantum chromosomes
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 10 / 20

38. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary quantum chromosomes
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 10 / 20

39. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary quantum chromosomes
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 10 / 20

40. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary quantum chromosomes
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 10 / 20

41. State of the art New Theory Order-2 QIGA Experiments Conclusions
Binary quantum chromosomes
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 10 / 20

42. State of the art New Theory Order-2 QIGA Experiments Conclusions
The Algorithms Spaces
Space Properties Situation λ, r
binary strings X
nite, discrete set
X = {0, 1}N
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
λ = 0
schemata ΩH
nite, discrete set
ΩH = {0, 1, ∗}N
000...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) = N 000...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 spaces
λ → 1
r → N
quantum register
state space H
complex Hilbert space,
dim(H) = 2
N
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
λ = 1
r = N
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 11 / 20

43. State of the art New Theory Order-2 QIGA Experiments Conclusions
The Algorithms Spaces
Space Properties Situation λ, r
binary strings X
nite, discrete set
X = {0, 1}N
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
λ = 0
schemata ΩH
nite, discrete set
ΩH = {0, 1, ∗}N
000...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) = N 000...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 spaces
λ → 1
r → N
quantum register
state space H
complex Hilbert space,
dim(H) = 2
N
000...0 010...0 100...0 110...0 111...1
x
0
20
40
60
80
100
f(x)
λ = 1
r = N
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 11 / 20
Higher-Order Quantum-Inspired Algorithms

44. State of the art New Theory Order-2 QIGA Experiments Conclusions
Order-2 Quantum-Inspired Genetic Algorithm
for Combinatorial Optimization
QIGA2
1: t ← 0
2: Initialize quantum population Q(0)
3: while t ≤ tmax do
4: 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
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 12 / 20

45. State of the art New Theory Order-2 QIGA Experiments Conclusions
Order-2 Quantum-Inspired Genetic Algorithm
for Combinatorial Optimization
Quantum-Inspired Genetic Algorithm:
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 13 / 20

46. State of the art New Theory Order-2 QIGA Experiments Conclusions
Order-2 Quantum-Inspired Genetic Algorithm
for Combinatorial Optimization
Order-2 Quantum-Inspired Genetic Algorithm:
(quantum modelling of interactions in pairs of genes)
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 13 / 20

47. State of the art New Theory Order-2 QIGA Experiments Conclusions
Order-2 Quantum-Inspired Genetic Algorithm
for Combinatorial Optimization
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 then
3: p ← 00
4: else if r |α0|2 + |α1|2 then
5: p ← 01
6: else if r |α0|2 + |α1|2 + |α2|2 then
7: p ← 10
8: else
9: p ← 11
10: end if
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 14 / 20

48. State of the art New Theory Order-2 QIGA Experiments Conclusions
Chromosomes in QIGA2
000...0 010...0 100...0 110...0 111...1
x
20
40
60
80
100
f(x)
0.000
1.000
0.000
0.500
|
0.800
0.400
0.800
0.200
|
0.500
0.600
0.700
0.800
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 15 / 20

49. State of the art New Theory Order-2 QIGA Experiments Conclusions
Chromosomes in QIGA2
000...0 010...0 100...0 110...0 111...1
x
20
40
60
80
100
f(x)
0.500
1.000
2.000
0.500
|
0.800
0.400
0.800
0.200
|
0.500
0.800
0.700
0.500
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 15 / 20

50. State of the art New Theory Order-2 QIGA Experiments Conclusions
Chromosomes in QIGA2
000...0 010...0 100...0 110...0 111...1
x
20
40
60
80
100
f(x)
0.500
1.000
0.000
0.500
|
0.800
0.400
0.800
0.200
|
0.500
0.800
0.700
0.500
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 15 / 20

51. State of the art New Theory Order-2 QIGA Experiments Conclusions
Numerical experiments
Performance of four algorithms has been compared:
SGA, QIGA1, GPU-tuned QIGA1, QIGA2
Recognizable benchmark of 20 deceptive combinatorial
optimization problems has been used has been used
(knapsack + SATLIB benchmark)
1 Knapsack problem, problem size N = 100, . . . , 1000
2 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.
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 16 / 20

52. State of the art New Theory Order-2 QIGA Experiments Conclusions
Numerical experiments – results
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 17 / 20

53. State of the art New Theory Order-2 QIGA Experiments Conclusions
Numerical experiments – results
0 1000 2000 3000 4000 5000
Fitness evaluation count (FE)
1300
1320
1340
1360
1380
1400
1420
1440
1460
1480
Averagefitnessofthebestindividual Algorithms performance comparison
Problem: knapsack250, size N = 250
QIGA-2
QIGA-1 tuned
QIGA-1
SGA
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 17 / 20

54. State of the art New Theory Order-2 QIGA Experiments Conclusions
Numerical experiments – results
0 1000 2000 3000 4000 5000
Fitness evaluation count (FE)
620
640
660
680
700
720
740
760
Averagefitnessofthebestindividual Algorithms performance comparison
Problem: bejing-252, size N = 252
QIGA-2
QIGA-1 tuned
QIGA-1
SGA
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 17 / 20

55. State of the art New Theory Order-2 QIGA Experiments Conclusions
Numerical experiments – results
0 1000 2000 3000 4000 5000
Fitness evaluation count (FE)
5300
5350
5400
5450
5500
5550
5600
5650
5700
5750
Averagefitnessofthebestindividual Algorithms performance comparison
Problem: knapsack1000, size N = 1000
QIGA-2
QIGA-1 tuned
QIGA-1
SGA
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 17 / 20

56. State of the art New Theory Order-2 QIGA Experiments Conclusions
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
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 18 / 20

57. State of the art New Theory Order-2 QIGA Experiments Conclusions
Conclusions
In 17 out of 20 test problems (85%), the authors'
QIGA2 algorithm presented on average a better solution
than 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)
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 19 / 20

58. State of the art New Theory Order-2 QIGA Experiments Conclusions
Our Recent Papers on QIEA algorithms
1 Nowotniak, R., Kucharski, J., GPU-based Tuning of Quantum-Inspired Genetic Algorithm for a
Combinatorial 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 in
The Polish Grid Infrastructure, 2nd Scientic Session of TUL PhD Students, ISBN
978-83-7283-490-4
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 Genetic
Algorithm, 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 the
FIMB PhD students conference, ISSN 2082-4831
6 Nowotniak, R., Kucharski J., GPU-based Tuning of Quantum-Inspired Genetic Algorithm for
a Combinatorial Optimization Problem, XIV International Conference System Modelling and
Control, 2011, ISBN 978-83-927875-1-8
7 Nowotniak, R., Quantum-Inspired Evolutionary Algorithms in Search and Optimization, I
Wyjazdowa Sesja Naukowa Doktorantów PŠ, Rogów, ISBN 978-83-7283-411-9
8 Nowotniak, R., Kucharski J., GPU-based massively parallel implementation of metaheuristic
algorithms, 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, INotice
2011, ISBN 978-83-7283-407-2
10 Je»ewski, S., Šaski, M., Nowotniak, R., Comparison of Algorithms for Simultaneous Localization
and Mapping Problem for Mobile Robot, 2010, Scientic Bulletin of Academy of Science and
Technology, Automatics, ISSN 1429-3447
11 Jopek, Š., Nowotniak, R., Postolski, M., Babout, L., Janaszewski, M., Application of Quantum
Genetic Algorithms in Feature Selection Problem, 2009, Scientic Bulletin of Academy of
Science and Technology, Automatics, ISSN 1429-3447
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms 20 / 20

59. State of the art New Theory Order-2 QIGA Experiments Conclusions
Thank you for your attention
Robert Nowotniak, Jacek Kucharski Higher-Order Quantum-Inspired Genetic Algorithms