Quantum Meta-Heuristic Algorithm Based on Harmony Search

International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org ||Volume 4 Issue 10|| October 2015 || PP.13-18
www.ijesi.org 13 | Page
Quantum Meta-Heuristic Algorithm Based on Harmony Search
Essam Al Daoud
(Computer Science/ Zarka University, Jordan)
ABSTRACT: Harmony search is meta-heuristic optimization algorithm. It was inspired by the observation
that the aim of music is to search for a perfect state of harmony. A drawback of the harmony search algorithm
cannot find the global minimum easily and becomes very slow near the minimum points, moreover an exhaustive
search method should be implemented around the minimum points to get high accuracy. Therefore a modified
quantum search algorithm is suggested to handle the candidates points. The estimated results show that the
suggested algorithm outperform the previous harmony search algorithm and its variants.
KEYWORDS - Global minimum, harmony search, heuristic, optimization, quantum algorithm.
I. INTRODUCTION
Optimization problems arise in several applications such as manufacturing system, electrical engineering,
control engineering, molecular modeling, economics etc. In literature, various NP-hard combinatorial
optimization problems have been studied. Combinatorial problems such as assignment problem, closure
problem, knapsack problem, minimum spanning tree and traveling salesman problem [1]. In the most of the
optimization problems (continuous or discrete) there is more than one local solution. Therefore, there is a need
for efficient and robust optimization algorithm. The best results can be obtained in an optimization problem is to
check all search space. but, checking all the solutions is forbidden, especially when the search space is large [2].
The meta-heuristic algorithms introduce a suitable solution although it is not the most accurate one. Most of the
existing meta-heuristic algorithms imitate natural, scientific phenomena, e.g. human memory in tabu, evolution
in genetic algorithms, swarm intelligence in particle swarm optimization, Ant colony optimization, bees
algorithm, bat algorithm and wild dogs [3, 4]. Harmony Search (HS) is based on natural musical performance
processes that happen when a musician searches for a better state of harmony. In harmony search and its variant
several operations as improvisation and harmony memory update. However, the drawback of the harmony
search algorithm and other meta-heuristic algorithms is very slow around the minimum points specially with
multimodal functions [5]. On the other hand, Quantum Algorithm allows for superposition of classical
algorithms and, due to interference effects can exhibit different features and offer advantages when compare to
the classical case, but the observation of the superposition of states makes it collapse to one of the states with a
certain probability. Thus, the best known quantum searching algorithm has a complexity O( ) is invented by
Grover [6]. In this paper, we will integrate the harmony search with modified quantum algorithm such that the
harmony search will be used for exploiting good locations (related to intensification), and quantum algorithm
will be used to explore the search area (related to diversification).
II. QUANTUM ALGORITHMS BASICS
The second postulate of quantum mechanics describes the evolution of a closed system by the
Schrödinger equation [7]:



 || H
t
ih (1)
where H is the Hamiltonian operator and h is Planck’s constant. In quantum physics, it is common to use a
system of measurement where h = 1, so the discrete-time solution of Schrödinger equation is:
| > = U |0> (2)
where U is an unitary matrix. A general 2-dimensional complex unitary matrix U can be written as:
U= eitH
(3)
The basic unity information in the quantum computer is the qubit, which has two possible states |0> or
|1>, This can be realized by the spin of a particle, the polarization of a photon or by the ground state and an

Quantum Meta-Heuristic Algorithm…
excited state of an ion. Unlike classical bits, a qubit can be forced into a superposition of the two states which is
often represented as linear combination of states:
|> =  |0> +  |1> (4)
for some  and  such that ||2
+ ||2
= 1. There is no good classical explanation of superpositions: a quantum
bit representing 0 and 1 can neither be viewed as between 0 and 1 nor can it be viewed as a hidden unknown
state that represents either 0 or 1 with a certain probability. However; the processes in the quantum computer
are governed by Schrödinger equation which has no classical explanation.
The quantum states can be represented as vectors in Hilbert space rather than classical variables such that:
|0> = 







0
1
and |1>= 







1
0
(5)
and the superposition state is
|> =  







0
1
+  







1
0
=










(6)
The state of n qubits (a register) is represented by the tensor () product of the individual states of the
qubits in it. For example if we have two qubits in a register, and the both have the state 0 then the register
status is 00 , which corresponds to the vector
|0>  |0> = |00>=
1
1 1 0
0 0 0
0
 
 
         
 
   
 
 
(7)
The superposition of n qubits (or a register) allows each operation or quantum gate acts on all basis states
simultaneously, This type of computation is the basis for quantum parallelism which leads to a completely
new model of data processing. Shor’s algorithm is a good example of quantum superposition and parallelism.
Let | >=|0>|0> be the initial state of a quantum computer , then the Hadamard operation on the first register
leaves the quantum computer in the following superposition state [8]:
| >=
2 1
0
1
| | 0
2
n
n
i
i


  (8)
quantum parallelism exploited by applying a reversible function f on all states from |0> to |2n
-1>
simultaneously. In Shor’s algorithm f(x)=xi
mod n, and the computer state becomes:
| >= 



12
0
mod||
2
1
n
i
i
n
nxi (9)
However, the observation of the superposition of states makes it collapse to one of the states with a
certain probability. For example if we like to measure the quantum register:
 >= 



12
0
|
2
1
n
i
n
i (10)
then the superposition states will collapse to the state |x> with probability:

  ||)( x
t
x
MMxp (11)
and the state of the register after measurement
' |
|
| |
x
t
x x
M
M M


 

 
 
(12)
where Mx=|x><x|. Fortunately, quantum interference can be used to improve the probability of obtaining a
desired result by constructive interference and minimize the probability of obtaining an unwanted result by
destructive interference. Thus The challenge is to design quantum algorithms which utilize the interaction of the
superposition states to maximize the chance of the interesting states [9, 10].
III. HARMONY SEARCH
Harmony Search is a popular meta-heuristic optimizer, which was introduced by Geem et al. in 2001
[9-10]. The main steps in the HS are constructing a new vector from the previous vectors and replacing the
worst one. After initializing the harmony memory, the HS algorithm can be described as follows:
Repeat until termination condition is fulfilled
1- for each component i do
if HMCR  rand
i
newx = i
jx
if PAR  rand
i
new
x =
i
new
x  rand  bw
else
i
new
x = rand
2- if the new vector is better than the worst, replace the worst vector
where j 1 is the size of the harmony memory (HMS), HMCR is the harmony memory considering rate, PAR is
the pitch adjusting rate, and bw is the bandwidth. Mahdavi et al. introduced an improved version of the HS
where the bw and the PAR are updated as follows [11]:







iter
MaxIter
h
ebwtbw max
)(
(13)
where
m in
m ax
ln
b w
h
b w
 
  
 
(14)
and
( )PA R t = m ax m in
m in
PA R PA R
PA R iter
M ax Iter

 (15)
MaxIter is the maximum number of iterations, bwmin and bwmax are the minimum and maximum
bandwidths, PARmin and PARmax are the minimum and maximum PARs. Another development was introduced by
Omran and Mahdavi, where a global best pitch was used to enhance the ith
component in the pitch adjustment
step instead of the random bandwidth. Wang and Huang updated the pitch adjustment by removing bw and
using the maximum and the minimum values of the harmony memory. Most of the other HS variants attempt to
find a dynamic solution for parameter selection. However, the same situation arises as for PSO, as there is no
conscious connection between the selection of the parameters and the progress in the fitness function.
Initialize the optimization problem, which is the maximum weight submatrix (F) and HS algorithm
parameter:
























HMS
N
HMS
N
HMSHMS
HMS
N
HMS
N
HMSHMS
NN
NN
xxxx
xxxx
xxxx
xxxx
HM
121
11
1
1
2
1
1
22
1
2
1
2
1
11
1
1
2
1
1





(16)
where F represents the objective function and x denotes the set of each decision variable. Each row presents the
candidate solution for our problem; therefore, xi (from 1 to N) is the index of genes at the mutation matrix. N
pertains to the number of candidate solutions; it is the multiple of sample size. Under this context, the HS
algorithm parameters that are required to solve the optimization problem are also specified in this step. The
number of solution vectors in harmony memory (HM) is the size of the HM matrix [12].
IV. THE PROPOSED ALGORITHM
Although HS very efficient optimization method, but it cannot find the global minimum easily and
becomes very slow near the minimum points, therefore, the HS will be used to find the candidates minimum
points ( x ). The area of the minimum points can be detected by calculating the difference between two values of
the objective function at two sequent points, if the difference is less than  then the point will be handled by
quantum search algorithm. The quantum algorithm will be used to search all the binary combination of length m
around the detected point. Where m << n and n is the length of the complete vector x. In the following Quantum
harmony search algorithm two small numbers are used: 1 and 2 , where 1< 2.
Repeat until termination condition is fulfilled
1- Let xold = xnew
2- for each component i do
if HMCR  rand
i
newx
= i
jx
if PAR  rand
i
new
x
=
i
new
x
 rand  bw
else
i
new
x = rand
3- if the new vector is better than the worst, replace the worst vector
4- if | f(xold)-f(xnew)| >1 then Goto step 1.
Let y be a sub-vector of xnew of length m
5- Use two registers of length m
6- Let xold = xnew
7- Convert the registers to the superposition states: |1> and |2 >
8- Let s= ( * ) / 4m 
 
9- Repeat the steps 9-12 s times
10- Change the state |1> to -|1> if and only if
| f(|1>)-f(|2 >)| < 2 and | f(|1>)-f(|2 >)| ≠ 0
11- |1 >=
m
 H |1 >
12- Change the state |1> to -|1> if and only if 1=0
13- |1 >=
m
 H |1 >
14- Observe the register |1> and call it xnew
15- if | f(xold)-f(xnew)| <2 then break.

V. EXPERIMENTAL RESULTS
The suggested quantum harmony search algorithm (QHS) is compared with three meta-heuristic algorithms;
namely, the HS [13], global-best harmony search (GHS) [14], self-adaptive harmony search (SAHS) [15], Table
1 shows the used parameters for each of the tested algorithms.
Table 1. The parameters of the tested algorithms.
Algorithm Parameters
HS HMS=10, HMCR=0.92, PAR=0.35 and bw=0.01
GHS HMS=10, HMCR=0.92 and 0.01  PAR  0.99
SAHS HMS=50, HMCR=0.99 and 0  PAR  1
QHS 1=0.5, 2=10-10
, m=225
, HMS=10, HMCR=0.92, PAR=0.35 and bw=0.01
All the results are obtained by averaging 10 runs. However QHS cannot be implemented using the conventional
devices, therefore the time is estimated using equation (17), where v is the number of the number of candidates
minimum points and t is the required time using HS to explorer the candidates minimum points (HT), the total
QHS time is HST-HT+QT.
QT =v * ( * ) / 4t 
 
(17)
Table 2 shows and estimates the required time in (seconds) to find the minimum points for eight famous
optimization functions. which has several features such as regularity, multimodality, continuity, reparability, and
difficulty. The dimension of the tested function is 30. The suggested method faster than the previous methods
for all tested functions, moreover the differences become more clear if the dimension of the test problems
bigger.
Table 2. Comparison of three algorithms and QHS
function HS GHS SAHS QHS
Rosenbrock 205 206 150 83
Sphere 181 175 97 62
Ackley 166 143 124 56
Griewank 84 90 72 23
Schwefel 68 53 61 22
Step 55 57 42 11
Rotated-h-e 198 192 204 72
Rastrigin 180 175 179 64
VI. CONCLUSION
Although meta-heuristics algorithms are different in the sense that some of them are population-based,
and others are trajectory methods, but all the meta-heuristics algorithms based on exploring and exploiting.
However some algorithms perform better than others, which depends on the trade off between the
intensification and diversification. This study hybridized the harmony search and modified quantum algorithm
such that the harmony search will be used for exploiting good locations and quantum algorithm will be used to
explore the search area. The estimated results show that the suggested algorithm outperform the previous
harmony search algorithms. Moreover, the proposed algorithm can be used with any classical meta-heuristics
algorithms.
VII. ACKNOWLEDGEMENTS
This research is funded by the Deanship of Scientific Research in Zarqa University /Jordan.
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Quantum Meta-Heuristic Algorithm Based on Harmony Search

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