Hybrid Quantum Scatter Search Algorithm for Combinatorial Optimization Problems

(1)

! " # $% $& ! " # $ # %$ & ' " # $ # # ( # " # " ( ) ( * # ' * ( ( # # "( ' ! ( ( ( + + ( ,+ ( ( ! ' ( " ! # * " '()* % +$ #- - . "/ / # , 0 " ) * (" ! ) ( - # # # -' ! % & 10 23 4- 325 6 ) # ( - 7 ! ( #" ( # ! ! ( ! ! " ) (

(2)

- (" ( # # # - ) $ # ! # * * # 18 46 $ * ' * ( $ ( # # ( * " $ ( # " ) " # 9 "-# "-# ! # # ! " * #1:; /. <-2 =6 # ( * # ( ( " ! " # * "- * # ! # $ 3 # $3 1:; <- /. <6- $ 7 / ,! $/,12 =- 2 6 5 - "( # $ # %$ & ( ) ( # ( # * # # # * $ # * " ) (> ! $ # ! " # # + # ! 5 ! # ?@7 ( A 0 + B7 ( 12 6 ! ( # ( ( " # * "- # ) A ( * # + $ ' ( .// B ( < C + - (" #! # ! - ! ! . $ # %$ & # # ! A " - # #-

(3)

"-$ * # # ' # ) * " * '" ( ( # * - # # * ( ." ' # -* ( * ( ' ! -( #! (" 1325 6A % & 5 D D ( ! ! "E + ( % & ( ( " * ( % & % & ' ( + ( * ( # * ' - " * # ' * # , * 7 # * "- 18 46 * # " # * * " $ # * # * F - * ." * ( - # ( * # GG - # # * ! " ( ! ( ! ( # + ( 2 ' " ,! # - $ ,! # $, ! - ! " " $, * " #

(4)

#! ( * ! ! # # 1 /. <- 2 6 / , % & # ) 10 23 4- 325 6 ! # # " ( # + # # # - 7 ! # ( # ! ! ( ! ! " ) ( - - (" ( # # # - ) @ ( " # ! A # # B ! ( # H( I < " -( ( # ( C #" $ ( #! ( (" ( 3 0 ( # ! # # # ( ( ( # - "( ! " # $ %$ & # * # * ( -* - # * ( * #

(5)

" " * ( ) ( - * ( " ("* -' ' # # # - ! # * #- ( * # # # ( , # * ( ( " # ( ( " ! " # * ( - ( "! # ( ( - ( ( ( ! ! " * "-( * # % # & ( * ( ! ( ( # 5 ( '- ! "

GG

<<

==

GG

<<

0 / ! 5 ! # '- * # ) / / -&&& 0

(6)

-%%% " (" > * ! ( "! % #B& - ( # . " * "- " ! # ! # # ' # ( " ! * ( # ( ( ( " + (

-"

&&&

/ / -0 1 %%%& '" " ( ( " ! # * ( # ( ! ( # " ( ( (" # ! # ( ! # ( % # <& ! # ! ! # + " ! ( # J ! ( " +% ( & 0 2& $ -& 3 ( 9 ( ! # K K LMN K K 7MN K O 7MN K O LMN O K 7MN O K LMN O O LMN O O 7MN

(7)

%%) " ! " ! # # ( + # ! ( 5 # " * A *'" " ( * ( # " # * + - * ( " # ( ( "% # C&

GG

<<

==

GG

<<

0 4& , 56 6 *'" " # " * ( # ( ( "- + ' ( * ( J # # GG << == == GG <<

<<

GG

==

GG

<<

0 7& , 56 6 1 # (" . " / / # .// 1 32G<6 # ' " ( .// ,+ #

(8)

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

(9)

0 8& # # +

0 9&$. # %- -B& %- -<&

! +:- :/ :1 :2; + +:- :1 :/ :2 2 $ , ! ( $ ) ( ( * V V VB V V V VB V< VB V VB V< V

(10)

( * " + (" # * (" # * " % # G& ? - ( # * * ( ( # "( # #! ( 3 ! .// ! ( ( - -? # 23 # # # ( # # * " U # ( " # ! ( # 0 <& = # 3 ("/ ! 0 9 @ ( (" ( 3 0 @ (" ( 0 ! (" ! 0 F ,?/ ? $ ' ("9 P 0 $ "

(11)

( ! # ! ( ( * # # " # * "- " ! # 5 > * 7 ( ( * % # & ! 0 ! - # ! ( .// # " + . # + . 0 ->& (6 5 # ! ! 0 % 0 & # ! ! # -! ! ! * ! - " * * " ( 5 ! ! ( .// -! " + - ! $ ( # # ! ( (" # % # & 5 " # * ! ! ( -! ( ) U.// * ' 5 @ 0 !

(12)

-- # #"1 FG 6 # # ( .// ) .//J ) 1 4 2 - 7 / 1 - 4 4 / 0 --& $ * ! " # # ( # * " ! ! 3 "- J + # ! ( ' # ( # ( # * " ! ! # # -! ( * " ! " -9 ! # A9 ( 9 ( 9 ( * "-("* " ! " * " (" # ) .// ( ! ( " $ # " ( ( , # ( -- ' " # 9 9 # ! ( - ( # (

(13)

% $ ( (" # ( * % # & # - ! ( 5 " * - # * + ' " * ( - # (" * ( - #! * ( 30 ( 1 # A3 0 04 1( 04 11% 0 -/& : 6 7 , (! $ .// >! C B 3:W 3. " 5 ! ' # 8.// 15 XX6 .// " " ! + # + ' " # # ?/-VU9 ?U ' # 5 ! $ .// # ! " # ( # A $3 .// ( * # # 12 =6- $, ( * ! " # #

(14)

* $, 5 @ ( * ! " # * 12 6 0 ! - ! 5 ?B , ( ! # ) 'QB # # .// ' # . "19 GB6 "- 5 + 7 # 7 ' # " ) ( ( ' - ( ! ( -(" + # 4 - ( # % ) & # -# ( $ .//-$, $, 5 @ - $, # + % # B& P "-$, ! " # + U - # $ .// ( $, 5 @ "-5 ?B , $3 .// + - 5 ?B , $3 .// 0 -1& $ 0 ?@>&>4A+ . #

(15)

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

(16)

) ( ! ' - - * - ( ( ( ! ! : ! - ! ! # "- # - Z( ( ! ! " * # ' ( "-# " ( ! (" # ! % 1. "G 6 %& (& # 7 5 # 6 # "# . 7 6 5 / - 0 # ! " - <-? B- GB7B - GG 1.5 G46 #& # ,& * 7 / # / 76 82 # ( ,,, - <C%G&AGGB - GG4

1 /0 6 & & & C & " & " 7 + # /

. ### / #/ @ #

@/@ J

-1 32G<6 (& " & 3 & = & (& 7 # 9 /

( 0 @ # #2 # #

" - 4%C& AC C< - ( GG<

1/.G 6 %& #& # 3 7 6 5 / & 5

# ( ; @ ( - GG

1 ; 6 " & 0 D & 0 D . B& ' . 7 , #"

6 # 5

/ Z /- # C- G

1 U:GB6 & 0 D =& & 7 # / #

/ 5 / Z

(17)

1325 6 0& = ! & E33 & & * 7

2 #" AU9 #

0 #- U ) A

U 9 - , 0 2 #

8 2 3 )[ )7 - GG B

=-1:; <6 '& & F& & ' 7 3" " * , #" 5

:#/ . 8 . & $ ( ;< (

&. 2 - ,,, ,! "

- ,,,@ - ! - - C47 4G- <

1 FG 6 B& , & . & ) D 7 = 5

/ - / # Z

/- # <= <=C- GG

18 46 =& F 7 3" " : 7 . . A

# 4

12 =6 & & (& 7 3" " :#/

6 5 / 7 /2 # #

8 ? '

"=%G&- B BC- =

12 6 & & (& 7 3" " #, #"

:#/ + # 7 / 2 # 6 5

/ @ # B

J - # A

- 4- G< G<C-0

12 6 & & (& (&5=& $ 5 : 8 . 3" "

:#/ :> )* :&2 # % %

#. 9 $ #- 5 ##/

: # 8 " #$ #- 5 ? - @ 0

P ! " @ - ? 4C7 < 4- C7 9

? ! (

10 23 46 %& " " & 0& = ! 5 2 # -, 8 U 9 4G % 4&- BCG B=

(18)

19 GB6 %& % 7 5 # / 5

/ - Z /- # < <=- GGB

1 /. <6 & $ & " & # 7 : 3" " *

#, #" :#/ / / /

# < 7< - <

15 XX6 F& * 7 ?:+:6 ( ? 6 5 /