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E. RIEFFEL FX Palo Alto Laboratory, 3400 Hillview Avenue, Palo Alto, CA 94304, USA W. POLAK 1021 Yorktown Drive, Sunn...

Author: Rieffel E. | Polak W.

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E. RIEFFEL

FX Palo Alto Laboratory, 3400 Hillview Avenue, Palo Alto, CA 94304, USA

W. POLAK

1021 Yorktown Drive, Sunnyvale, CA 94087, USA

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% )", (L<), " ! . ")+ * " % p( ) ")( . =( " ) , ( (( " , ")( /$ . = L< )" % ( pp( , " " ! "* ! * . " " % )" 3 " M 19954 : 1996] , %+ ")( , ." " . 6 , ( ")( " p$ , % . 6)")( $ )+ % , ")(

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0 " , + " , , $ , . . $, ( /)"$ . ( . /)"$ () . 6 " ( " " " , . () " " " " . : " ( * " $ ) ( " " (/" ( , ! 6 " 6 " 1958]. --" jxi (% "- ($ ( )% " . 0 (" haj (% * "-" jxi. , ( fj0i j1ig ( % "" f(1 0)T (0 1)T g. G%()% " " )% )% "( $ % j0i j1i: aj0i + bj1i * , "" (a b)T . $ , ( " ( " * . = , * j0i "" (0 1)T j1i "" (1 0)T , * !. -( $ hxjyi ( ) ( ") ) ". = , j0i | ", h0j0i = 1. " j0i j1i h0j1i = 0. -( $ jxihyj | jxi hyj. = , j0ih1j ( , " () j1i j0i j0i (0 0)T , .". j0ih1jj1i = j0ih1j1i = j0i 0 j0ih1jj0i = j0ih1j0i = 0j0i = 0 : F ) j0i = (1 0)T h0j = (1 0) j1i = (0 1)T h1j = (0 1) j0ih1j * ." $ 1 0 1 j0ih1j = 0 (0 1) = 0 0 : 12

' ( % ) ( ( " / "$ ( " ), ( " ( . 4) = , ( , %+ j0i j1i, ! $ X = j0ih1j + j1ih0j: , " . ( ) / X : j0i ! j1i j1i ! j0i " ! ) ( ( ". 3.

- ( ")( | . " 2- " " " , " / " " ( fj0i j1ig. , ( j0i j1i * j "i j !i j %i j -i $ / C D, C D .". - ! ")( " (+, ( fj0i j1ig,

" )* , ( . < () , ( (, . ( ( . " ( (% j0i j1i, ( C D " " ( 0 1. =, " " ( , ")( ) ) $ j0i j1i, , aj0i + bj1i, a b | " " , " jaj2 + jbj2 = 1. ) $ /, " ) $ ( fj0i j1ig, , j0i jaj2, , j1i | jbj2. K " ( * ( * ) $ , )! * " ( " " /$ . F ")( ( , (% ." $ /. 1" "" "* " " ) ) , . . " ) ( " ) , , "" " " ,

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)% ) " "% E E 19874 E . 1992]. ,(, " "% ) $ ) / " "%, , RSA. )$ %, " Y E( " "%. : )+ ()

) ) ") ) ") "). ' ( " ) % N, " ) . : )$ (* )" *.

- " Y E() $ ( , /) " E(), " * . N "* * . $ E(). 6 , ( $ " "% Y E() ( , " ) "* ( " / )%+ (: " " "* ( ) ( 0 ! j "i 1 ! j !i 0 ! j -i 1 ! j %i: E( /, ( "* ) ( . ( . 0 "" ( , E( Y (+% ) )) ") "), "" ( " ( " ) ( ( !) /. ,( " /$ , ) , "" ( , . . ( , " ( !. ' ( () ) " "%, () ) (. Y E( () ) 50% ( , . ., "% () ( . 14

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0 ( / " " " " (9", " n , %+ . C " * %, "* %+ ! . 6 " * n $, ) ) , " "% , " , (+ )

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$, ( #! aj0i + bj1i,

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" " n $, "* $ % " . - * % . : * " $ ( % * A. 6 "" * ) " , " % "% " . 0) V W | 2- " " " ( fv1 v2g fw1 w2 g . F " ()

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dim(X) dim(Y ). : j00i+j11i " , " ")( . * ; 6) , " a1 a2 b1 b2 , (a1j0i + b1 j1i) a2 j0i + b2j1i = j00i + j11i, . ".

;a j0i + b j1i ;a j0i + b j1i = a a j00i + a b j01i + b a j10i + b b j11i 1

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'" 2.1.2. ), "" ( ")( " ) " ( , %+ ) . ) , $ . , , $ 2-")( . : %( ) ")( * "" aj00i + bj01i + cj10i + dj11i, a b c d " " , " , jaj2 + jbj2 + jcj2 + jdj2 = 1. 0 * ")( ( fj0i j1ig. 6 ) ( )%+ (: ; ; aj00i + bj01i + cj10i + dj11i = j0i aj0i + bj1i + j1i cj0i + dj1i = ; ; = uj0i a=uj0i + b=uj1i + vj1i c=vj0i + d=vj1i : p p 1" "" u = jaj2 + jbj2 v = jcj2 + jdj2, c=vj0i + d=vj1i | " . M ( " . G" , "" ) ( ) . F ; ( % u2 = jaj2 + jbj2 j0i, " ) , j0i a=uj0i +b=u j1i , % v; = jcj2 + jdj2 ;

j1i;, " ) j1i c=vj0i +d=vj1i . 1. ". j0i a=uj0i +b=uj1i j1i c=vj0i + d=vj1i % " , " () . F ")( . 6 " ( *

"" ( ( . * ) . = , * , % ")( " , ")( . = ( ." ) ( , " ) ")( , .) " . ) ")( " , + , ")( j0i, , ")( j1i. G%( " *

) ) ", ) "* . F k ")( ( 2k * ) mi . G%( ) , %+ k ")( n-")( , 2n - H " S1 : : : S2k , H = S1 : : : S2k . . k ")( mi , Si " i. O ( Si % " ) " Si . 6 " ) .) " ), )

. 6) , , ) , ) " ( % ) ( ", . .

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))%. = , p j00i + j11i ), .". , 2 ( j0i 1=2, ) , ( . , " ( ( , ( "" j0i 1 0, , "" ( ( ( : "" j0i j1i . : , ) ( ; 1 ( . : ) , p j00i + j01i ), .".

; p

= j0i p1 ;j0i + j1i.2 Y %( ( j0i 2

1 j00i + j01i 2

, ( ( . Y (, ( ( "" j0i, , ( ( . , , ) , "" $ ) ), ) )+) % ) , "" , " $. 3.4. &'

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Y E( ) ). 0 * , Y % $) ()* j0i. ' , (+ () j00i. Y E( % $), "* ()* j0i. Y (, Y j1i, ) E( () * ) . M , (+ " , , $ )

) ). 0) , . Y E() ) )) , " % ( " . 6 18

"*, , * ) ) $ )+ ) , Y E( ! * ) . /$ . :)+ ) "" , " )! ! +! %

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") ), ) "" E. , ", ) ." " , E . 1" (, ")% ") (9 + % " . E ! E )%+ ." * ( > M* " 1997]. ( . = , ( ", , )+ Y , , " E(. 1 , . " "* , , "" ', 0 " , )(" . <* * )%+ $ '0: (% (% Y , E(. . (% E(, Y . ) * ( (% , .. (% (% * . '" ) " * (9 + % E( $ Y , (. ' , E(, Y ) '0

( " . 1" ( )+ )%+ " . !, * " , . , Y E( () ) (% " ) ) .

, , " %+ " $ , ) , '0 * () ) " )+ /$ , " ! ! () " .

19

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6 " " , " " . '%$ * " " ) 3! . 0 . % . G ( " " " , " % , % ) . G%( ( " " " * $. 0) $ M ) M )%+ " " * . < $ M ) (.. ) ( ) " , " MM = I. G%( ) ( ) .%$ " (. O ( * "" " " " . * , " ( ) , ( . 1.. " * ( ( . 8 * " " " , , "" " E, " 1// , ) ( ( . E ( ( . " " CG"$ D 1996]. 4.1. (

" " ( 1-")( " . ) , ( % % ( ". : )%+ $ ( . 1 0 I : j0i ! j0i j1i ! j1i 0 1 0 1 X : j0i ! j1i j1i ! j0i 1 0 0 1 Y : j0i ! ;j1i j1i ! j0i ;1 0 1 0 Z : j0i ! j0i j1i ! ;j1i 0 ;1 : ,( . ( % (+ . I | * ( , X | $ , Z | $ /, Y = ZX "( $ ). 0( X )* 2.2. 0 , . ) , ( ) . = , 0 ;1 0 1 Y Y = 1 0 ;1 0 = I: 20

CONTROLLED-NOT Cnot, ) ")( )%+ (: ")( , $, ! ( , )%. " j00i j01i j10i j11i ()% ( )")( { ! " " " . 6 , ( ( . / ( ( / * ) . ! " " . N , " / )), . ) . 1" / j00i j01i j10i j11i ( " * " (1 0 0 0)T (0 1 0 0)T (0 0 1 0)T (0 0 0 1)T . 1 ( Cnot 01 0 0 01 j00i ! j00i BB0 1 0 0CC : j01i ! j01i @0 0 0 1A j10i ! j11i j11i ! j10i 0 0 1 0 =C 0( Cnot ) , .". Cnot not Cnot Cnot = I. M , Cnot , "" ) ( ( . O ( ( " / " , ( ), " " " ( . Cnot ( % )%+ =" ")*" ( ) %+ ")( , " | ) $ ! ")( . =" " ")*" (% ) ) 0, ) ( , " ) %+ ( 0. (+ ), ) %+ ")( * ( " ". CONTROLLED-CONTROLLED-NOT $ ")( !, " ), ( ")( $. >/ " * "" " )". , ")( $ / " (*% + % " , "" " *.

4.1.3. { . 6) * -

( ( | . ( Y , "" H : j0i ! p1 (j0i + j1i) 2 j1i ! p1 (j0i ; j1i): 2 0( H ( * . 6 ) j0i, H ) $ p1 (j0i + j1i). 0 H " n ")( 2

21

E. Rieffel, W. Polak

, ) ) $ % 2n * , (

0 2n ; 1. (H H : : : H)j00 : : :0i = n ;1 2X ; 1 1 = p n (j0i + j1i) (j0i + j1i) : : : (j0i + j1i) = p n jxi: 2 2 x=0 0( W, " H " n ")( ( O{Y . N * ") W1 = H Wn+1 = H Wn : (4.1)

4.1.4. ! " . ," , ) " " " . 0 " * " ( ) ) M)" ) M)" 1982]. , "% ) ( . 0 * , U | . " )%+ ) ( , " U(ja0i) = jaai jai. 0) jai jbi | " . U(ja0i) = jaai U(jb0i) = jbbi. ! jci = p1 (jai + jbi). 1 , ) , ) 2 ; ; ; U jc0i = p1 U(ja0i) + U(jb0i) = p1 jaai + jbbi : 2 2 = .". U " )%+ ( , ; ; U jc0i = jcci = 21 jaai + jabi + jbai + jbbi 1 p (jaai + jbbi . 1" (, ) $ , " 2 * " " , )+ ). : , " * , .". )" , %+ ( " ) . - * , "" " ) " , "" . -" " , " " *. , " " $ * " . 0 n $ ) aj00 : : :0i + bj11 : : :1i aj0i + bj1i *. -* . $ () ( " ( fj00 : : :0i j00 : : :01i : : : j11 : : :1ig, ( ( " , (; ""- )

; ( . : n $ aj0i +bj1i : : : aj0i +bj1i aj0i + bj1i *. 22

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. " ) : " $ . 0 " ) " " ( ) '0 . 1" "" '0 * ( ,

) ( /$ () / " ) $). ' * /", " "), "" 3, ")( )! * " " " ( /$ . 1 $ | . $ , * ) " %. M ")( , "" , )% " " ( . 1 $ * ) $ * " , .". " .

-% " $ ) . 0 ) " " ( $ . 0) , , Y E( ) )) /$ %. -* ) ) $ '0 ; 0 = p1 j00i + j11i : 2 0 * , Y )% $), E() | )%. F", " $ % , " Y * ( $, " E( | .

4.2.5. . Y " " ( , "-

)* E(), " ) 0 3. . Y ( fI X Y Z g ")( ) 0. 0( ")( ) , ) ")( ( . 0) " ( $:

23

E. Rieffel, W. Polak

M 0( = ; 0 0 = (I I)0 p1 j00i + j11i 2 ; 1 1 1 = (X I)0 p j10i + j01i 2

2 = (Y

I)0

3

3 = (Z I)0

2

;

p12 ;j10i + j01i ; p12 j00i ; j11i

M Y ")( E(). E( CONTROLLED-NOT " ) . 0 CONTROLLED-NOT 0 ")( B ; ")( ; ; 1 1 1 0 = p j00i + j11i p j00i + j10i j0i p j0i + j1i

; p12 ;j11i + j01i 2 2 ; ; 1 1 2 = p ;j10i + j01i p ;j11i + j01i 2 ; 2; 3 = p1 j00i ; j11i p1 j00i ; j10i 2 2 2

1 = p1 j10i + j01i

2

;

p12 j1i + j0i ; p12 ;j1i + j0i ; p12 j0i ; j1i

j1i j1i j0i

, , E( * ")( , )+ " . N ) j0i, . , " ( 0 3, ) j1i, | 1 2. 6 E( Y H " ) ")( ). 0 - 0 ")( H ( ")( ) ; ; 0 p12 j0i + j1i p12 p12 (j0i + j1i + p12 (j0i ; j1i) = j0i ; ; 1 p12 j1i + j0i p12 p12 (j0i ; j1i) + p12 (j0i + j1i) = j0i ; ; 2 p12 ;j1i + j0i p12 ; p12 (j0i ; j1i) + p12 (j0i + j1i) = j1i ; ; 3 p1 j0i ; j1i p1 p1 (j0i + j1i) ; p1 (j0i ; j1i) = j1i 2

2

2

2

F "$, ")( , ) 0 3 1 2.

4.2.6. % &. N! $ % " -

$, " " " ( , " ) ). 0 " ") " " , ) $ ( () ). 1 $ % ")( ." )+ E) . 19974 = . 19984 E . 1998]. 24

Y + ) ! ")( . ,

. ")( ' = aj0i + bj1i E() " " ". -" " , ) Y E( ) ")( ) ) 0 = p1 (j00i + j11i): 2 Y " )%+)% $ )) " " ")( ), " )* , " $ ) . : )%+ Y / " ")( ', "* % ) ) . : ; ' 0 = p1 aj0i (j00i + j11i) + bj1i (j00i + j11i) = 2 ; = p1 aj000i + aj011i + bj100i + bj111i 2 " Y ) ) ")( , E( | . 6 Y " Cnot I H I I " .) %: (H I I)(Cnot I)(' 0 ) = ; = (H I I)(Cnot I) p1 aj000i + aj011i + bj100i + bj111i = 2 ; = (H I I) p1 aj000i + j011i + bj110i + bj101i = 2 ; = 12 a(j000i + j011i + j100i + j111i) + b(j010i + j001i ; j110i ; j101i) = ; = 21 j00i(aj0i + bj1i) + j01i(aj1i + bj0i) + j10i(aj0i ; bj1i) + j11i(aj1i ; bj0i) : M Y ")( , ) % j00i j01i j10i j11i. ) " ")( E( " ) aj0i +bj1i aj1i +bj0i aj0i; bj1i aj1i; bj0i

25

E. Rieffel, W. Polak

. Y ) E() ) " " ( . , , Y ( ")( '. F . , " $ $ ) * " . E(, " " ( Y , , "" $ ) ")( Y 0 ")( : /" 00 aj0i + bj1i I 01 aj1i + bj0i X 10 aj0i ; bj1i Z 11 aj1i ; bj0i Y 0 )%+ ( " ) , E( * ")( Y '. M , . " )%+ " . 5. *

+ ( )* , "" " " * ( , ! " " , " " " %. , 4 , ( " ( . - " NOT (=N) * ( , "" AND (F), OR (FGF) NAND (2F-=N) " % . : , c + % " ( * " " . % ( ( , " ) )+ %( " " " " %. E , a * % ( , + % " * " . " " . 5.1.

- " ) $ ) ( + % (- ".( . 6 ) ) ( U1 U2 C) D ( j0ih0j U1 + j1ih1j U2 * ) .

CONTROLLED-NOT (--=N) * ( !, "" Cnot = j0ih0j I + j1ih1j X: 1!")( CONTROLLED-CONTROLLED-NOT 1// , 4, "* . ) : T = j0ih0j I I + j1ih1j Cnot : 26

, 1// T * ( ( () ", " "" + % * AND NOT: T j1 1 xi = j1 1 :xi T jx y 0i = jx y x ^ yi: F ) 1// * %()% " ")% ")% ). = , )%+ " * , ) 1// CONTROLLED-NOT ( . 12)

x y . ( , s | ) )% 2, c | , c0 | . ( , E" '" '" . 1996] )% ( * . " " C) (D, , "" F = j0ih0j I I + j1ih1j S S | $ , + ( S = j00ih00j + j01ih10j + j10ih01j + j11ih11j: 8 * , F T, % " " " . 6 " 6 1985], * ( " %( " " /)"$ . " " * ) " 1 % E 1997]. 0 . ), CD 1 % * ( ( " ")( . G%( " " /)"$ f m k ( * ( " " %, , )+ ) " " , " f. (m + k)-( ( Uf : jx yi ! jx y f(x)i, ( ( "%%+-FGF ( ) " )). - Uf , " (, ) %( /)"$ f. 6 , ( f(x) Uf " jxi, )*) k ) jx 0i. 0 " ") f(x) f(x) = 0, Uf Uf = I. >/ " ( Uf : jx yi ! jx y f(x)i (* )%+ (

27

E. Rieffel, W. Polak

K T F % ) " " " , " ( . 6 , ( )+ ) ( c % / * ((+ / / " ), ( + ( . E" . E" . 1995] " , Cnot ( ( " ) (. 6 * )%+ ( ( cos sin ei 0 0 e;i : ; sin cos

0 6 6 2, "* Cnot1 , ( ) ) ( " . -" () , " ( % "% )+ " . 5.2.

8 , Uf " ) %, " ) $ ? , ) : " ") Uf | . ( , " ( " ) $ , ) $ % ) . 1" ( * f(x) n ) x " Uf . 1" .//" " . 6 " "% )+ " ) . 1", ( " )%+ /)"$ ) $ . j00 : : :0i n-")( . 6 ( O{Y W 4.1.1. ) ) $

X ; jxi: p1 j00 : : :0i + j00 : : :1i + : : : + j11 : : :1i = p1 2n 2n x=0 6( k-( j0i, , ) % , )2n ;1

X X ; 1 X Uf p1 n jx 0i = p1 U jx 0i = p n jx f(x)i 2 x=0 2n x=0 f 2 x=0 2n ;1

2n ;1

2n ;1

1 < # $ #" C not #! . | . .

28

f(x) | )%+ /)"$ . : ) , " ") n ")( % ( 2n , " ( - " " , " "" * ( ." $ (9! / " . CONTROLLED-CONTROLLED-NOT 1// T, " %"$ ) : = ) $ % * ( -"( $ x y, " x = 1, y = 1: ; H j0i H j0i j0i = p1 j0i + j1i 12 (j0i + j1i) j0i = 2 ; 1 = 2 j000i + j010i + j100i + j110i :

0 T " ) $ , ( ) ) $ % ) : ; ; T H j0i H j0i j0i = 12 j000i + j010i + j100i + j111i : ) )%+)% ) $ % * "" ( $) "9%"$ "" / /)"$ . = x y x ^ y ) " (, ) ) ") ( $ , , ." , ") / /)"$ . 0 , ")( )% ) , ")( | )%+) % f. = , )+ " " , " ") * ) " ) , " ) * * ( , "" ) ) . :) %( " | . (, + %, " ) ) " " (, ( * ) ( %. O " " " () $ . , " : O )* . , . ( "% ( " (, ( )%+ ( )% ) ) , , ( ")% ( . 0 () ) 7. =* (+ f(x). ' ( 3, ) " ( ) ) f. 6. ,- .

1994 ) 0 3, ( 6 : ( )( " * : 1997]), " {

29

E. Rieffel, W. Polak

* * n- " " %. = , % +) .//" * $ . = ( .//" " " , , G G G G 1993], " ." $ ) . | . ( $ / M, %+ n log M. G% ( " ) , .//" * )+ ), ( " / " , RSA, " % *

. (. ) 3 ( %+ ( )!, () " " ( ( " . ( , "% 3, ) ( * " " /)"$ . 3 ) " ) ) $ /)"$ . M " ( ) , ) ", "" " " ( ) , /)"$ , ) " " , ( ). : " % + , " % , )* * $ M. ( " , " ) " , ) +! . = ( ) "% , " ( ) ( ( ) , " (, " ( ) ( ). 6 " ( ) , ( 3. 6.1. /#*

(+ ) ( ) )% ( . 1", ( ) ()% /)"$ r /)"$ , ) " , ), % " " 2r . 6 " ( ) (DFT, 60) ) N ) (" ) 0 2) " N, /)"$ %, ( | . $ 0 N ; 1. 6 " ( ) /)"$ r | . /)"$ , "$ " , " Nr . N r N ( ", ) () /)"$ , ) " , ), % " ", " Nr . ) ) () ( *!, ) , ( " " " Nr . E ( ) (FFT, E0) % DFT, N | " . - ( ) (QFT, -0) DFT, , "*, "" FFT, )% " . --

30

( ) ) ) " , ""

X x

g(x)jxi !

X c

G(c)jci

G(c) | . " ( ) g(x), x c )% , "" $ 0 N ; 1 ( ). N ( , "" ( ) , , ) jci, ( ( jG(c)j2. 0 " ( ) P " " /)"$ g(x) r, " c G(c)jci, G(c) )% , c " , " Nr . 1" (, " , ) % , " Nr , j Nr . =, "" )* ( , " ( ) % ( )

, " % % ), .. , "

N. , " ( " , " ( ( , " $ . - ( ) UQFT ( N = 2m ""

X 2icx UQFT : jxi ! p1m (6.1) e 2m jci: 2 c=0 6 , ( 3 ( , ( ( " ( ) .//" . 3 ", " ( ) ( 2m * " m(m2+ 1) ( . ' ) 2 . , | $ )* " ( Y H ( Hj ( ( Y , " j-) ( )), ) | $ )( ( 01 0 0 0 1 B1 0 0 0 CC Sj k = B @0 0 1 0 A 0 0 0 eik;j 2m ;1

k;j = =2k;j . ' ( ) k- j- ( ( . - ( ) * H0S0 1 : : : S0 m;1 H1 : : :Hm;3 Sm;3 m;2 Sm;3 m;1 Hm;2Sm;2 m;1 Hm;1 " ) (+ ( . N FFT ) , "" 3, (+ ( " " (.

* ( 3 1997].

31

E. Rieffel, W. Polak 6.2. - .

0 3 * )%+ , * M = 21. 1. . 0 ( a. N a c M, )* M. ) )% . 0) m () " , M 2 6 2m < 2M 2 ' ( ( " (, ( " $ , 3 /)"$ , % " , () , (

(]. F ) " , "" 5.2. f(x) = ax mod M $ 0 2m ; 1. 0) , " ( 1m X jx f(x)i: (6.2) 2 x=0 . 0 * a = 11 ( ( ). 1.". M 2 = = 441 6 29 < 882 = 2M 2, m = 9. 1" (, 14 ")( , 9 x 5 f(x), ()% ) $ (6.2). 2. , , f. - ( ) ) /)"$ % ) , . 8( " ( ) ) /)"$ f, ( , /)"$ ) * , f. 6 " , log2 M] ")( (6.2), " " f(x). 0) ) u. : ( u "" 4 ) " * ) $ . ' " ) , .) 2m ;1

C

X x

g(x)jx ui

c % " * C,

(

g(x) = 1 f(x) = u, 0 ). , , " , " ), , g(x) 6= 0, % ) ) , " ), , g(x) . /)"$ , ")% +. N ( ) " ), ( . = " * %, " " / " % " . 32

. 0 * , ) ) $ ) (6.2) 8. : . (= . 2 " 9 ( 4 ( f(x) ) ) f. 3. ! ". 8 jui ( () , .) . 0 " ( ) " %, )) 2.

UQFT :

X x

g(x)jxi !

X c

G(c)jci

F ) ), , " r /)"$ g(x), 2, " , ) " ( )

X 2m cj jj r i j

) )%, " ", " 2m =r. - r 2m , ( , m ! ( ) ( $ , " 2r .

. 2

P

x C jx 8i )x2X 2, X = fxj211x mod 21 = 8gg. " ( ) . . = . 3 (*! ) " ( ) " %, )) 2. =( , . 3 | . / " E 0( ) /)"$ , " . 2. . ) /)"$ f 2m .

33

E. Rieffel, W. Polak

. 3

4. $ . F ( ) ) v. ), " % " , " ( , " 2m=r, *m . . ) v = j 2r " j. ( ) j r () ) , . ) "+ ( 2vm (= rj ) ( , q r. 6 , (+ ) " ( ) ! " " ( , ) * ) ) . - % " , * ( ) $" , " * ( ")% ( 2vm ' " " ! * B. . 6 ) , ! ) v = 427. 0 " ") v 2m , r " () 2m .) * * ( ")% ( , * B. : )%+ ( $ * , * B, i ai pi qi "i 0 0 0 1 0:8339844 1 1 1 1 0:1990632 2 5 5 6 0:02352941 3 42 211 253 0:5 " " 6 = q2 < M 6 q3. 1" (, q = 6 " /)"$ f. 5. &' M. - ) q !, ) N" .//" " , % aq=2 + 1 aq=2 ; 1 $ (+ M. 0 , " aq=2 + 1 aq=2 ; 1 ) (+ $ M )%+. N q

34

/)"$ f(x) = ax mod M, aq = 1 mod M, " ") aq ax = ax x. N q | !, *

(aq=2 + 1)(aq=2 ; 1) = 0 mod M: : , " ") aq=2 + 1, aq=2 ; 1 % " M, aq=2 +1 aq=2 ; 1 % $ (+ M . . 0 " ") 6 ! , ( a6=2 ; 1 = 113 ; 1 = 1330, ( a6=2 +1 = 113 +1 = 1332 () ) (+ M . . =,6(21,1330)=7 =,6(21,1332)=3. 6. !' . =" ) , .) $ *

M: m 2 (1) M v ( " " ") r . (2) 0 r * j ) (+ * , . q , . (3) 3 5 ! M, "" M. (4) 0 /)"$ f(x) = ax mod M !. 3 ", ( * M " %.

6.2.7. ( ) *+ 2 * , . ," , -

2 * % ) . E E 1997] " , * (* . N "% 2, () ) $ " " " /)"$ . -* * . " , " ( ) " ) $ ( ) . /)"$ . -* . ( ) )%+ u , , / )% ) ). F . ( . 0 , "" . )* * ((+ , % ) " " ( " ) $ . 0 " ( 2nP ;1 ) , UQFT I, " C jx f(x)i ) x=0

C C0

0

X X

2n ;1 2m ;1 2ixc 2m j

x=0 c=0

e

X X X u xjf (x)=u c

e

c f(x)i

2ixc 2m j

c ui

(6.3) 35

E. Rieffel, W. Polak

u f(x). 1, ) , ) $ ) , ) 3, * u. - ( ) % " ) /)"$ gu , " ) u,

(

gu = 1 f(x) = u 0 ), ! ) " . =( , ) u " / )% ( " % ) *% ) ) ). 0( UQFT I * "" UQFT I : C

n ;1 X 2X

u2R x=0

gu (x)jx f(x)i ! C

0

n ;1 2n;1 X 2X X

u2R x=0 c=0

Gu(c)jc ui

Gu (c) | . " ( ) gu(x), R | f(x). F c 4 5 "" * . 7. 1

E " * "" " C x * * , ", )* P (x) | D. 6 ( ": " /$ ( " " /. = , ) " " / * "" " " ( $ , )* C "%+ % $D . Y " * ( , "" " " , " )* C " x " *D

( . " )") ( )* P . = , P(x0) "" /$ * P (x1) x0 6= x1.

" * /$ % " ( )* P . = , " / ) ") ) ", / " ) . ) ), "*, , )* ( (3- SAT " " /) )")) * . " , " ( % " ). = (+ ), " ) ) ", ), * " " )! | . "* )* P(xi). 6 " N . ( ) " () O(N) " P. = " * " %, "" " >, ) ) p " * ( % " O( N) 36

" P . 1" (, " > > 1996] ( .//" , %( ) ", " " " %. ," , ) " > E . 19974 E) . 19964 M" 1997]. , " ( " () ) ". 0 " ") )+ )% " " . " , )%+ ) , * * , )+ )% "* .//" "

) ". :/ . :/ . 1998] )% > " " " ) . " , ( " , * " ) " , " ! NP- * . E . E . 1998], >, " , (+ " " . " " " " ) " . N " * , " ) )+ ) ) " ( , " . *, ( ()% , " "%% " " " . N 3 " " * , * )* , "

." $ ) " , ) )")) ( % ) (, ) " " . 1 K "* ( " " . " " ) ". N " " , ! )% " . N ) "% , , "" . " , ) ) *!, . ) ) $ . : , " +! , " " .//" K. " " .//" . " $ + % . " . =, " "), " $ " " " % (% ." $ , . " " *, ( "% . Y K " " ) ", ( .//" , >, ) " . . : " " | . , " " | * ) " . . 6 , " () ( " " % ) ! " , .//" () .

37

E. Rieffel, W. Polak 7.1. - 2

Y > " . x, " " )* , ) " N. 0) n () " , 2n > N, ) Up () " , " " ")% /)"$ % P (x), %+)% )* ( ) 1). Up : jx 0i ! jx P(x)i: 0 | " , 5.2. P * xi UP " ), nP ;1 *+) ) $ % p1 n jxi 2n * x 2 x=0 , ) 0. : ., ! "

1 X jx P(x)i: (7.1) 2n x=0 :* ) . ) $ . 6 %( x0 , " P (x0) | , jx0 1i () % ) $ ) 2. 1. ". ) " p1 n ,

2 , ) ) $ x0 { 2;n. K

, ( " ). 2 " (, ) " jx0 1i, " P , %, ) " jx 0i, " P * | ) % . 0) " ( " ( , " ")( " , " P(x). 1" "" ) , )%+ ( ) , )+ ) " , ) 1. . )Pk " ) ). 2 p1 k jxi 1i, 2 i=1 k | . 6 ( . . N ")( P (x) 0, $ , ) $ % ). 2 . Y > )%+ : (1) 0 " , *+ ) $ % * xi 2 0 : : :2n ; 1]. (2) P(xi) . . (3) F ) aj ;aj " xi, P (xj ) = 1 ('//" . $ 7.1.2). >/ " ) . : (4) 0 ) ) xj , )%+ P(xj ) = 1 (- .//" 7.1.1). ) )%+ ) "" p

38

n;1

Y ) xi, "pP(xi) = 0 ( *. (5) 0 2,3,4- 4 2n . (6) : ) . E . E . 1996] $ >. , " , . ( % * ), ) " * ) " ( . , "* " , , p )+ ) " x0 , " P(x0), 8 2n $ p 2,3 4 (" 0,5. 0 4 2n $ (" * 2;n. F, p , $ ) % !. = , 2 2n $ (" ( * " 1. :)+ ) * " " , $ ) ) ) ) . 0 " $ ), * )) ) , ! " ) )) . - $ ) | . ) ( , " % + " " . 1" (, " ( " *) %, , , () ) " ) % * . : , ( ) ) ) " ( , , " . E . E . 1998] % >, " ) ( ) " $ . ' (+, ) * . > ) " ) " " " , , " "" /)"$ > 1998]. F ) ( !, > "* ", " " , " " " " % % O(log N), ) ( O(1) " " %. 0 " > * ) " "

39

E. Rieffel, W. Polak

. E . E . 1998] " , > * p ) , . * " O( N).

7.1.8. . " *. 8( )+ %

" " %, * (

p ) ( . E , ( ) O( N),

* ( .//" .; -" "$ () ", % * ) O(n) = O log(N) " . =) , ( NX ;1 i=0

ai jxii !

NX ;1 i=0

(2A ; ai )jxii

A ( aj , $ N N

02 2 BB N 2; 1 2 N : : : D=B BB N N ; 1 : : : @ : 2: : : 2: : : : :

1 N C 2 C N C C: ::: C A 2 2

::: N ; 1 1" "" DD = I, D | ) , , * ( " . 1 " ) ( .//" $ ; . ( . 0"*, * * O(n) = O log(N) " . : ) ( >, D * "" D = WRW, W | ( O{Y ( 4.), N

N

01 0 : : : 0 1 B0 ;1 0 : : :CC : R=B @0 : : : : : : 0 A 0 :::

0

;1

8( )( , D = WRW , * , R = R0 ; I, I | * , 02 0 : : : 0 1 B0 0 0 : : :CC : R0 = B (7.2) @0 : : : : : : 0 A 0 ::: 0 0

40

1 WRW = W(R0 ; I)W = WR0 W ; I. G" ,

1 02 2 2 : : : BB N2 N2 2 N CC B N N N : : :CC : WR0 W = B BB 2 : : : : : : 2 CC NA @ N2 2 2 N :::

1" (, WR0 W ; I = D.

N

N

7.1.9. . . M "* (+ )

( ) , " P(x) = 1. 0) UP " , " ( UP : nP ;1 jx bi ! jx b P (x)i. 0 UP " ) $ ji = p1 jxi (n 2

x=0

b = p ( , " " x, ". P(x) = 1 () !, b . 6 ( ) ., * X0 = fxjP (x) = 0g, X1 = fxjP(x) = 0g UP X X X X UP (j bi) = p 1n+1 UP jx 0i + jx 0i ; jx 1i ; jx 1i = 2 x2X1 x2X0 x2X1 Xx2X0 X X X jx 0 0i+ jx 0 1i; jx 1 0i; jx 1 1i = = p 1n+1 2 X0 x2X1 x2X0 x2X1 x2X X X X = p 1n+1 jx 0i + jx 1i ; jx 1i ; jx 0i = 2 x2X0 x2X1 Xx2X0 X x2X1 = p1 n jxi ; jxi b: 2 x2X0 x2X1 Y ) X1 , ( . 1 j0i ; j1i, 2

7.2. &

7.2.10. 1 2 3 { . :)+ ) ) ( O{Y , 4.1.1, , "" * . ( " . n-( ( O{Y * $ W 2n 2n . Wrs , r s )% 0 2n ; 1. < "*, Wrs = p1 n (;1)rs 2

41

E. Rieffel, W. Polak

r s . (+ ( r s. 8( . , , ; X W jri = Wrs jsi: s

0) rn;1 : : : r0 () r, sn;1 : : :s0 | s, ; ; ; W jri = H : : : H jrn;1i : : : jr0i = ; ; = p1 n j0i + (;1)rn;1 j1i : : : j0i + (;1)r0 j1i = 2 n ;1 2X s=0 X 1 =pn (;1)sn;1 rn;1 jsn;1i : : : (;1)s0 r0 js0 i == p1 n (;1)sr jsi: 2 s=0 2 2n ;1

. 4. ! "

CSP

(# $ %$).

7.2.11. 4 * 5**.

- n V = fv1 : : : vn g, " ) m X = fx1 : : : xm g, C1 : : :Cl . - * xi " vj V X. :)+ ) ! )"), . = )" 20 " U

42

)") n = 2 m = 2 x1 = 0 x2 = 1. , , !" "% "" , " ( . F ) * ) * ) . , " 1 n- ) "% % n- ., 0 | "% %, . * * ( " " . = , )" 5 (* !") )" 4, )% " ( , . ( )%+ ": v1 = 0 v1 = 1 v2 = 0 v2 = 1.

. 5. ! " -'

N ( ) , !" , + , * )% . 0 , " K " , )%+. : ) j0 : : :0i, $ !" * " ) * , ) * , " )% . : ) , 3 >, " % /)"$ ) $ , . - )). K ) K 19964 K 1998] ) $ * ) !". < ( (, , "" ) ) C D * . 6! +7 8: 1. :)+ ) ( , " ) ) * " ) * . G%( ) , ) * " * .. G%( ) , . * " ). * , " * . . . ( . )" !"

43

E. Rieffel, W. Polak

. * ).

= ( (

p j000i ! 1= 3(j001i + j010i + j100i) p j001i ! 1= 3(j011i + j110i + j101i)

:::

"" $ . ( ("" (, ( " * )

00 B p13 B B B B p13 B B B 0 B B B B p12 B B B B 0 B B B @0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 1 0 0 0 0 0 0

p12 p12 0 0 0 0 0 0

p13 0 0

0 0

p12 0

0 0 0 p1 2 1 0 p 2 1 0

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

1 CC CC CC CC CC CC CC 0C CC CC A

- * % . ( ) . K K 1996] /", ( * ( )%+ ") ) $ Um " $ M * ( * M = UDV T , D | $, U V ) $. 0 UM = UV T ( *)% " M ) )% $). 0 " ") Um ( " " M, Um () ( "" M, " ( () ) ) * " ) * . 44

6! +7 8: 2. . ) K 1998] ( O{ Y . K , () $ /) WDW, W . ( O{ Y , D |

$, . " * . K . D, " " ) * ) * " ) ) * ). . ) Wrs = p1 (;1)jrsj = p1 (;1)jr\sj N N ) 7.2.1. :* +7 ;88< ! . 8( ) * , " )% , K ) / * , , " ) ) . 0 . ) , ! * , " ( % C D * ) * , ) , ! * C D * ) . "" ( () ) (

( .//" . ( % / C D * , " " "+ ) ) * , + C D C D * . ' / * ( ( ), "" . > (7.1.2) " P, " , ) ) . 6) * , , ( / C D * , ", ( " ) ) ) * C D * ( ). * ) . F- , "+ , "%+ ) / , , ) )

). E " " ( ." , * ": $ " ! * ." $ , , ) ) ". F " () ( " " % ) !

" , .//" () *. 8. $ (

F$ " | (, )%+ % " " %. $, %+ ")( , )+ " , ) " , () ") (, * ) . : : 1998] $ , " %( * 7 " ( , ( ( 3 , *+ 130 . , ", ( " ""$ (" * " , ! * )+ 3 ( .

45

E. Rieffel, W. Polak

= , " ""$ (" * " ")%, * ( ()* (". =, ", " ""$ (" *, , " . - ""$ (" * " " . 1) * % " " . , ", " , . ) , " (" !-" * . 8.1. 4 (

E) , (" % ) " ")( ")*%+ . * (" "* ")( , () "( $ : (I) ( ) (" ), (X) ( ), (Z)(/ ("), (Y ) ( / ("). 1" (, (+ * ")( (" " ( : e1 I + e2 X + e3 Y + e4 Z. ")*%+ () ")( * % ji ! (e1 I

+ e2 X + e3 Y + e4 Z)ji =

X i

ei Ei ji:

6 ( " (" " * *% "( $ ) (" Ei. ' % (" ")( fI X Y Z g ( (+ ")( (". %( ), (") *P "" ei Ei. i

8.2. -

" " ), "" )%+ " " ( (" Ei ( C, " (* n ( (n + k)( " , " * ( (" Sc , " (*; (n + k)-( " " ; i "" ) (" Ei,

i = Sc Ei(C(x)) . N y = Ej C(x) " "" ) (" , Sc (y) * C(x), ES;1C(y) (y) = C(x). 1 " ). - , * ( ) $ ( ". , (" * ( "( $ "" ) (" Ei . 0 . " , " " ! * *. M "" )%+ " C ( (" Sc . 1" (, n-( " ji " (n+k)-( " j'i = C ji. 46

P

6 ) , " " % i ei Ei j'i " "( $ "" ) (" Ei. 0 j'i * )%+ ( (1) 0 " (" Sc " ") % ( " j0i ")( SC

X i

ei Eij'i

j0i =

X i

ei Eij'i jii :

- ) $ % (" ( " ) i). (2) F " ) jii ) . ' ")% ( ))%) ) i0 " ) Ei0 j' i0 i: (3) 0 ( ( (" Ei;1 " n + k 0 ")( Ei0 j' i0 i, ( ) j'i. M , (2) ) $ " " (" " ) )% ("). : , (3) ( " ( ( (" . 8.3. (

"" )%+ " C, (*%+ j0i ! j000i j1i ! j111i. : + % C * "" (") ")( E = fI I I X I I I X I I I X g: , ("

S : jx0 x1 x2 0 0 0i ! jx0 x1 x2 x0 xorx1 x0 xorx2 x1 xorx2i )%+ ""$ (" , ( $ * ( , . Ei = Ei;1 ). F ( 0 " (" -"$ (" | j000i | 0 j110i X I I 1 j101i I X I 2 j011i I I X " ( ji = p1 (j0i ; j1i), " ) )%+ 2 ( ; C ji = j'i = p1 j000i ; j111i 2

47

E. Rieffel, W. Polak

(")

E = 54 X I I + 53 I X I: : ("

4

E j'i = 5 X I I + 35 I X I

!

1 p (j000i ; j111i) = 2

!

= 54 X I I + 35 I X I p1 (j000i ; j111i) = 2 ; ; = p4 X I I j000i ; j111i + p3 I X I j000i ; j111i; = 5 2 5 2 ; ; = p4 j100i ; j011i + p3 ; j010i ; j101i 5 2 5 2

;

M " * % E j'i j000i " (" :

; SC ((E j'i) j000i) = SC p4 j100000i ; j011000i + p3 (j010000i ; j101000i = 5 2 5 2 ; ; 4 3 = p j100110i ; j011110i + p j010101i ; j101101i 5 2 5 2 ; ; 4 = p j100i ; j011i j110i + p3 j010i ; j101i j101i: 5 2 5 2 F ( . , ) ( j110i, ( j101i. O * ) , "" 1; p j100i ; j011i j110i: 2 F ( ) (" ), " . , (" * ( " ! ( ( (" X I I, )%+ ) % j110i. 1 ) 1; p j000i ; j111i = C ji = j'i: 2 9.

- | . , ( " , * . (" ! " " % | * ) $ . - 48

) / " $ "% $ * . 0 ( , () !, / ( , ) ) . < ( * + ( " ) $ , ) %+ " ) % ( )" . < " "- " .//", " , "" ." $ , ) ,

" ( , " % " *.1 = , " * ( ( , %( " " * ( " " . = + + . , .. ." $ , * ( " " . 1" ( . < 3 * * , " ) ( " . Y 3, )%+ " " %, ( ! " / " . Y ", ( >, * ( ) " , . /" ", " " % + " " . = , > , )+ ) * ( ( ) ". < " % " , * K. :)+ ) +! " " " , " . 6* < " 6* < " 1998] $ % 2- ( " " %, + " * , ( /)"$ . > > 1998] ( .//" $" " * . 0* >, : 1 * " ( . = . , !-" +! " , * " " % " ". :* " , " ( ." $ ) " ) , " /" $ ? ' ". : / " * , " ( ) ( " . 6 + ! ." )% " " " , ( * *. Y(1

# . H! $ ! , ! #"

, !" (, ! . >., , D.Aharonov, A. Kitaev, N. Nisan ?Quantum Circuits with Mixed States@ (29806029

http://xxx.itep.ru/quant-ph). | . .

49

E. Rieffel, W. Polak

G Y( G 1998] " , * (

* ( * NP- " " % . 0 ) , . , | . /) / " $ , .//" )% " / " . O " " " % % " * . = , " BQP | . ( ", (( " 1 % " %. 6 , ( " * , " . 8 , * " , ) )%+)% /$ % E 1997] ) 1998]. -) 1998] * ) ( " * . -, " )+ )% * / " (, " )* () . E . () " " (%+ " " %. 6" , .. "* " - ")*%+ | "% (. C " ""$ (" ( ) " ( , ( " , / ". < )* " ""$ (". 6 * ( " ""$ (" ) " (" () ) "* * " " %, "" ) " ( . 9.1. ' # #

,( ' % : C- D : 1998] / ". < " ) .) %, ( " " . , ( * ) /$ " , "* % ""$ " (", ( ! . 1 ( / " , " ( " " %. ' ( % 1997 . ( (, , " . 1"* ". 6) , ( , ( , * E 1997]. ) * 1996] * ") "$ -' , 1985], " * ) %. "* ( )* " , " ( /$ . - - O :" -) $ ! ' O -) 1998] (* 50

/ " CMathematicaD, " )% " " , , 3. *) CSIAM Journal of ComputingD "( 1997 * 6 "$ " , "% "$ , " )* E . 1997] E $ 1997] 3 1997] : 1997]. E )% , ) (, "* ) , * G Y :1 http://xxx.lanl.gov/archive/quant-ph. : " " " ))% /$ % " * $ F: http://www.pocs.com/qc.html. 6 #

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, ( ( ( 1 K - <" ( " " + . . < "* ( G -( ), 6 ) > (), G) >), =) K , ) 0), <") /% ." , * , ( " " . F " ( % ( ) " FXPAL, " .)

* ) (). " ,. 8

1 () n- k- " nk- ". : , A B % ( n- k- " , A B 1 ( nk- " . 0 ( " . ( ( . K/ 1974]). 6 $ () " " ( " , ( ! . 6 $ A B C D U, " u x y " a b % )%+ ) : (A B)(C D) = AC BD (A B)(x y) = Ax By (x + y) u = x u + y u u (x + y) = u x + u y A Bax by = ab(x A Uy) B U C D U = C U DU 1

T .

55

E. Rieffel, W. Polak

, " a b c d a b aU bU c d U = cU dU : - " * () , .. (A B) = A B < $ U ) , $, * , ( " : U U = I. 1 " " $ ) , " , " "* $ . ) c % " . 0) U = A1 A2 : : :An . 1 $ U ) , Ai Ai = ki I i ki = 1. U U = (A1 A2 : : : An)(A1 A2 : : : An ) = = A1 A1 A2 A2 : : : An An = = k1 I : : :knI = I "* I * $ )%+ . = , () " )%+ : (a0 j0i + b0j1i) (a1j0i + b1j1i) = = (a0 j0i a1j0i) + (b0j1i a1 j0i) + (a0 j0i b1 j1i) + (b0 j1i b1j1i) = = a0 a1((j0i j0i) + b0a1 (j1i j0i) + a0b1 (j0i j1i) + b0 b1(j1i j1i) = = a0 a1(j00i + b0 a1j10i + a0 b1j01i) + a0 b1j11i . 9 - .

(+ ), " r 2m , v, 4- 3, ( % () ( " " ") ), m m 2 2 ") r , j r . 7 % r v. 3 ", ( % v ) m 1 2 (, 2 j r . 1" (,

2m 1 v ; j r < 2

" j, ") ) v j 1 2m ; r < 2 2m < 2M1 2 : 56

0 $ * ) ) ( qp pq0 , , M, :

p p0 pq0 ; p0q 1 q ; q0 = qq0 > M 2 :

1" ( )+ ), " , ( pq q < M, " 2vm ; pq < 1 2 . ( ), " v ) M

( r . ( () rj . N ( M, .. ) ( 1 2 2vm , * ( ) ./" ( + % M * ( ")% ( 2vm . F ) 1 2

m j2 ,

h i a0 = 2vm

0 = 2vm ; a0 h i an = 1; 1

n = 1; 1 ; an n n p0 = a0 p1 = a1 a0 + 1 pn = anpn;1 + pn;2 q0 = 1 q1 = a1 qn = an qn;1 + qn;2

)% ( pqnn ")%, qn < M 6 qn+1. 6" . . %( " . ( ), " 2vm ) ( M1 2 jr , " 1r , ( , )-

)+ $ ), rj , . ". , M. < q ) ( " ) , " () , " j r | .

57

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