E. RIEFFEL
FX Palo Alto Laboratory, 3400 Hillview Avenue, Palo Alto, CA 94304, USA
W. POLAK
1021 Yorktown Drive, Sunn...
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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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1982] , ! "- " $ " " " %1 . ' (% ")
, , *, $, ) ( ( )" , )+ % + " $ . , " " " % , " )% ( " $ , " *. - ) * ( )p , " .//" ) "p . 0.) p "p . 1" *
c ACM (Association for Computing Machinery). ACM Computing Surveys, V. 32, 23, September 2000 y c & (.: 5. 6. , 7. 8. . 1 ) 1980 (# ! !$ ! $ 6. . = 6. . , 1980, . 15. | . .
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1994 , " 0 3 ) , " p* $ * 3 19944 3 1997]. 1 ( " ) * . ' " () " % "" ." , %+ " " %, " ", %+ " . 6 " .) ) ( ( " " "%, "* + ) , ) ." " $ $ !")( " " %. 7 + ( "% , (
) $ , %+ ." ( " , p $ ) ( , %+ " $ , "* " . . 6 (+ * . , .". ) " , * . (+-, , " ( )+ ! , * ) , ) $ . 8( ." $ ) , () ." $ ) p$ p, , , (9! / " . 1 "" " ." $ ) p , p() ) (9! ( / " . ' p " 6 6* 1992]. :)+ ) +! * ( . 0" " , ) " ) . 0$ ) " ) | p$ , " )+ " , "* . ( " ) p " n . O (pn) .p p$ " " %. - " " % (% ) O(n), .) " ) p " " " % .//" , * " " " " %. = ! * ) "p , ." $ , p 3p. E
", > ( p " ) " " " %. = p
" " () ) p ", , " "% ")% ) ) p , ) ( ( p. 1 K, ) ) )! , ")% * . 7 " ! ". 6 " , * + " " ") . , | ) " " % NP - | ! ". =, *), * ! " " %. :)+ ) ( * * % " "
7
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% )", (L > 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 " > * ) " "
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. 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
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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. (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 ( , () !, / ( , ) ) . , * ( ) " , . /" ", " " % + " " . = , > , )+ ) * ( ( ) ". 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 " ""$ (" ( ) " ( , ( " , / ". ), =) K , ) 0), 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.