Квантовые вычисления. Ожигов Ю.С. - 91 стр.

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МГУ | Москва

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f
0
∈ S
b
B
0
= {j | f
0
(e
j
) = 1}
l
j
f
0
l
j
= card {j | j ∈ L
j
∩B
0
}.
El
j
= b
ˆ
l
j
/N f
0
S
b
ζ =
P
j
l
j
c
j
f
0
Eζ =
X
j
c
j
El
j
=
X
j
c
j
b
ˆ
l
j
N
= O(1)b/N = o(1) (N −→ ∞)
P (ζ ≥ 0.9) ≤
10
9
Eζ
P (ζ ≥ 0.9) −→ 0 (N −→ ∞).
card(G)/card(S
b
) =
0
= const
X
0
t
=
P
j
λ
0
j
e
j
f
0
f
0
∈ G
0
0 ≤
X
j /∈B
0
|λ
0
j
|
2
≤ p
err
.
ξ
2
−→ 0 (N −→ ∞).
ξ
2
= kX
t
− X
0
t
k
2
=
P
j∈B\B
0
|λ
j
− λ
0
j
|
2
+
P
j∈B
0
\B
|λ
j
− λ
0
j
|
2
+
P
j /∈B∪B
0
|λ
j
− λ
0
j
|
2
+
P
j∈B∩B
0
|λ
j
− λ
0
j
|
2
.
P
j∈B
0
\B
|λ
0
j
|
2
= q
0
,
P
j /∈B∪B
0
|λ
0
j
|
2
= z
0
P
j∈B
0
\B
|λ
j
|
2
= q,
P
j /∈B∪B
0
|λ
j
|
2
= z,
P
j∈B∩B
0
|λ
j
|
2
= r,
P
j∈B∩B
0
|λ
0
j
|
2
= r
0
.
q ≤ ε
0
≤ p
err
z ≤ p
err
ka − bk ≥
|kak − kbk| a, b
δ = |
√
q
0
−
√
q|
2
|
√
z
0
−
√
z|
2
N
ξ
2
< p
err
.
N q
0
< 4p
err
q
0
≥
4p
err
p
err
> δ ≥ (
√
q
0
−
√
p
err
)
2
≥ p
err
z
0
< 4p
err
N −→ ∞
P
j /∈B
|λ
0
j
|
2
= q
0
+ z
0
<
                                                                                                                   
           %&  f 0 ∈ S $  B 0 = {j | f 0 (e ) = 1}                                                            
                                      b                                 j
$      l     f 0
                   j

                                                      lj = card {j | j ∈ Lj ∩ B 0 }.
       El = bl̂ /N      f 0          
               j    j               
Sb  &     ζ =
                            lj cj   '   $$   $     $  f 0  
                           P
                           j
    '   

                                        X                   X cj bl̂j
                                Eζ =            cj Elj =                   = O(1)b/N = o(1) (N −→ ∞)
                                           j                 j
                                                                    N

            )  P (ζ ≥ 0.9) ≤                                              
                                                                                             
                                                                                                                                               
                                                                                   10
                                                                                   9 Eζ

                                                     P (ζ ≥ 0.9) −→ 0 (N −→ ∞).                                                                 

        '   card(G)/card(S ) =  = const 
                                          b    0
    X 0 = P λ0 e                                                                 $ %&              f0
           t      j  j
                      j
 f 0 ∈ G                      

                                                                                                                                               
                                           0

                                                                                                                                                
                                                    X
                                                 0≤   |λ0j |2 ≤ perr .
                                                                     / 0
                                                                   j ∈B

                        

                                                            ξ 2 −→ 0 (N −→ ∞).
                                                                                                                                               
                                                                                                                                                
     

                                                                                                                                               
                                                        P                      P
                                ξ 2 = kXt − Xt0 k2 =          |λj − λ0j |2 +        |λj − λ0j |2 +
                                   P                  j∈B\B
                                                        P
                                                            0                   0
                                                                             j∈B \B                                                             
                                       |λj − λ0j |2 +         |λj − λ0j |2 .
                                j ∈B∪B
                                  /    0                         j∈B∩B 0

   '            P
                              |λ0j |2 = q 0 ,
                                                  P
                                                           |λ0j |2 = z 0 
                                                                               P
                                                                                        |λj |2 = q,
                                                                                                          P
                                                                                                                 |λj |2 = z,
                                                                                                                                 P
                                                                                                                                         |λj |2 = r,
                   j∈B 0 \B                     j ∈B∪B
                                                  /    0                     j∈B 0 \B                 j ∈B∪B
                                                                                                        /    0                 j∈B∩B 0
  P
          |λ0j |2 = r .
                     0
j∈B∩B 0
              q ≤ ε ≤ p     
                           0    err     z ≤ perr           ka − bk ≥
|kak − kbk|     #     a, b        *      
   $       )   δ = |√q 0 − √q|2    $     )  
 √     √
| z 0 − z|2   N  )$  

                                                                     ξ 2 < perr .
                                                                                                                                               
                                                                                                                                                



 0
  $ N            q 0 < 4p   $      q 0 ≥
4perr          p > δ ≥ (√q 0 − √p )2 ≥ p
               
                              err
                                   
                                               err
                                                   err
                                                       err
                                                         
                                                                   
                                                                                           
                                                                         P |λ0 |2 = q 0 + z 0 <              
                                                                                                                                   
z < 4perr                                       N −→ ∞                        j
                                                                                                                        j ∈B
                                                                                                                          /

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