Ожигов Ю.С. Квантовые вычисления
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π
X
l>j>k≥0
a
0
j
a
k
k
2
j−k
(j − k)
+ π
X
l>j>k≥0
a
0
j
b
k
(j − k)
2
j−k
(j − k)
+ π
X
l>j≥0
a
0
j
b
j
+ π
X
l>j>k≥0
b
j
b
k
(l − j − 1 )
2
j−k
(j − k)
.
A B
|ai |bi
j l − 1 − j
π
P
l−1>k+j≥0
a
j
b
k
2
j+k
2
l−1
+ π
P
l−1≥j≥0
a
l−1−j
b
k
= 2π
P
l>k+j≥0
a
j
b
k
2
j+k
2
l
= 2πS + 2π
P
l>k,j≥0
a
j
b
k
2
j+k
2
l
=
2πS + 2π
ab
2
l
S
A B
A B
A A
j,k
= c
j,k
a
0
j
a
0
k
c
j,k
j k a j k
p, q, q > p
∆t d
p,q
∆t a
0
p
a
0
q
d
p,q
a
0
p
, a
0
q
d
p,q
= e
−|q−p|
/|q − p|
q 1 − a
0
q
∆t d
p,q
∆t a
0
p
(1 −a
0
q
)
q
d
p,q
∆t a
0
p
p
p, q p
q p
p A
p
0 < t
p
1
< t
p
2
< . . . < t
p
m
p
< 1
λ 1 A
p
p
t
p
m
p m
p
λ A
p
p
a
0
p
1 −a
0
p
j
1
2
d
p,j
a
0
p
a
0
j
+
1
2
d
p,j
(1 −a
0
p
)a
0
j
1
2
d
p,j
a
0
j
q 6= p
p q λ
a
0
p
, a
0
q
a
0
p
, 1 − a
0
q
1 − a
0
p
, a
0
q
1 − a
0
p
, 1 − a
0
q
1
4
d
p,q
[a
0
p
a
0
q
+ a
0
p
(1 −a
0
q
) + (1 −a
0
p
)a
0
q
+ (1 −a
0
p
)(1 −a
0
q
)] =
1
4
d
p,q
p /∈ {j, k}
1
2
[
X
p/∈{j,k}
d
p,j
a
0
j
+
X
p/∈{j,k}
d
p,k
a
0
k
] +
1
4
X
p,q /∈{j,k}
d
p,q
.
d
j,k
a
0
j
a
0
k
∆t
∆t d
j,k
a
0
j
a
0
k
−∆t d
j,k
a
0
j
a
0
k
j
j −∆t d
j,k
a
0
k
c ·a
0
j
a
0
k
c
X
j,k
c
j,k
a
0
j
a
0
k
c
j,k
QF T
−1
A B QF T
−1
λ
d
j,k
δt kδt
k p = 1/λ λ T
M = T/δt
δt D D
p 1 − p O(
√
M)
T/
√
M T = O(log N)
M = O(
log
2
N
)
j k
A) H
j,k
=
E
j,k
1
0 0 0
0 E
j,k
2
0 0
0 0 E
j,k
3
0
0 0 0 E
j,k
4
, B) H
j,k
=
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 E
j,k
, E
j,k
> 0.
H
0
j,k
a 0 0 0
0 a 0 0
0 0 b 0
0 0 0 b
,
α 0 0 0
0 β 0 0
0 0 α 0
0 0 0 β
.
U
j,k
= exp(−iH
j,k
)
j k
U
j,k
j
jkδt k δt
δt n
2
U
j,k
j, k
∆E = E
1
−E
2
−E
3
+ E
4
∆E
π
/∈ Q
∆E
π
CNOT =
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
E
E =
exp (iE
1
) 0 0 0
0 exp (iE
2
) 0 0
0 0 exp (iE
3
) 0
0 0 0 exp (iE
4
)
A =
1 0
0 exp (i (E
1
− E
3
))
,
B =
exp (−iE
1
) 0
0 exp (−iE
2
)
,
E U
U = E (A
O
B) =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 exp (i∆E)
.
∆E
π
∀ε > 0∃m ∈ N∃n ∈ N : |∆En − π(2m + 1 )| < ε,
ε n = n(ε) U
n
Π
Π =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 −1
(I
O
H)Π(I
O
H) = CN OT,
I
I =
1 0
0 1
,
H
H =
1
√
2
1 1
1 −1
1
√
2
1 1 0 0
1 −1 0 0
0 0 1 1
0 0 1 −1
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 −1
1
√
2
1 1 0 0
1 −1 0 0
0 0 1 1
0 0 1 −1
=
1 0 0 0
0 1 0 0
0 0 0 1
0 0 1 0
(I
O
H)
E (A
O
B)
n
(I
O
H)
n
ψ(r
1
, r
2
, . . . , r
n
) = ψ
j
1
(r
1
)ψ
j
2
(r
2
) . . . ψ
j
n
(r
n
) {ψ
j
}
j
s
1, 2, . . . , J r
j
ψ
j
1
(r
1
)ψ
j
2
(r
2
)) . . . ψ
j
n
(r
n
)
n
Ψ =
1
√
n!
ψ
j
1
(r
1
) ψ
j
1
(r
2
) . . . ψ
j
1
(r
n
)
ψ
j
n
(r
1
) ψ
j
n
(r
2
) . . . ψ
j
n
(r
n
)
,
ψ
j
s
s = 1, 2, . . . , n
ψ
k
k ∈ {1, 2, . . . , J} j
s
ψ
ψ
k
|¯n
Ψ
i = |n
1
, n
2
, . . . , n
J
i n
k
k
¯n
P
¯n
λ
¯n
|¯ni λ
1/2
Ψ =
1
p
(2n)!
ψ
j
1
(r
1
)α
1
ψ
j
1
(r
2
)α
1
. . . ψ
j
1
(r
2n
)α
1
ψ
j
1
(r
1
)β
1
ψ
j
1
(r
2
)β
1
. . . ψ
j
1
(r
2n
)β
1
ψ
j
n
(r
1
)β
n
ψ
j
n
(r
2
)β
n
. . . ψ
j
n
(r
n
)β
n
,
α
j
β
j
j
1/2 j −1/2 j
k
Ψ
R
hΨ| H |Ψid¯r
α
∗
j
α
j
= β
∗
j
β
j
= 1, [α
j
α
k
] = [β
j
β
k
] = [α
j
, β
k
] = 0
j 6= k α
j
α
j
= β
j
β
j
= 0 j
a
j
j a
+
j
a
j
|n
1
, . . . , n
J
i = δ
1,n
j
(−1)
σ
j
|n
1
, . . . , n
j−1
, n
j
− 1, n
j+1
, . . . , n
J
i σ
j
=
n
1
+ . . . + n
j
a
+
j
a
k
+ a
k
a
+
j
= δ
j,k
a
j
a
k
+ a
k
a
j
= a
+
j
a
+
k
+ a
+
k
a
+
j
= 0
H = H
one
+ H
two
V
1
(r)
V
2
(r, r
0
)
H
one
=
P
k,l
H
k,l
a
+
k
a
l
H
two
=
P
k,l,m,n
H
k,l,m,n
a
+
l
a
+
k
a
m
a
n
H
k,l
= hψ
k
| H
one
|ψ
l
i =
R
ψ
∗
k
(r)V
1
(r)ψ
l
(r)dr,
H
k,l,m,n
= hψ
l
, ψ
k
|H
two
| ψ
m
ψ
n
i =
R
ψ
∗
k
(r)ψ
∗
l
(r
0
)V
2
(r, r
0
)ψ
m
(r)ψ
n
(r
0
)drdr
0
.
ψ
i
H =
P
i
H
i
ext.f.
+
P
i,j
(H
i,j
diag.
+ H
i,j
tun.
)
H
i
ext.f.
= α
i
a
+
i
a
i
, α
i
∈ R,
H
i,j
diag.
= β
i,j
a
+
i
a
i
a
+
j
a
j
, β
i,j
∈ R,
H
i,j
tun.
= γ
i,j
a
+
i
a
j
+ γ
∗
i,j
a
+
j
a
i
.
k
F
k
F
j j
j
0
j
j F F =
F
1
N
F
2
N
. . .
N
F
k
F
j
j
F
j
F
j
j
0
j j j
0
|1i
j
|0i
j
F = F
1
N
F
2
N
. . .
N
F
k
F θ
H F
θ(|ξ
1
, ξ
2
. . . ξ
n
i) = |ξ
1
i
1
N
|ξ
2
i
2
N
. . .
N
|ξ
n
i
n
ξ
j
θ
H
H H
0
+ H
1
H
0
=
d
1
0
0 d
2
, H
1
=
0 d
¯
d 0
.
ξ ∈ H |0i
2
|1i
2
|010i
e
e
e
e
e
u
u
u
u
u
θ(ξ)
F
3
2
1
0
1
0
2
0
3
0
˜
H
0
= d
1
a
+
k
a
k
+d
2
a
+
k
0
a
k
0
˜
H
1
= da
+
k
a
k
0
+
¯
da
+
k
0
a
k
˜
H
i
θ = θH
i
i = 0, 1
θ (
˜
H
0
+
˜
H
1
)θ = θH
U e
−iH
H
θ θ
−1
H
s
θ = (θ
−1
Hθ)
s
s
U
F
k
N
F
j
- -
- -
6 6 6 6
θ θ θ θ
H H H H
˜
F
˜
F
˜
F
˜
F
˜
F
H H
˜
F = F
j
N
F
k
.
F F
- - -
- - -
6 6 6
H H H
(−1)
σ
j
a
+
j
a
j
a
+
k
a
k
a
+
j
a
j
0
F (−1)
σ
0
σ
0
=
j
0
−1
P
s+j
n
s
= j
0
− j |¯ni ∈ F
j j
0
1
√
2
|000 . . . 0i +
1
√
2
|111 . . . 1i
|0i −→ |1i, |1i −→ |0i |0i −→ |0i, |1i −→ |0i
ξ
0
=
1
√
2
|100 . . . 0i+
1
√
2
|011 . . . 1i ξ
0
=
1
√
2
|000 . . . 0i+
1
√
2
|011 . . . 1i
λ|0, 0, 0i+ µ|1, 1, 1i
|0, 0, 0i |1, 1, 1i