Ожигов Ю.С. Квантовые вычисления
Подождите немного. Документ загружается.
U
1
√
r
r−1
P
j=0
exp(−2πikj/r)|y
j
(mod q)i
exp(2πij/r)
j/r
j
ξ
j r
r
q
y
U
cond
U
α
y α q
U
log
2
q y
α
y
α
y y, y
2
, y
4
, . . .
q y 2
l
1
q/2
l
1
y
q q log
2
q
U O(log
3
q)
l
2
n
n
n
O(1/N)
t
e
−iHt
ψ
0
H = H
p
+ H
q
H
p
=
p
2
2m
H
q
= V (q) p =
1
i
∂
∂q
V (q)
t
H
q
e
iH
q
H
p
∆t
e
−iH
≈ (e
−iH
q
∆t
e
−iH
p
∆t
)
1/∆t
.
e
H
q
QFT : f −→
1
√
2π
R
+∞
−∞
e
−ipq
f(q) dq ∂/∂ q ip
e
−iH
p
= FT
−1
e
−ip
2
∆t/2m
FT
−p
2
/2m
t t
2
q
t
t
0
< t
V (t, X) t X P
H =
P
2
2
+ V (t, X)
P =
1
i
∂
∂X
i
∂Ψ
∂t
=
P
2
2
+ V (t, X), Ψ(0) = Ψ
0
.
Ψ(t, X) = exp(−i
t
R
0
H(t, X)dt)Ψ
0
e
−iHt
ψ
0
H = H
p
+ H
q
H
p
=
p
2
2m
H
q
= V (q) p =
1
i
∂
∂q
V (q)
t = 1
∆t
e
−iH
≈ (e
−iH
q
∆t
e
−iH
p
∆t
)
1/∆t
.
H
q
FT : f −→
1
√
2π
R
+∞
−∞
e
−ipq
f(q) dq ∂/∂ q
ip
e
−iH
p
= FT
−1
e
−ip
2
∆t/2m
FT
−p
2
/2m
ψ(q)
(−A, A) FT ψ (−B, B)
∆q ∆p
2A/∆q
P
a=0
ψ(q
a
)δ
a
δ
a
(q)
(q
a
, q
a
+ ∆q) q
δ
a
1
√
2π
∆q
2B/∆p
P
b=0
e
−ip
b
q
a
σ
b
(p) σ
b
(p)
δ
a
d
a
(q) = δ
a
(q−A) s
b
(p) = σ
b
(p−B)
d
a
−→
1
√
2π
∆q
2B/∆p
X
b=0
e
−i ba∆q ∆p
s
b
∆q ∆p
q
a
p
b
q
a
a = 0, 1, . . . , N = 2
l
|ai l
∆q = ∆p =
√
2π/
√
N A = B =
p
πN/2
QFT : |ai −→
1
√
N
N−1
X
b=0
e
−
2πi ab
N
|bi,
−p
2
∆t/2m −πb
2
∆t/mN
s
1
s
1
H
1
= H
1
kin
+ V (µ, ν, η), H
1
kin
(µ, ν, η) =
J
2
x
2I
x
+
J
2
y
2I
y
+
J
2
z
2I
z
µ, ν, η
I
¯
J = [¯r × ¯p] r p
r
U = exp −iHδt H
j
= H
0
j
+ H
1
j
H
0
j
=
P
2
j
2m
j
+ V (s
j
, X
j
) X
0
j
H
1
j
j X
j
x
p [0, 1) P =
p
2π/Np
X =
p
2π/Nx V
0
(X)
X
H
p
x −→
X
p
e
−2πi xp
N
|pi −→
1
√
N
X
p
e
−2πi xp
N
e
iP
2
δtπ/N
|pi
θ
p
(κ) = a P P + A
A =
p
2π/Na
1
N
X
y
λ
y
|yi, λ
y
=
X
p
e
ip
2
δtπ/N
e
−2πi xa/N
e
ip(−2πX+2πAδt+2πY )
N
|yi
λ
y
= 0 y
Y = X − Aδt P
X
0
= ∂H/∂P
H
x
= V (X)
|pi QFT H
x
QFT
−1
|pi −→
1
√
N
X
x
e
2πixp
N
|xi −→
1
√
N
X
x
e
2πixp
N
e
iv(x)
|xi −→
1
N
X
p
1
λ
p
1
|p
1
i,
λ
p
1
=
X
x
e
2π ix(
p
N
+
v(x)δt
2πx
−
p
1
N
.
λ
p
1
6= 0
P
1
= P + δt
V (X)
X
,
P
0
=
∂H
∂x
.
U w ∈ [0, 1)
U
U U
cond
Rev
P
k
x
k
|ψ
k
i ψ
k
w Rev ω
k
w = ω
k
Rev
−1
w
Rev w w
U
U
cond
U
α, β, γ
δa
+
k
a
+
j
+ δ
∗
a
k
a
j
.
a = a
0
+ a
0
2 + . . . + a
l−1
2
l−1
|a
0
a
1
. . . a
l−1
i = |ai
b
j
b = b
0
+b
0
2+. . .+b
l−1
2
l−1
QF T
−1
O(l
2
) N = l
2
A) H =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 ρ
, ρ > 0, B) H =
ρ
1
0 0 0
0 ρ
2
0 0
0 0 ρ
3
0
0 0 0 ρ
4
,
ρ = ρ
0
e
−br
r
b = const r ρ
1
+ ρ
4
6= ρ
2
+ ρ
3
l j
k H
j,k
b = 1
QF T : |ai −→
1
√
N
N−1
X
b=0
e
−
2πi ab
N
|bi, QF T
−1
: |ai −→
1
√
N
N−1
X
b=0
e
2πi ab
N
|bi.
i
i
i
i
i
a
4
a
3
a
2
a
1
a
0
b
0
b
1
b
2
b
3
b
4
U = e
−i
˜
H
˜
H =
P
l>j>k≥0
˜
H
j,k
˜
H
j,k
ρ
0
= π r = j − k
ρ
0
π r = j − k
U
QF T
−1
|ai
|bi
1/
√
L
a
0
j
= a
l−1−j
, j = 0, 1, . . . , l − 1
j k ≤ j
a
0
j
a
0
k
a
0
j
a
0
k
j > k a
0
j
b
k
j > k
b
j
b
k
j > k k j −k l −1 −j