Download can you see?

quantum computing
quantum-bit (qubit)
|0  
a
a1 |0 + a2 |1 = a1
2
|1  
preparation
|Y0
Vorlesung Quantum Computing SS ‘08
calculation

H

U
read-out
 -1
H

Y|A|Y
time
time
1
from classic to quantum
we live in Hilbert Space H
the state of our world is |y
Vorlesung Quantum Computing SS ‘08
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can you see?
Don Eigler (IBM, Almaden)
48 Fe atoms on Cu(111)
http://www.almaden.ibm.com/vis/stm/gallery.html
Vorlesung Quantum Computing SS ‘08
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double slit experiment
classically:
number of electrons measured has a broad distribution
Vorlesung Quantum Computing SS ‘08
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double slit experiment
quantum mechanically:
wave function
y = y (r,t)
coherent superposition
|y = c1|y1 + c2|y2
probability density: probability
of finding a particle at sight r
r(r,t) = |y(r,t)|2
interference pattern is observed
→ particles are described as waves
Vorlesung Quantum Computing SS ‘08
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double slit with electrons
Vorlesung Quantum Computing SS ‘08
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double slit with electrons
http://www.hqrd.hitachi.co.jp/global/movie.cfm
Vorlesung Quantum Computing SS ‘08
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Double slit with larger objects
O. Nairz, M. Arndt, and A. Zeilinger: Am. J. Phys. 71, 319 (2003)
Vorlesung Quantum Computing SS ‘08
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state and space of the world
a particle is described by a vector |y in Hilbert–Space
complex functions of a variable, y(r), form the Hilbert–Space:
y *(r) y(r) dr = y |y  < ∞
H is a linear vector space with scalar (inner) product
j |y  = j *(r) y(r) dr = a
,aC
y |j  = j |y * = a *
Vorlesung Quantum Computing SS ‘08
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the space of the world
the scalar product is distributive
j |y1 + y2  = j |y1 + j |y2
j |cy  = c j |y 
cj |y
and thus
 = y |cj * = c* j |y 
it is positive definite and real for y |y  ≥ 0 ,  
Vorlesung Quantum Computing SS ‘08
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quantum computing
quantum-bit (qubit)
|0  
a
a1 |0 + a2 |1 = a1
2
|1  
preparation
|Y0
Vorlesung Quantum Computing SS ‘08
calculation

H

U
read-out
 -1
H

Y|A|Y
time
time
11
vector bases
every vector |j  can be decomposed into linear
independent basis vectors |yn: |j  = cn|yn , cn C
n
orthogonality can be written as
ym *(r) yn(r) dr = ym|yn = dmn
ym|j = c

y
|yn = cn dmn
m
n
n
n
cm = ym|j
|j = |yn yn|j
n
Vorlesung Quantum Computing SS ‘08
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euclidic representation
Vorlesung Quantum Computing SS ‘08
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our world H
is normed with respect to finding a particle of state |j anywhere
P = j *(r) j (r) dr =  ||j (r)||2 dr = j |j  = 1
can be divided into sub-spaces connected by the vector product

|

|

H = H1  H2  H3    HN
HQC
|ym|y1 mno
 |y=n
|yo|y3 m=|1ym|1y|y
||cmno
cmno
yoo33
QC=cmno
2
2 ||y
QC =
nn
2 
m,n,o
m,n,o
 we can find (or build) a quantum computer in our world
Vorlesung Quantum Computing SS ‘08
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endohedral fullerenes
atom inside has an electron
spin that can serve as qubit
4Å
mS
|+1/2
mI
|+1/2
|-1/2
|-1/2
|-1/2
|+1/2
10 Å
source: K. Lips, HMI
Vorlesung Quantum Computing SS ‘08
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quantum computing
quantum-bit (qubit)
|0  
a
a1 |0 + a2 |1 = a1
2
|1  
preparation
|Y0
Vorlesung Quantum Computing SS ‘08
calculation

H

U
read-out
 -1
H

Y|A|Y
time
time
16
boolean algebra and logic gates
classical (irreversible) computing
in
1-bit logic gates:
out
gate
identity
x
0
1
Id
0
1
NOT
x
0
1
x
Vorlesung Quantum Computing SS ‘08
NOT x
1
0
NOT x
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quantum logic gates
1-bit logic gate:
x
0
1
NOT (a1| 0  + a2| 1 ) = a1|1  + a2| 0 
NOT x
1
0
manipulation in quantum mechanics is done by linear0 operators
1
matrix representation
for the
NOT representation
gate:
X≡ 1 0
operators have
a matrix
X
Vorlesung Quantum Computing SS ‘08
a1
a2
=
0
1
a1
1
0
a2
=
a2
a1
18
manipulation in our world
because of the superposition principle |y = c1|y1 + c2|y2,
mathematical instructions (operators) have to be linear:
^
^ |y  + L
^ |y 
L (|y1 + |y2) = L
1
2
^ (c |y ) = c L
^ |y 
L
1
1
1
1
examples:
(c + d/dx)
(c + d/dx) (f(x) + g(x)) = cf + d/dx f + cg + d/dx g
 dx
 dx (f(x) + g(x)) =  f dx +  g dx
()
X
(f(x) + g(x))2 ≠ f2 + g2
2
Vorlesung Quantum Computing SS ‘08
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linear operators
^ + M)
^ |y = L
^ |y + M
^ |y
(L
^ M)
^ |y = L
^ (M
^ |y)
(L
however, generally
^M
^ |y ≠ M
^L
^ |y
L
^ ^ =L
^M
^ –M
^ L
^
commutator: [L,M]
^ ^ =L
^M
^+M
^L
^
anticommutator: [L,M]
^^ =
 [L,M]
, |y
^^
– [M,L]
+
^ ^ = [L,1]
^ ^ = [L,L
^ ^-1] = 0, [L,aM]
^ ^ = a [L,M],
^ ^
[L,L]
^ +L
^ ,M]
^ = [L
^ ,M]
^ + [L
^ ,M],
^
[L
1
2
1
2
^L
^ ,M]
^ = [L
^ ,M]
^ L
^ + [L
^ ,M]
^ L
^
[L
1 2
1
2
2
1
Vorlesung Quantum Computing SS ‘08
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vectors and operators
|j = |yn yn|j
n

1 = |yn yn|
n
^ |y  y |) = 
^ 1 = |y  y | ( L
1L
m
m
n
n
m
n
with matrix elements
Vorlesung Quantum Computing SS ‘08

Lmn |ym yn|
m n
^ |y 
Lmn = ym| L
n
21
quantum dynamics
free particle
wave packet traveling in a potential
http://jchemed.chem.wisc.edu/JCEWWW/Articles/WavePacket/WavePacket.html
movement of ion-qubits in a trap
Vorlesung Quantum Computing SS ‘08
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quantum dynamics
the state vector |y (r,t) follows the Schrödinger equation:
2
^

p
r
iħ |y (r,t) =
+ V(r) |y (r,t)
2m
t
ħ 
^
pr =
i r
• analogue to mechanical wave equations
• instead of the Hamilton Function H = T + V,
the Hamilton Operator is used
2
2
^ ^
p
ħ
2 + V(r)
H = r + V(r) = -
2m
Vorlesung Quantum Computing SS ‘08
2m
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time evolution
(t) |yn evolve in time?
how does a state |y(t) = c
n
n
?
^(t) |y(0)
|y(t) = U
^ (t): time evolution operator
U
insert into Schrödinger equation:
^ U(t)
^ |y(0) = H
^ |y(0)
iħ  U(t)
t
^
U’(t)
iH
^
=
^
ħ
U(t)

Vorlesung Quantum Computing SS ‘08
^t
-iH
^ =e ħ
U(t)
24
unitary operators
^ =1
^+ U
^+
^ -1 = U
U
U
unitary operators transform one base into another without
loosing the norm (e.g., a rotation is a unitary transformation)
^
the time evolution operator is unitary because H is hermitian
^ – t ) - i H(t
^ –t )
i H(t
0
0
^
^ =e ħ
ħ
U+(t) U(t)
e
= e0 = 1
manipulation in quantum computing is done by unitary operations
 quantum computing is reversible!
(as long as one does not measure)
Vorlesung Quantum Computing SS ‘08
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logic operations
1-bit logic gate:
NOT (a1| 0  + a2| 1 ) = a1|1  + a2| 0 
matrix representation for the NOT gate:
X
a1
a2
=
X
Vorlesung Quantum Computing SS ‘08
0
1
a1
1
0
a2
X-1 =
1
0
0
1
=
X≡
0
1
1
0
a2
a1
26
quantum computing
quantum-bit (qubit)
|0  
a
a1 |0 + a2 |1 = a1
2
|1  
classical bit
1  ON  3.2 – 5.5 V
0  OFF  -0.5 – 0.8 V
preparation
|Y0
Vorlesung Quantum Computing SS ‘08
calculation

H

U
read-out
 -1
H

Y|A|Y
time
time
27
measurement
^
a physical observable is described by a hermitian operator A
an adjoint (hermitian conjugated) operator is defined by:
^ |j 
|y  = A
^+
y | = j | A
^ +|y  = y | A
^ |j *
j | A
^+ = A
^
for a hermitian operator: A
Vorlesung Quantum Computing SS ‘08
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measurement
|y  = a1 |0 + a2 |1
probability that the measurement outcome is 0 or 1:
^ | y  = |a |2
p(0) = y | A
0
1
^ | y  = |a |2
p(1) = y | A
1
2
state after the measurement:
A0 | y 
a1
=
|0
|a1|
|a1|
A1 | y 
a2
= |a | |1
|a2|
2
Vorlesung Quantum Computing SS ‘08
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hermitian operators
an example: the momentum operator
ħ 
^
px =
i x
ħ *
y =  dx i xj
=  dx ─ ħ  j* y
i x

= ─ ħ j*y |∞ +  dx j* ħi x
i
-∞
p^xj |y  =  dx (pxj)*
(
(
)
y
)
(
y) = j |p^xy 
wavefunctions vanish at infinity
Vorlesung Quantum Computing SS ‘08
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measurement
^
a physical observable is described by a hermitian operator A
^ if
a state |y  is an eigenstate of an operator A,
^ |y  = a |y 
A
the eigenvalues a are real
vector is invariant under sheer
^
y | A |transformation
y  = a y→|y

eigenvector
of the transformation
A^y |y  = a* y |y 
0 = (a – a*) y |y 
Vorlesung Quantum Computing SS ‘08

31
measurement
^|y 
the mean value of A is given by y | A
^
^ | y  =  a |c |2 = A
y | A
n n
n
|y  = cn|yn
n
the probability measuring eigenvalue an is given by |cn|2
^ | y  is the mean value of A
^
y | A
^ ^ = 0,
If the operators of two observables A and B commute, [A,B]
they can be measured at the same time with unlimited precision.
^ ^ ≠ 0, [A,B]
^ ^ is a measure for the uncertainty of a and b:
For [A,B]
^ ^ |
Da·Db ≥ ½ | [A,B]
y
Vorlesung Quantum Computing SS ‘08
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measurement
^
a physical observable is described by a hermitian operator A
eigenvectors of different eigenvalues are orthogonal
^ |y  = a |y 
A
m
m
m
^ |y  = a |y 
A
n
n
n
^y  = A^y |y  = a y |y 
an ym|yn = ym|A
m
n
m n
m n
 (an – am) ym|yn = 0  ym|yn = 0
an ≠ a m
hermitian operators share a set of eigenvectors if they commute
^ ^ =0
[A,B]
^ are diagonal in the same base
 A^ and B
Vorlesung Quantum Computing SS ‘08
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