1. quantum computing |Y0 U H H-1 Y|A|Y quantum-bit (qubit) 0  
0  
1  
a1 0 + a2 1 = a1 a2
preparation
|Y0
calculation
read-out
time  U   H
H-1
Y|A|Y 
time

2. from classic to quantum
we live in Hilbert Space H
the state of our world is |y

3. can you see? Don Eigler (IBM, Almaden) 48 Fe atoms on Cu(111)

4. double slit experiment
classically:
number of electrons measured has a broad distribution

5. double slit experiment
quantum mechanically:
coherent superposition
|y = c1|y1 + c2|y2
wave function y = y (r,t)
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

6. double slit with electrons

7. double slit with electrons

8. Double slit with larger objects
O. Nairz, M. Arndt, and A. Zeilinger: Am. J. Phys. 71, 319 (2003)

9. 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 *

10. cj |y  = y |cj * = c* j |y 
the space of the world
j |y1 + y2  = j |y1 + j |y2
the scalar product is distributive
j |cy  = c j |y 
and thus
cj |y  = y |cj * = c* j |y 
it is positive definite and real for y |y  ≥ 0 ,  

11. quantum computing |Y0 U H H-1 Y|A|Y quantum-bit (qubit) 0  
0  
1  
a1 0 + a2 1 = a1 a2
preparation
|Y0
calculation
read-out
time  U    H
H-1
Y|A|Y
time

12. 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 = cnym|yn = cn dmn n
cm = ym|j
|j = |yn yn|j n

13. euclidic representation

14. 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
|cQC = cmno|ymno = cmno |ym1  |yn2  |yo3
m,n,o
|cmnoQC = |ym1  |yn2  |yo3 = |ym1 |yn2 |yo3
 we can find (or build) a quantum computer in our world

15. endohedral fullerenes
4 Å
atom inside has an electron
spin that can serve as qubit
|+1/2
|-1/2 mS mI
10 Å
source: K. Lips, HMI

16. quantum computing |Y0 U H H-1 Y|A|Y quantum-bit (qubit) 0  
0  
1  
a1 0 + a2 1 = a1 a2
preparation
|Y0
calculation
read-out
time  U    H
H-1
Y|A|Y
time

17. boolean algebra and logic gates
classical (irreversible) computing
gate in
out
1-bit logic gates:
identity
NOT x Id 1 x
NOT x 1 x
NOT x

18. quantum logic gates X ≡ X = 1-bit logic gate: NOT
(a1| 0  + a2| 1 ) =
a1|1  + a2| 0  x
NOT x 1
manipulation in quantum mechanics is done by linear operators
operators have a matrix representation
X ≡ 1
matrix representation for the NOT gate: X = 1 a1 a2

19. manipulation in our world
because of the superposition principle |y = c1|y1 + c2|y2, mathematical instructions (operators) have to be linear:
L (|y1 + |y2) = L |y1 + L |y2 ^
L (c1 |y1) = c1L |y1 ^
examples:
(c + d/dx)
 dx
()2
(c + d/dx) (f(x) + g(x)) = cf + d/dx f + cg + d/dx g
 dx (f(x) + g(x)) =  f dx +  g dx X
(f(x) + g(x))2 ≠ f2 + g2

20. [L,L] = [L,1] = [L,L-1] = 0, [L,aM] = a [L,M],
linear operators
(L + M) |y = L |y + M |y ^
(L M) |y = L (M |y) ^
however, generally
L M |y ≠ M L |y , |y ^
commutator: [L,M] = L M – M L ^
[L,M] = – [M,L] ^ 
anticommutator: [L,M]+= L M + M L ^
[L,L] = [L,1] = [L,L-1] = 0, [L,aM] = a [L,M],
[L1 + L2,M] = [L1,M] + [L2,M],
[L1L2,M] = [L1,M] L2 + [L2,M] L1 ^

21. vectors and operators  |j = |yn yn|j 1 = |yn yn|
1 L 1 = |ym ym| ( L |yn yn|) =   Lmn |ym yn| n m ^
with matrix elements Lmn = ym| L |yn ^

22. quantum dynamics movement of ion-qubits in a trap
free particle
wave packet traveling in a potential
movement of ion-qubits in a trap

23. quantum dynamics ħ  ^ pr = i r 2 + V(r) iħ  t
the state vector |y (r,t) follows the Schrödinger equation: iħ  t
|y (r,t) = V(r) |y (r,t) ^
pr2 2m
pr =
ħ 
i r ^
analogue to mechanical wave equations
instead of the Hamilton Function H = T + V,
the Hamilton Operator is used ħ2
H = V(r) = - ^
pr2 2m
2 + V(r)

24. time evolution ?  U(t) = e how does a state evolve in time  iħ t i
|y(t) = cn(t) |yn n
evolve in time
|y(t) = U(t) |y(0) ^
U(t): time evolution operator ^
insert into Schrödinger equation:
U(t) |y(0) = H U(t) |y(0) ^ iħ  t
= - H i ħ
U’(t)
U(t) ^
 U(t) = e ħ
- i H t ^ ^

25. unitary operators  quantum computing is reversible! U-1 = 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
U+(t) U(t) = e e = e0 = 1 ^ ħ
- i H(t – t0)
i H(t – t0) ^ ^
manipulation in quantum computing is done by unitary operations
 quantum computing is reversible!
(as long as one does not measure)

26. logic operations X ≡ X = X X-1 = 1-bit logic gate: NOT
(a1| 0  + a2| 1 ) =
a1|1  + a2| 0 
X ≡ 1
matrix representation for the NOT gate: X = 1 a1 a2
X X-1 = 1

27. quantum computing |Y0 U H H-1 Y|A|Y classical bit
1  ON  – 5.5 V
0  OFF  – 0.8 V
quantum-bit (qubit)
0  
1  
a1 0 + a2 1 = a1 a2
preparation
|Y0
calculation
read-out
time  U    H
H-1
Y|A|Y
time

28. measurement j | A+|y  = y | A |j *
a physical observable is described by a hermitian operator A ^
an adjoint (hermitian conjugated) operator is defined by: ^
|y  = A |j  ^
y | = j | A+ ^ ^
j | A+|y  = y | A |j * ^ ^
for a hermitian operator: A+ = A

29. measurement = 0 = 1 |y  = a1 0 + a2 1 ^
probability that the measurement outcome is 0 or 1:
p(0) = y | A0 | y  = |a1|2
p(1) = y | A1 | y  = |a2|2 ^
state after the measurement:
A0 | y 
= 0
|a1| a1
A1 | y 
= 1
|a2| a2

30. hermitian operators ħ  ^ px = i x
an example: the momentum operator
ħ 
i x ^
px =
pxj |y  =  dx (pxj)* y =  dx ( j)* y
=  dx (─ j*) y
= ─ j*y | +  dx j*( y) = j |pxy 
ħ 
i x ħ i ∞ - ^
wavefunctions vanish at infinity

31. measurement  y | A | y  = a y |y  Ay |y  = a* y |y  ^
a physical observable is described by a hermitian operator A ^
a state |y  is an eigenstate of an operator A, if
A |y  = a |y  ^
vector is invariant under sheer
transformation → eigenvector of the transformation
the eigenvalues a are real
y | A | y  = a y |y 
Ay |y  = a* y |y  ^
0 = (a – a*) y |y  

32. For [A,B] ≠ 0, [A,B] is a measure for the uncertainty of a and b:
measurement
the mean value of A is given by y | A | y  ^
y | A | y  =  an |cn|2 = A ^ n ^
|y  = cn|yn n
the probability measuring eigenvalue an is given by |cn|2
y | A | y  is the mean value of A ^
If the operators of two observables A and B commute, [A,B] = 0, they can be measured at the same time with unlimited precision. ^
For [A,B] ≠ 0, [A,B] is a measure for the uncertainty of a and b:
Da·Db ≥ ½ | [A,B]y | ^

33. measurement  (an – am) ym|yn = 0  ym|yn = 0
a physical observable is described by a hermitian operator A ^
eigenvectors of different eigenvalues are orthogonal
A |ym  = am |ym  ^
A |yn  = an |yn  ^
an ym|yn = ym|Ayn = Aym|yn = am ym|yn ^
 (an – am) ym|yn = 0
an ≠ am
 ym|yn = 0
hermitian operators share a set of eigenvectors if they commute
[A,B] = 0 ^
 A and B are diagonal in the same base ^