1. ( ( ) quantum bits conventional bit
off <=> V <=> 0
quantum mechanical bit (qubit)
| 0  <=> <=>
| 1  <=> <=> 1 (
superposition: a1 a2 ( )
a1| 0  + a2| 1  =

2. 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

3. 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

4. quantum gates X ≡ X = X X-1 = 1-bit logic gate: NOT
(a1| 0  + a2| 1 ) =
a1|1  + a2| 0 
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
X X-1 = 1

5. quantum parallelism a1 |00> + a2 |01> a3 |10> a4 |11>
input
b1 |00> +
b2 |01>
b3 |10>
b4 |11> =
output
a1 F |00> +
a2 F |01>
a3 F |10>
a4 F |11> F

6. how to create superposition
≡ | 0  ≡ 1
start in ground state
manipulation with a unitary transformation
Hadamard Gate
H = 1 -1 √2
H |0 = 1 √2
(|0 + |1)

7. 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

8. Bloch Sphere |y  = cos( ) |0 + eij sin( )|1 |0 NOT: |1  |0 H:
the 2 dimensional Hilbertspace of a single qubit can be represented by the Bloch-Sphere
operations on a single qubit are represented by rotations on this sphere
|y  = cos( ) |0 + eij sin( )|1 q 2
|0
|1
source:
NOT:
|1  |0 H:
|0  |0 + |1
|1  |0 - |1

9. Bloch Sphere  y |y  = 1 |y  = a1|0 + a2|1 |a1|2 + |a2|2 = 1
|y  = r1eij|0 + r2eiJ|1
polar coordinates:
multiply with global phase e-ij:
|y  = r1|0 + r2eif|1
f = (J - j)
|y  = r1|0 + r2eif|1 = r1|0 + (x + iy)|1

10. Bloch Sphere  3 dim unit sphere |y  = z |0 + (x + iy)|1
|y  = r1|0 + r2eif|1 = r1|0 + (x + iy)|1
with normalization constraint:
|r1|2 + | x + iy |2 = r12 + (x – iy) (x + iy)
= r12 + x2 + y2 = 1
 3 dim unit sphere
|y  = z |0 + (x + iy)|1
= cos q |0 + sin q (cos f + i sin f)|1
= cos q |0 + eif sin q |1
x = r sin q cos f
y = r sin q sin f
z = r cos q
|y  = cos( ) |0 + eif sin( )|1 q 2
0 ≤ q ≤ p, 0 ≤ f ≤ 2p

11. infinitesimal unitary transformation
finite transformations can be decomposed
in successive infinitesimal transformations
U(e) = I + ieF ^
(I + ieF) ^
(I - ieF*) = I ^ ^
F hermitian,
e infinitesimal small and real ^
with
F can be determined by the change dL of an observable L ^
L’ = L + dL = U(e)LU(e)* = L + ie[F,L] ^
U = eieF ^

12. how to rotate a qubit ^ UY (x) = Y (R-1x) ≈ Y (x+ey, y-ex, z)
rotation about z-axis ^
UY (x) = Y (R-1x) ≈ Y (x+ey, y-ex, z)
≈ Y (x,y,z) + e (y Y /x – x Y /y)
= Y (1 – ie/ħ [xpy – ypx]) = Y (1 – ie/ħ J3) ^ ^
and finite angle q i ħ
- qJ3 ^ ^
i q
ħ n
U(q) = (U(e))n = (1 – J3)n → e

13. spin as basis Sz = = Z Sx = = X Sy = = Y 1 ħ 2 -1 |0 ħ 2 1 -i ħ |1 2
Pauli spin matrices form a complete base
 using spin as basis is very convenient for all implementations
Sz = = Z 1 -1 ħ 2
|0
|1
Sx = = X 1 ħ 2
Sy = = Y i -i ħ 2 p 2
NOT : e = = e i ħ
- pSx -i
- i 1

14. Superposition of one and more qubits
H = 1 -1 √2
e = i ħ
(Sx+ Sz) p √2 1 -1
i√2
H2=H2H1 |00 = √2
(|0 + |1)
= (|00 + |01 + |10 + |11) 1 2

15. entanglement Bell states |cQC = |Y1  |Y2√2
states that can be factorized:
live in subspaces H1 and H2
Bell
states 1 √2
| c → (|01 - |10)
states that cannot be factorized:
live in product space HQC only

16. Bell states
The Bell State has the property that upon measuring the 1st qubit one obtains two possible results.
- 0 with probability ½ leaving the post measurement state
- 1 with probability ½ leaving the post measurement state
- The measurement of the 2nd qubit always gives the result depending on the measurement of the 1st qubit.
- ie: The measurements are CORRELATED
|Y‘ = |01
|Y‘ = |10

17. Bell basis |01 |10 |11 |00 |Y = (|01  |10) |Y = (|00  |11)
superpositions connected by the outline
Bell states connected by diagonals
|Y = (|01  |10) 1 √2
|Y = (|00  |11) 1 √2

18. Bell’s inequalities -1 1 1 1 1 1 -1 -1 -1 1 a’
restrictions due to assumption of hidden classical variables
 inequalities are violated by quantum mechanics z’ x’ z x
no influence between measurements,
they are done at different spacetime points
g,g’ =  1
a,a’ =  1
f := (a+a’)g – (a-a’)g’ -1 1 1 1 1
(a and a’ are either equal or opposite)
f := S p(a,a’,g,g’) f ≤ 2 1 -1 -1 -1 1 a’
ag + a’g – ag’ + a’g’ ≤ 2 a’ a’ a g’ g g’ g g a

19. Bell’s inequalities |Y = | A | A | G | G | A | A | G | G 1 √2z’ x’ z x 
g,g’ =  1 
a,a’ =  1
ag = Y | a  g |Y  = 1 √2
a’g = a’g’ = 1 √2
ag’ = 1 √2
ag+a’g – ag’+ a’g’ = 2 √2 1 -1
a =
a’ = 1 -1
g = – √2
g’ =

20. experiment source of entangled photons
(use spontaneous parametric down conversion of a non-linear, birefringent crystal)

21. experiment spacetime separation measurement apparatus
measurement time: 100 ns
physical random number generator
G. Weihs et al, Phys. Rev. Lett. 81, 5039 (1998)

22. uncorrelated measurements
random number generator
measurement
apparatus
measurement
apparatus

23. boolean algebra and logic gates
2-bit logic gates: x y
x OR y x y
x AND y y
x OR y 1 x y
x AND y 1 x
all other operations can be constructed from NOT, OR, and AND
x XOR y = (x OR y) AND NOT (x AND y)

24. classical binary addition
two one-bit digit in, one two-bit digit out: a0 + b0 = c0 + c1 X
& a0 b0
c0 (add mod2)
c1 (carry bit) 
fanout
0 1
truth table + X 1 2 3 4 a3 b1 a2 b2 a1 b3 a0 b0 c1 c2 c0 c3 c4
more than
one bit...

25. 2 qubit gates |00 = |01 = |10 = |11 = 1 CNOT: 1 00 01 10
base vectors of a two–qubit register:
|00 = 1
|01 =
|10 =
|11 =
a, a  b
a, b
00
01
10
11 1
CNOT:

26. 2 qubit gates switch on the interaction Hamiltonian
use free evolution of the system
|1
|0
00  00
01  01
10  11
11  10
CNOT:
source:

27. the CNOT gate
control
target i ħ p 2 e Sy Sy i ħ p 2 e Sz

28. create entanglement  1 |00 = |01 = |10 = |11 = Ca= |1 |a  |b
Cb= |1 H 1 √2
(|0 - |1) 
|a  |b 1
UCNOT = = - √2
|00 = 1
|01 =
|10 =
|11 =
|a  |b → (|01 - |10)  no factorization into product states possible 1 √2

29. no-cloning theorem 1 1 1 1 is a “no-copying theorem”
classical: copy with XOR y
x XOR y 1 x
(x,0) → (x,x) 1 1 1 1
quantum mechanical: copy with CNOT ?

30. no-cloning theorem |Y  |0 = a0 |0  |0 + a1 |1  |0 1 =
“control” qubit is used as source
“target” qubit is initialized to |0
try to copy |Y = a0 |0 + a1|1
|Y  |0 = a0 |0  |0 + a1 |1  |0
Bell state 1 = a1 a0 a1 a0
= a0 |0  |0 + a1 |1  |1
≠ |Y  |Y

31. toward n qubits |Y = S ai |i |Y = a0|00 + a1|01 + a2|10 + a3|11
two–qubit state:
|Y = a0|00 + a1|01 + a2|10 + a3|11
n–qubit state:
2 -1 n
|Y = S ai |i
Hilbertspace: 2n  2n
i=0
e.g., n = 5:
|00101 = |5
|00110 = |6

32. universal computing |a |b |c |c(a  b) Toffoli gate
all possible operations can be done by using
1-qubit-rotations, phase-shifts and the CNOT gate
→ this set of gates is therefore called “universal”
(in a classical computer NOT and NAND are a universal set)
single universal gate: Toffoli gate (3 qubits)
|a
|b
|c
|c(a  b)
Toffoli
gate

33. Toffoli gate 1 UTF = Table of Truth 000 Matrix 001 010 011
|c
|c(a  b)
Toffoli
gate
a, b, c (ab)
a, b, c
000
001
010
011
100
110
111
101
Table of Truth
Matrix 1
UTF =