vorlesung quantum computing ss ‘08 1 quantum bits conventional bit on 3.2 - 5.5 v 1 off -0.5 - 0.8...
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Vorlesung Quantum Computing SS ‘08
1
quantum bits
conventional bit
on <=> 3.2 - 5.5 V <=> 1
off <=> -0.5 - 0.8 V <=> 0
quantum mechanical bit (qubit)
| 0 <=> <=>
| 1 <=> <=>
10(
(
01(
(
a1| 0 + a2| 1 = a1
a2( )
superposition:
Vorlesung Quantum Computing SS ‘08
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quantum computing
H H-1
calculation
U
preparation
read-out
|A|
time
time
quantum-bit (qubit)
0 1
a10 + a21 =a1a2
Vorlesung Quantum Computing SS ‘08
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boolean algebra and logic gates
classical (irreversible) computing
gateinout
1-bit logic gates: identity
x NOT x
0 11 0
x Id
0 01 1
NOT
x NOT x
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quantum gates
1-bit logic gate: NOT a1|1 + a2| 0 (a1| 0 + a2| 1 ) =
manipulation in quantum mechanics is done by linear operators operators have a matrix representationX ≡
0
01
1matrix representation for the NOT gate:
X =0
01
1a1
a2
a1
a2
=a2
a1
X X-1 =1
10
0
Vorlesung Quantum Computing SS ‘08
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quantum parallelism
a1 F |00>+
a2 F |01>+
a3 F |10>+
a4 F |11>
}{a1 |00>
+a2 |01>
+a3 |10>
+a4 |11>
}{input
b1 |00>+
b2 |01>+
b3 |10>+
b4 |11>
}{=
output
F
Vorlesung Quantum Computing SS ‘08
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how to create superposition
start in ground state ≡| 0 ≡1
0
manipulation with a unitary transformation
H = 1√2
H =1
-11
11√2
Hadamard Gate
Vorlesung Quantum Computing SS ‘08
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quantum computing
H H-1
calculation
U
preparation
read-out
|A|
time
time
classical bit
1 ON 3.2 – 5.5 V
0 OFF -0.5 – 0.8 V
quantum-bit (qubit)
0 1
a10 + a21 =a1a2
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NOT:|1 |0
Bloch Sphere
the 2 dimensional Hilbertspace of a single qubit can be represented by the Bloch-Sphere
H:|0 |0 |1|1 |0 |1
|0
|1source: http://www.c3.lanl.gov/~knill/qip/nmrprhtml/node5.html
operations on a single qubit are represented by rotations on this sphere
| = cos( ) + eisin( )
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Bloch Sphere
| = a1 + a2
| |a1|2 + |a2|2 = 1
| = r1ei + r2ei polar coordinates:
multiply with global phase e-i | = r1 + r2ei
(
| = r1 + r2eir1 + (x + iy)
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Bloch Sphere
| = r1 + r2eir1 + (x + iy) with normalization constraint:
|r1|2 + | x + iy |2 = r12 + (x – iy) (x + iy)
= r12 + x2 + y2 = 1
3 dim unit sphere
x = r sin cos y = r sin sin z = r cos
| = cos( ) + eisin( )
= z + (x + iy) = cos + sin (cos + i sin ) = cos + ei sin
0 ≤ ≤ 0 ≤ ≤ 2
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infinitesimal unitary transformation
finite transformations can be decomposed in successive infinitesimal transformations
U() = I + iF^ ^ ^
(I + iF)^ ^ (I iF*) = I^ ^ ^ F hermitian, infinitesimal small and real
^with
F can be determined by the change L of an observable L ^ ^ ^
L’ = L + L = U()LU()* = L + i[F,L]^ ^ ^ ^ ^ ^ ^ ^ ^
U = eiF̂^
Vorlesung Quantum Computing SS ‘08
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how to rotate a qubit
rotation about z-axis
U(x) = (R-1x) ≈ (x+y, y-x, z)
≈ (x,y,z) +(y /x – x /y)
= (1 – i/ħ [xpy – ypx]) = (1 – i/ħ J3)
and finite angle
^
^ ^
U() = (U())n = (1 – J3)n → ei ħ n
iħ
J3^ ^
Vorlesung Quantum Computing SS ‘08
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spin as basis
Pauli spin matrices form a complete base using spin as basis is very convenient for all implementations
Sz = = Z
1
-10
0ħ2
ħ2
Sx = = X
0
01
1ħ2
ħ2
Sy = = Y
0
0i
-iħ2
ħ2
|0
|1
2
NOT : e = = e iħ
Sx 0
0-i
-i i 0
01
1
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Superposition of one and more qubits
H =1
-11
11√2
e =
iħ
(Sx+ Sz)
√2 1
-11
11i√2
H2=H2H1 = √2
√2
= 12
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entanglement
QC
→1√2
1√2
states that can be factorized:
live in subspaces H1 and H2
1√2
→states that cannot be factorized:
live in product space HQC only
BellBellstatesstates
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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
‘
‘
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Bell basis
– superpositions connected by the outline
– Bell states connected by diagonals
= 1√2
= 1√2
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Bell’s inequalities
restrictions due to assumption of hidden classical variablesinequalities are violated by quantum mechanicsz
x
a,a’ = 1
z’x’
g,g’ = 1no influence between measurements,they are done at different spacetime points
f := (a+a’)g – (a-a’)g’
(a and a’ are either equal or opposite)
f := p(a,a’,g,g’) f ≤ 2
aa’ g’gga’aa’
g’g
1
1-1-1-1
-1
1
111
ag + a’g – ag’ + a’g’ ≤ 2
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Bell’s inequalities
z
x
a,a’ = 1
z’x’
g,g’ = 1
1
-10
0
0
01
1
a =
a’ =
1
-11
1g = – 1√2
1
-1-1
-1g’ =
1√2
AGAG
1√2 AAG G
ag = | a g | = 1√2
a’g = a’g’ = 1√2
ag’ = 1√2
ag+a’g – ag’+ a’g’ = 2 √2
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experiment
source of entangled photons
(use spontaneous parametric down conversion of a non-linear, birefringent crystal)
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experiment
G. Weihs et al, Phys. Rev. Lett. 81, 5039 (1998)
spacetime separation measurement apparatus
measurement time: 100 ns
physical random number generator
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uncorrelated measurements
measurementapparatus
measurementapparatus
random number generator
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boolean algebra and logic gates
2-bit logic gates:
y x OR y
0
1
0
1
0
0
1
1
x
0
1
1
1
y x AND y
0
1
0
1
0
0
1
1
x
0
0
0
1
x
yx OR y
x
yx AND y
all other operations can be constructed from NOT, OR, and AND
x XOR y = (x OR y) AND NOT (x AND y)
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classical binary addition
two one-bit digit in, one two-bit digit out: a0 + b0 = c0 + c1
X
&
a0b0
c0 (add mod2)
c1 (carry bit)
fanout0 1
0 00 011 01 10
truth table
+
++
X
++
X
++
X
1 2 3 4
a3
b1
a2
b2
a1
b3
a0
b0 c1
c2
c0
c3
c4
more thanone bit...
Vorlesung Quantum Computing SS ‘08
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2 qubit gates
base vectors of a two–qubit register:
a, a ba, b
00 0001 0110 1111 10
CNOT:
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2 qubit gates
– switch on the interaction Hamiltonian– use free evolution of the system
|1
|0
00 0001 0110 1111 10
CNOT:
source: http://www.c3.lanl.gov/~knill/qip/nmrprhtml/node7.html
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the CNOT gate
control
target
iħ2
eSy
iħ2
e- Sy
iħ2
eSz
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create entanglement
Ca= 1
Cb= 1
H1
√2
ab
ab
ab→ no factorization into product states possible1√2
UCNOT = =
1√2
1√2
1√2
1√2
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no-cloning theorem
is a “no-copying theorem”
classical: copy with XOR
y x XOR y
0
1
0
1
0
0
1
1
x
0
1
1
0(x,0) → (x,x)
quantum mechanical: copy with CNOT ?
Vorlesung Quantum Computing SS ‘08
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no-cloning theorem
“control” qubit is used as source“target” qubit is initialized to
try to copy a0a1
a0 a1
=
a1
a0
a1
a0
= a0 a1
≠
Bell stateBell state
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toward n qubits
a0a1a2a3
two–qubit state:
n–qubit state:
Hilbertspace: 2n 2n ai ii=0
e.g., n = 5:
2 -1n
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universal computing
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
a
b
c(a b)
Toffoligate
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Toffoli gate
a, b, c (ab)a, b, c
000 000001 001010 010011 011100
110111
101100101111110
Table of Truth
Matrix
UTF =
a
b
c c(a b)
Toffoligate
b
a