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prime factorization
algorithm found by Peter Shor 1994
generalized to an algorithm for order finding
key step is the quantum Fourier transform (QFT)
Vorlesung Quantum Computing SS ‘08
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discrete Fourier transform (DFT)
Lieven Vandersypen, PhD thesis: http://arxiv.org/abs/quant-ph/0205193
DFT
unitary transform that converts a string of N complex
numbers xj to a string of N complex numbers yj
yk =
1
√N
N-1
∑xe
j
i2pjk/N
j=0
inverts periodicity (input string period r, output string N/r)
10001000
10101010
converts off-sets into phase factors
01000100
Vorlesung Quantum Computing SS ‘08
1 0 -i 0 -1 0 i 0
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quantum Fourier transform
Lieven Vandersypen, PhD thesis: http://arxiv.org/abs/quant-ph/0205193
number of qubits n = log2 N (for N=8, three qubits)
amplitude of |000 represents first complex number, |001 the 2nd
states |000, |001, ... , |111 are labelled |0, |1 , ... , |7
N-1
e
∑
√N
|j  → 1
i2pjk/N
|k
k=0
01000100
(|1 + |5)√2
Vorlesung Quantum Computing SS ‘08
1 0 -i 0 -1 0 i 0
(|0 - i|2 - |4 + i|6)/2
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quantum Fourier transform
N-1
e
∑
√N
|j  → 1
i2pjk/N
|k
k=0
j = j122 + j221 + j320 (short: j = j1j2j3)
0.j1j2j3 = j1/2 + j2/4 + j3/8
reverse order of output qubits: j1in = j3out, j3in = j1out
|j1 j2 j3 → 1n/2
2
1
√2
2pi 0. j13
|0 +e
2pi 0. j2j13
|1 |0 +e
2pi 0. j31j2j13
|1 |0 +e
|1
j3in =1
1 1
1 -1 j in
3
H
Vorlesung Quantum Computing SS ‘08
1 1 1
1 0
0 eip/2 j in√2 1 -1 j in
2
2
R90°
23
H
4
QFT quantum circuit
Weinstein et al.: quant-ph/99060059v1 (1999)
| j 3
| j 1
H
| j 2
90°
R
| j 2
H
| j 1
45°
R
H=e
i pI
ħ x
-i p
ħ 2
Rjk= e
e
e
Iyj,k
e
i pI
ħ 2 y
R1 = e
e
i pI j i J t I j I k
z z
ħ q ħ
Vorlesung Quantum Computing SS ‘08
R
- i f Iz
ħ
i f I j,k i p I j,k
ħ x ħ 2 y
e
90°
e
i pI j
ħ q
| j 3
H
-i f
2
=
e
e
- i p Iy
ħ 2
= e
e
if
2
i fI i p I
ħ x ħ 2 y
e
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QFT implementation
Weinstein et al.: Phys. Rev. Lett. 86,1889 (2001)
13C
labeled spins of Alanine as qubits (T1>1.5s, T2>400ms)
Preparation of pseudo-pure state |000
Preparation of input state |000+|010+ |100+|110
Apply QFT => |000+|001 and SWAP => |000+|100
|111 000|
|000000|
|111 111|
|000111|
Vorlesung Quantum Computing SS ‘08
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Order finding
Lieven Vandersypen, PhD thesis: http://arxiv.org/abs/quant-ph/0205193
M rooms with one entrance and one exit
Connected by subcycles
How often to change room until back at start?
p1(5)=4, p2(5)=2, …
0
6
y={2,4,5}
r=3
7 y={0,1,3,7}
4
r=4
y=6
r=1
5
2
1
Vorlesung Quantum Computing SS ‘08
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Order finding: quantum circuit
Lieven Vandersypen, PhD thesis: http://arxiv.org/abs/quant-ph/0205193
|x= |x2x1x0, x=4x2+2x1+x0
M=4, |y1y0= |y, y  M-1
|x|y → |x|px(y) = |x|p4x 2 (y)|p2x 1(y)|px 0 (y)
y=3 p1(3)=1, p2(3)=3, …
|0
H
|0
H
|0
H
||y11
QFT
p (y)
||y10
p2(y)
p4(y)
|y|y
=(|0+|1+|2+|3+|4+|5+|6+|7)|11
|y3=(|0+|4)|1+(|0-|4)|3
|y0=|000|11
21=(|0+|2+|4+|6)|1+(|1+|3+|5+|7)|3
Vorlesung Quantum Computing SS ‘08
measurement yields multiple of 2n/r => r=2
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Order finding for n qubits
Lieven Vandersypen, PhD thesis: http://arxiv.org/abs/quant-ph/0205193
register 1
number of transitions between “rooms”
register 2
number of starting room
1) |0|0
n
2) Hn →
2 -1
S
1
|j|0
n
√2 j=0
n
3) f(j)=pj(0) →
2 -1
S
1
|j|f(j)
n
√2 j=0
n
n
n
2 -1 2 -1
n
2 -1 2 -1
S S
S S
4) QFT r1 → 1
ei2pjk/N |k|f(j) → 1
ei2pjk/N |k|f(j)
√2n j=0 k=0
√2n k=0 j=0
n
5) Measure the amplitude of |k →
Vorlesung Quantum Computing SS ‘08
2 -1
S
e
j=0
2
i2pjk/N
(large for k=c*N/r)
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Shor’s algorithm
L. Vandersypen et al.: Nature 414, 883 (2001)
one factor is gcd(ar/21,N)
number to factorize: N
(e.g. N=15)
(probability >0.5)
integer a does NOT divide N
(a=2,4,7,8,11,13,14)
f(x,y)=px(y)=axy mod N
“easy”
“easy?”
a2 mod 15 = 1, a  {4,11,14}
a4
f(x) needs x0 only
2k
mod 15 = 1, a  {2,7,8,13} => a mod 15 =1, k ≥ 2
f(x) needs x0 and x1
register 1
needs 2 qubits (take 3)
n=2m
register 2
needs 4 qubits
m=(log2N)
Vorlesung Quantum Computing SS ‘08
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Implementation of Shor’s algorithm
3
5
6
1
2
C
C
7
4
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Measurement
thermal spectra
a=11
effective pure state
calculated
measured
simulated
x1=0
x0=0
x=|000+|100=|0+|4 =>r=2n/4=2
Vorlesung Quantum Computing SS ‘08
x2=|0+|1
gcd(112/21,15)=3,5
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Pulse sequence
total ~300 pulses
y
90°pulse
180°refocusing pulses
-x x
x -x
z rotation
Vorlesung Quantum Computing SS ‘08
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