1. prime factorization algorithm found by Peter Shor 1994
generalized to an algorithm for order finding
key step is the quantum Fourier transform (QFT)

2. discrete Fourier transform (DFT)
Lieven Vandersypen, PhD thesis:
DFT
unitary transform that converts a string of N complex numbers xj to a string of N complex numbers yj
yk = xj ei2pjk/N 1 √N ∑
j=0
N-1
inverts periodicity (input string period r, output string N/r)
converts off-sets into phase factors
1 0 -i i 0

3. quantum Fourier transform
Lieven Vandersypen, PhD thesis:
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
|j → ei2pjk/N |k 1 √N ∑
k=0
N-1
1 0 -i i 0
(|1 + |5)√2
(|0 - i|2 - |4 + i|6)/2

4. quantum Fourier transform
|j → ei2pjk/N |k 1 √N ∑
k=0
N-1
(short: j = j1j2j3)
j = j122 + j221 + j320
0.j1j2j3 = j1/2 + j2/4 + j3/8
reverse order of output qubits: j1in = j3out, j3in = j1out e
|j1 j2 j3 → 1
2n/2
|0 +
2pi 0.
|1 j3 j1
j2j1
j2j3
j3j2j1
j1j2j3 1 -1 √2
j3in 1 -1 √2
j2in 1
eip/2
j2in
j3in =1 H
R23
90° H

5. QFT quantum circuit |j3 |j2 |j1 |j1 |j2 |j3 H R R H R H e H = p
Weinstein et al.: quant-ph/ v1 (1999)
|j3
|j2
|j1
|j1
|j2
|j3 H R
90° R
45° H R
90° H -i f 2 e i
H = i ħ Ix p e Iy 2 i ħ Iz f e
R1 = = i ħ
Iyj,k p 2 e
Iyj,k
Ixj,k f i ħ Iy p 2 e Iy Ix f =
Rjk= i ħ
J t Izj Izk e
Iq j p

6. 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|

7. Order finding M rooms with one entrance and one exit
Lieven Vandersypen, PhD thesis:
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, … 6
y={2,4,5} 7
r=3
y={0,1,3,7} 4
r=4
y=6
r=1 5 2 3 1

8. Order finding: quantum circuit
Lieven Vandersypen, PhD thesis:
|x= |x2x1x0, x=4x2+2x1+x0
M=4, |y1y0= |y, y  M-1
|x|y → |x|px(y) = |x|p4x (y)|p2x (y)|px (y) 2 1
y=3
p1(3)=1, p2(3)=3, …
|0 H
QFT
p4(y)
p2(y)
p (y)
|y1
|y0
|1
|y2=(|0+|2+|4+|6)|1+(|1+|3+|5+|7)|3
|y1=(|0+|1+|2+|3+|4+|5+|6+|7)|11
|y3=(|0+|4)|1+(|0-|4)|3
|y0=|000|11
measurement yields multiple of 2n/r => r=2

9. Order finding for n qubits
Lieven Vandersypen, PhD thesis:
register 1
register 2
number of transitions between “rooms”
number of starting room
1) |0|0
2) Hn →
√2n 1 S
|j|0
j=0
2 -1 n
3) f(j)=pj(0) →
√2n 1 S
|j|f(j)
j=0
2 -1 n
4) QFT r1 →
ei2pjk/N |k|f(j) → ei2pjk/N |k|f(j)
√2n 1 S
j=0
2 -1 n
k=0
5) Measure the amplitude of |k →
S ei2pjk/N
j=0
2 -1 n 2
(large for k=c*N/r)

10. Shor’s algorithm number to factorize: N (e.g. N=15)
L. Vandersypen et al.: Nature 414, 883 (2001)
number to factorize: N
(e.g. N=15)
one factor is gcd(ar/21,N)
(probability >0.5)
“easy”
integer a does NOT divide N
(a=2,4,7,8,11,13,14)
“easy?”
f(x,y)=px(y)=axy mod N
a2 mod 15 = 1, a  {4,11,14}
f(x) needs x0 only
a4 mod 15 = 1, a  {2,7,8,13} => a mod 15 =1, k ≥ 2 2k
f(x) needs x0 and x1
register 1
needs 2 qubits (take 3)
n=2m
register 2
needs 4 qubits
m=(log2N)

11. Implementation of Shor’s algorithm1 2 3 4 5 6 7 C

12. Measurement thermal spectra a=11 effective pure state calculated
measured
simulated
x0=0
x1=0
x2=|0+|1
x=|000+|100
=|0+|4
=>r=2n/4=2
gcd(112/21,15)=3,5

13. Pulse sequence total ~300 pulses 90°pulse 180°refocusing pulsesy -x x
90°pulse
180°refocusing pulses x -x
z rotation