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Vorlesung Quantum Computing SS ‘08 1 prime factorization algorithm found by Peter Shor 1994 generalized to an algorithm for order finding key step is the.

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Presentation on theme: "Vorlesung Quantum Computing SS ‘08 1 prime factorization algorithm found by Peter Shor 1994 generalized to an algorithm for order finding key step is the."— Presentation transcript:

1 Vorlesung Quantum Computing SS ‘08 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 Vorlesung Quantum Computing SS ‘08 2 discrete Fourier transform (DFT) DFT unitary transform that converts a string of N complex numbers x j to a string of N complex numbers y j y k = x j e i2  jk/N 1 √N ∑ j=0 N-1 inverts periodicity (input string period r, output string N/r) converts off-sets into phase factors i i 0 Lieven Vandersypen, PhD thesis:

3 Vorlesung Quantum Computing SS ‘08 3 quantum Fourier transform number of qubits n = log 2 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  → e i2  jk/N  k  1 √N ∑ k=0 N i i 0 (  1    5  )√2(  0   i  2    4   i  6  )/2 Lieven Vandersypen, PhD thesis:

4 Vorlesung Quantum Computing SS ‘08 4 quantum Fourier transform  j  → e i2  jk/N  k  1 √N ∑ k=0 N-1 ( short: j = j 1 j 2 j 3 )j = j j j j 1 j 2 j 3 = j 1 /2 + j 2 /4 + j 3 /8 e  j 1 j 2 j 3  → 1 2 n/2 00  i 0. 11 00 e 11 00 e 11 reverse order of output qubits: j 1 in = j 3 out, j 3 in = j 1 out j3j3 j2j3j2j3 j1j2j3j1j2j3 j1j1 j2j1j2j1 j3j2j1j3j2j √2 j 3 in √2 j 2 in 1 e i  /2 0 0 j 2 in j 3 in =1 H H R 23 90°

5 Vorlesung Quantum Computing SS ‘08 5 QFT quantum circuit H R 90° H R H R 45° Weinstein et al.: quant-ph/ v1 (1999) j3j3 j2j2 j1j1 j1j1 j2j2 j3j3 H = i ħ  I x  e i ħ I y  2 e R 1 = i ħ  I z  e = ii  2 e i  2 e i ħ  I y  2 e i ħ I y  2 e i ħ  I x  e= R jk = i ħ  I y j,k  2 e i ħ I y j,k  2 e i ħ  I x j,k  e i ħ  J t I z j I z k e i ħ  I  j  e i ħ  e

6 Vorlesung Quantum Computing SS ‘08 6 QFT implementation Weinstein et al.: Phys. Rev. Lett. 86,1889 (2001) Preparation of pseudo-pure state  000  Preparation of input state  000  010  100  110  13 C labeled spins of Alanine as qubits (T 1 >1.5s, T 2 >400ms) Apply QFT =>  000  001  and SWAP =>  000  100  |000   000| |111   111| |000   111| |111   000|

7 Vorlesung Quantum Computing SS ‘08 7 Order finding Lieven Vandersypen, PhD thesis: M rooms with one entrance and one exit Connected by subcycles How often to change room until back at start? y={2,4,5} y=6 y={0,1,3,7} r=3 r=1 r=4  1 (5)=4,  2 (5)=2, …

8 Vorlesung Quantum Computing SS ‘08 8 Order finding: quantum circuit M=4, |y 1 y 0  = |y , y  M-1 |x  |y  → |x  |  x (y)  = |x  |  4x (y)  |  2x (y)  |  x (y)  210 |x  = |x 2 x 1 x 0 , x=4x 2 +2x 1 +x 0 00 00 00 y1y1 y0y0 H H H   (y)   (y)   (y) QFT Lieven Vandersypen, PhD thesis: 11 11 y=3  1 (3)=1,  2 (3)=3, …  0  =|000  |11   1  =(|0  +|1  +|2  +|3  +|4  +|5  +|6  +|7  )|11   2  =(|0  +|2  +|4  +|6  )|1  +(|1  +|3  +|5  +|7  )|3   3  =(|0  +|4  )|1  +(|0  |4  )|3  measurement yields multiple of 2 n /r => r=2

9 Vorlesung Quantum Computing SS ‘08 9 Order finding for n qubits 1)  0  0  register 1 register 2 number of transitions between “rooms” number of starting room 2) H  n → √2n√2n 1  j0j0 j= n 3) f(j)=  j (0) → √2n√2n 1   j  f(j)  j= n 4) QFT r1 → e i2  jk/N  k  f(j)  → e i2  jk/N  k  f(j)  √2n√2n 1  j= n  k= n √2n√2n 1  k= n  j= n Lieven Vandersypen, PhD thesis: 5) Measure the amplitude of  k  →  e i2  jk/N j= n 2 (large for k=c*N/r)

10 Vorlesung Quantum Computing SS ‘08 10 Shor’s algorithm number to factorize: N (e.g. N=15) integer a does NOT divide N (a=2,4,7,8,11,13,14) one factor is gcd(a r/2  1,N) (probability >0.5) L. Vandersypen et al.: Nature 414, 883 (2001) f(x,y)=  x (y)=a x y mod N a 4 mod 15 = 1, a  {2,7,8,13} => a mod 15 =1, k ≥ 2 2k2k a 2 mod 15 = 1, a  {4,11,14} f(x) needs x 0 only f(x) needs x 0 and x 1 “easy” register 2needs 4 qubits m=(log 2 N) register 1 needs 2 qubits (take 3)n=2m “easy?”

11 Vorlesung Quantum Computing SS ‘08 11 Implementation of Shor’s algorithm C C

12 Vorlesung Quantum Computing SS ‘08 12 Measurement thermal spectra effective pure state a=11 x 0 =0 x 1 =0 x 2 =|0  +|1  calculated measured simulated x=|000  +|100  =|0  +|4  =>r=2 n /4=2 gcd(11 2/2  1,15)=3,5

13 Vorlesung Quantum Computing SS ‘08 13 Pulse sequence total ~300 pulses 90°pulse y -xx 180°refocusing pulses z rotation -xx


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