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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 quantum Fourier transform (QFT)

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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:

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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:

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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 00 i 0. 11 00 e 11 00 e 11 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°

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Vorlesung Quantum Computing SS ‘08 5 QFT quantum circuit H R 90° H R H R 45° Weinstein et al.: quant-ph/ v1 (1999) j3j3 j2j2 j1j1 j1j1 j2j2 j3j3 H = i ħ I x e i ħ I y 2 e R 1 = i ħ I z e = ii 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

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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|

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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, …

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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 00 00 00 y1y1 y0y0 H H H (y) (y) (y) QFT Lieven Vandersypen, PhD thesis: 11 11 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

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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 j0j0 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)

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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?”

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

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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

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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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