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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)

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**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

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**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

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**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

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**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

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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|

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**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

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**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

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**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) Hn → √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)

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**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/21,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)

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**Implementation of Shor’s algorithm**

1 2 3 4 5 6 7 C

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**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(112/21,15)=3,5

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**Pulse sequence total ~300 pulses 90°pulse 180°refocusing pulses**

y -x x 90°pulse 180°refocusing pulses x -x z rotation

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