Factorization of 15 on an NMR quantum computer

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1 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 Studet Presettio Fctoriztio of 15 o NMR qutum computer Bsed o L. M. K. Vdersype, M. Steffe, G. Breyt, C. S. Yoi, M.. Sherwood, I. L. Chug, Eperimetl reliztio of Shor s qutum fctorig lgorithm usig ucler mgetic resoce, Nture 414 (001) D-PYS, Qutum Systems For Iformtio Processig (QSIT) FS 013/

2 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, 013 / 1 Motivtio Prime Fctoriztio - Clssiclly O( l/3 ) - Shor O(l 3 ) Fctoriztio of lrge primes (such s ecryptio keys) fesible o idel qutum computer Wht bout Shor s lgorithm o ctul qutum computers? 15 fctorized usig compiled lgorithms - 001: Nucler spis o molecule [1] - 009: Photos itegrted o chip [] - 01: Phse of Josephso juctios [3] s well s 1-01: Photos [4]

3 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 Fctoriztio of 15 usig Shor s lgorithm 1 f () = mod 15 - Possible = [,, 4,,, 7, 8,,, 11,, 13, 14] Fid period r i of f () - mod 15 = 1 for = [4, 11, 14] r = - 4 mod 15 = 1 for = [, 7, 8, 13] r = 4 3 gcd( r ± 1, 15) - gcd(11 ± 1, 15) = [gcd(10, 15), gcd(1, 15)] = [5, 3] - gcd( 4 ± 1, 15) = [gcd( 3, 15), gcd( 5, 15)] = [3, 5] Lrgest period of r is 4

4 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 Qutum Implemettio m iit clculte f() clculte r 1 3 0> Iverse QFT 1> 1 mod N Period fidig i by iverse QFT Prlleliztio - Qbits i superpositio sttes to store d f () - Epoetil sclig of ilbert spce Clssiclly epoetil problem rus i polyomil time Emple: N = 15, = 11 r = 3 qbits for, 4 qbits for m = { } 1 + { } 11 3 { } 1 delt comb delt comb + { 0 4 } 11 smple shift lier phse Redout of 3 o results i superpositio of 0 d 4, i.e. 000 d 100 see lso lecture otes (the lst few slides, but vlues bove for m represet ctul f() vlue.)

5 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 NMR Qutum Computtio Bsics [5] Qbits - Esemble of distiguishble d coupled ucler spis - Coherece times > 1 sec - ighly mied groud stte, E kt 1 Gtes - Sigle Qbit - Spi-selective RF pulses, σ, σ y or combitio, 0. - ms (d composite σ z ) - Double Qbit - Cotrolled phse by evolutio uder J for t = 1 J 5 10 ms Redout - Wek esemble mesuremet of σ d σ y F F F e 7 13 C C 5 5 (CO) F spectrum 1 C J coupligs o 1 5 Zeem terms + chem. shift [z] Iterspi terms

6 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 Qutum Circuit for NMR Qbit ssigmet (N = 15) - r m = 4 requires = qbits for sufficiet, 3 qbits chose for ep - m requires log (N) qbits Iitiliztio - Temporl vergig for pseudopure stte Optimiztios - Gtes reduced for f () = mod N compiled circuits for = 11 d = 7 m 1: : 3: 4: 5: 6: 7: m T e m p or l iit clculte f() clculte r 1 3 0> Iverse QFT 1> 1 mod N v e r g i g A B C D E F G o o Specific optimiztios to reduce gtes 90 Temporl vergig: Cretio of pseudopure stte qbit swppig + vergig F F F e 7 13 C C 5 5 (CO) 1 C 36 5

7 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 Pulse Sequece iit clculte f() clculte r 300+ pulses - π -pulses - Gussi-shped profile - Compestio of J Selective ecittio - either or CNOT = -CPASE- - π-pulses - ermite-shped profile - Compestio of J 1: : 3: 4: 5: 6: 7: 1: : 3: T e m p or l v e r g i g A B C D E F G Selective refocusig - to rewid J, where uwted - z-rottios - Used for f() d iqft 4: 5: 6: 7: (0) (1) () (3) 00 ms 400 ms 10 ms

8 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 Pulse Sequece 300+ pulses - π -pulses - Gussi-shped profile - Compestio of J Selective ecittio F spectrum J coupligs o [z] J refocused durig free evolutio J compested uder RF pulse - either or CNOT = -CPASE- - π-pulses - ermite-shped profile - Compestio of J 1: : 3: Selective refocusig - to rewid J, where uwted - z-rottios - Used for f() d iqft 4: 5: 6: 7: (0) (1) () (3) 00 ms 400 ms 10 ms

9 Fctoriztio of 15 usig NMR Itroductio Adri Doll QSIT Studet Presettio NMR ep My 31, 013 Summry 8 / 1 Refereces Results for = 11 Epected result: { 0 i + i + 4 i + 6 i} 1 i + { 1 i + 3 i + 5 i + 7 i} 1 1 i 0> m 1> 1 1 Iverse QFT 3 mod N 3 { 0 i + 4 i} 1 i + { 0 i 4 i} 1 1 i therml Alysis of Spectr: For two spis I d S i pseudo-pure sttes d b, upo π pulse to I pseudo pure b obs ( 1 ) (I + ( 1 ) I Sz ) sim spectrl phse revels stte of observer spi ctul spectrl lie depeds o stte of coupled spi here: severl spectrl peks due to multiple spis detectio of observer spi stte vi spectrl phse desired superpositio of i d i, i.e. 0 i d 4 i + rtefcts ep sim + dec

10 Fctoriztio of 15 usig NMR Itroductio Adri Doll QSIT Studet Presettio NMR ep My 31, 013 Summry 9 / 1 Refereces Results for = 7 Epected result: 0> m 1> 1 1 Iverse QFT 3 mod N { 0 i + 4 i} 1 i + { 1 i + 5 i} 7 i + { i + 6 i} 4 i + { 3 i + 7 i} 1 3 i 3 { 0 i + i + 4 i + 6 i} 1 i + { 0 i i i 4 i + i 6 i} 7 i + { 0 i i + 4 i 6 i} 4 i therml pseudo pure + { 0 i + i i 4 i i 6 i} 1 3 i Alysis of Spectr: detectio of observer spi stte vi spectrl phse desired superpositio of i i i i i.e. 0 i, i, 4 i, 6 i + eve more proouced rtefcts sim ep sim + dec

11 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 Summry Successful i-priciple demostrtio Clcultio of f () = mod N epesive Compiled/optimized lgoritm, bsed o kow d r Simplest cse lredy proe to decoherece Eperimetl reliztio very demdig Mi resos: - Number of qbits - 3 log (N) for full-scle implemettio - Number of gtes - ( + 1)/ for iqft plus wy more for modulr epoetitio (Fctoriztio of N = 1 usig photos [4] ws chieved by itertive decompositio of the iqft. The eecutio time ws therefore icresed, wheres oly oe qbit ws required for )

12 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 Refereces [1] L. M. K. Vdersype, M. Steffe, G. Breyt, C. S. Yoi, M.. Sherwood, I. L. Chug, Eperimetl reliztio of Shor s qutum fctorig lgorithm usig ucler mgetic resoce, Nture 414 (001) [] A. Politi, J. C. F. Mtthews, J. L. O Brie, Shors qutum fctorig lgorithm o photoic chip, Sciece 35 (5945) (009) 11. [3] E. Lucero, R. Breds, Y. Che, J. Kelly, M. Mritoi, A. Megrt, P. O Mlley, D. Sk, A. Visecher, J. Weer, T. White, Y. Yi, A. N. Cleld, J. M. Mrtiis, Computig prime fctors with Josephso phse qubit qutum processor, Nt Phys 8 (01) [4] E. Mrti-Lopez, A. Lig, T. Lwso, R. Alvrez, X.-Q. Zhou, J. L. O Brie, Eperimetl reliztio of Shor s qutum fctorig lgorithm usig qubit recyclig, Nt Photo 6 (01) [5] J. A. Joes, Qutum computig with NMR, Prog Nucl Mg Res Sp 59 () (011) Figures dopted from [1] or lecture slides D-PYS, Qutum Systems For Iformtio Processig (QSIT) FS 013/

13 Fctoriztio of 15 usig NMR Adri Doll QSIT Studet Presettio My 31, / 1 Questios? D-PYS, Qutum Systems For Iformtio Processig (QSIT) FS 013/

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