Quantum Circuits. School on Quantum Day 1, Lesson 5 16:00-17:00, March 22, 2005 Eisuke Abe

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1 Qutum Crcuts School o Qutum D, Lsso 5 6:-7:, Mrch, 5 Esuk Ab Dprtmt of Appl Phscs Phsco-Iformtcs, CEST-JST, Ko vrst

2 Outl Bloch sphr rprstto otto gts vrslt proof A rbtrr cotroll- gt c b mplmt usg ol sgl qubt gts CNOT A rbtrr cotroll - gt c b mplmt usg sgl qubt gts CNOT Two-lvl utr gts r uvrsl Sgl-qubt gts CNOT r uvrsl Hmr, S, T, CNOT r uvrsl

3 Bloch sphr rprstto ψ ψ ψ ϕ s No obsrvbl ffct x ϕ ψ ϕ s

4 Importt sgl qubt gts Z Y X 4 T S H X HZH Y HYH Z HXH Y X Z Y X Z S T S I H Z Y X,,, },,{ ], [, L

5 Expotl oprtor xp Ax Ax! A I xp Ax x I s x A Ths oprtor s mportt bcus t ppr th soluto to Schrögr quto H t ψ t t xp ψ t h

6 otto gts s s s X I X x x s s s Y I Y xp xp s Z I Z

7 otto bout th xs ˆ s ˆ ˆ xp ˆ Z Y X I x σ Exmpl ˆ, Z X H Z Y X σˆ x ˆ x

8 o to uvrslt proof. A rbtrr cotroll- gt c b mplmt usg ol sgl qubt gts CNOT. A rbtrr cotroll - gt c b mplmt usg sgl qubt gts CNOT. Two-lvl utr gts r uvrsl 4. Sgl-qubt gts CNOT r uvrsl 5. Hmr, S, T, CNOT r uvrsl

9 o to uvrslt proof. A rbtrr cotroll- gt c b mplmt usg ol sgl qubt gts CNOT. A rbtrr cotroll - gt c b mplmt usg sgl qubt gts CNOT. Two-lvl utr gts r uvrsl 4. Sgl-qubt gts CNOT r uvrsl 5. Hmr, S, T, CNOT r uvrsl

10 Z-Y composto For rbtrr sgl qubt gt, thr xst rl umbrs,,, such tht s s s s Proof

11 Corollr C B A St A, B, C s AXBXC I ABC, Th W wll costruct rbtrr cotroll- gt usg A, B, C

12 Corollr I ABC Proof C B A C A XC XX AX AXBXC X X X X I XX

13 Phs shftr P P I P I

14 Cotroll- gt P C B A C B A ABC I P C B A AXBXC

15 o to uvrslt proof. A rbtrr cotroll- gt c b mplmt usg ol sgl qubt gts CNOT. A rbtrr cotroll - gt c b mplmt usg sgl qubt gts CNOT. Two-lvl utr gts r uvrsl 4. Sgl-qubt gts CNOT r uvrsl 5. Hmr, S, T, CNOT r uvrsl

16 Cotroll - gt V V V V VV I Exmpl T S S Z HSH I H X H

17 Cotroll - gt I V V VV I V V VV I V V V

18 Cotroll - gt,, r th orr rthmtc oprtos V V V V V V V

19 Cotroll - gt V V V V V V V 4 4 V V V V V V V V

20 Gt costructo b Gr co Gr co; Ol o bt chgs from o tr to th xt ptt b F. Gr V V V V V V V V V V V V V V

21 Cotroll - gt St V so tht mplmt th tt V k j k j j j L L L < < < C b prov for b ucto Proof for ] [ ] [ ] [ ] [ 4

22 Cotroll - gt Csc mplmtto cll Toffol Cotroll Csc rsur

23 o to uvrslt proof. A rbtrr cotroll- gt c b mplmt usg ol sgl qubt gts CNOT. A rbtrr cotroll - gt c b mplmt usg sgl qubt gts CNOT. Two-lvl utr gts r uvrsl 4. Sgl-qubt gts CNOT r uvrsl 5. Hmr, S, T, CNOT r uvrsl

24 Two-lvl utr gt Two-lvl utr mtrx I I tr mtrx whch cts otrvll ol two-or-fwr vctor compots j f c h b g vrslt; cs Brkg up to th prouct of two-lvl utr mtrcs

25 Two-lvl utr gts r uvrsl j f c h b g j f c h g * * b b b b I b b b b j f h g * * * c c c c c c c c

26 Two-lvl utr gts r uvrsl j f h j f h g * * * * j h f I For -msol, w rpt ths procur u u u u L M O M L u u L O M M L L

27 Two-lvl utr gts r uvrsl u u L O M M L L k L u u L M O M M M L L L L L k k I L L

28 o to uvrslt proof. A rbtrr cotroll- gt c b mplmt usg ol sgl qubt gts CNOT. A rbtrr cotroll - gt c b mplmt usg sgl qubt gts CNOT. Two-lvl utr gts r uvrsl 4. Sgl-qubt gts CNOT r uvrsl 5. Hmr, S, T, CNOT r uvrsl

29 Sgl qubt gts & CNOT r uvrsl Strtg To show tht sgl qubt & CNOT gts c mplmt rbtrr two-lvl utr mtrx h g f c b h h g f c b 8 8 ctg otrvll ol o Exmpl

30 Sgl qubt gts & CNOT r uvrsl ~ ~ W wt to ppl cotroll gt wth th trgt bt

31 Sgl qubt gts & CNOT r uvrsl h g f c b h g f h c b h g f c h b h g f c b h g f c b h g f c b h ~ ~

32 o to uvrslt proof. A rbtrr cotroll- gt c b mplmt usg ol sgl qubt gts CNOT. A rbtrr cotroll - gt c b mplmt usg sgl qubt gts CNOT. Two-lvl utr gts r uvrsl 4. Sgl-qubt gts CNOT r uvrsl 5. Hmr, S, T, CNOT r uvrsl

33 Dscrt st of uvrsl gts Wh scrt st of gts? It c b us to prform qutum computto rror-rsstt fsho Problm Th st of utr oprtos s cotuous Strtg To show tht scrt st c b us to pproxmt utr oprto to rbtrr ccurc

34 Approxmto b H & T T HTH H 4 H x 4 HZH X H H x 4 4 ˆ s ˆ x x ; rrtol multpl of 8 8 s 8 8 s

35 Approxmto b H & T [ ] s 8 8 s 8 8 s 8 8 s 8 8 s 8 8 xp 8 xp 4 4 ˆ x x Z Y X I Z Y X I X I Z I X Z ZX Y 8 8 s 8 8 ˆ 8 8 s 8 s 8 4 x

36 Approxmto b H & T Wl s thorm o uform strbuto Lt p b rrtol, th th squc {p, p, p,...} s uforml strbut moulo 5 5 { mo } Th pproxmto to ccurc ε s rl through Oε tms trtos O ε ˆ ˆ

37 H S T CNOT r uvrsl 8 8 s 8 8 ˆ m Z Y X HZH HYH HXH x ˆ ˆ m H H ˆ ˆ ˆ m ˆ ˆ ˆ m S hs ts ow rol og th pproxmto fult-tolrt fsho Is ths costructo ffct?

38 Effcc Totl ccurc ε L m Omε gts Solov-Ktv thorm Om ε gts c O mlog m ε c

39 Dscrt st of uvrsl gts H, S, T, CNOT H, S, CNOT, Toffol Dutsch gt x ; rrtol

40 Qu Costruct Toffol usg ol H, T, CNOT T T H T T T T T H

41 Aswr H Z H H S S S H S S S Z 4 TXT 4 X XTXT T T H T T T T T T T H

42 Aswr T T H T T T T T H T T H T T T T T H T T H T T T T T H

43 Aswr T T T H T T T T H T T T H T T T T H T T H T T T T T H

1. Stefan-Boltzmann law states that the power emitted per unit area of the surface of a black

1. Stefan-Boltzmann law states that the power emitted per unit area of the surface of a black Stf-Boltzm lw stts tht th powr mttd pr ut r of th surfc of blck body s proportol to th fourth powr of th bsolut tmprtur: 4 S T whr T s th bsolut tmprtur d th Stf-Boltzm costt= 5 4 k B 3 5c h ( Clcult 5