Quantum Computation and the Bloch Sphere

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Alyson Bryant
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1 Quanum Cmpuan and Blc Spr Frd Wllsd Jn Quanum Insu and Cnr fr Suprcnducvy Rsarc Dparmn f Pyscs Unvrsy f Maryland, Cllg Park, MD Marc 4, 8

2 Quanum Mcancs and Quanum Cmpung In prncpl, a cmpur can b bul a uss quanum mcancs prfrm usful calculans. A quanum cmpur wuld b bul frm quanum bs r "qubs", ndvdual quanum sysm w w bass sas, > and > T qubs ar cupld gr and lgc prans ar prfrmd by manpulang quanum sa f nr sysm. xampl: NOT n sngl qub: > > > > α> β> α> β> xampl: Pas ga n sngl qub: > > > φ > α> β> α> φ β> prans nd wrk n suprpsn sas!

quanum sas f a sysm, n sysm can xs n any lnar suprpsn f s sas wr α and β

3 Bu w dn s suc sas n vryday bjcs "Scrdngr's ca paradx" Scrdngr, 935 Quanum Mcancs and Quanum Cmpung T Prncpl f Suprpsn Supps > and > ar w allwd quanum sas f a sysm, n sysm can xs n any lnar suprpsn f s sas wr α and β ar cmplx numbrs ψ α > β > "macrscpc quanum suprpsn" f ru n macrscpc bjcs lv dad

4 Suprpsn Sa ψ α > β > - prbably ampluds α and β can b cmplx numbrs - sa mus b nrmalzd uny α β - an vrall pas facr as n ffc, s w can cs α b ral φ - n dfn α csθ β snθ α β cs θ φ sn θ cs θ sn θ - s w can always wr a suprpsn sa n frm: φ ψ α > β > cs θ > sn θ >

5 Suprpsn Sas as Pns n Blc Spr z > θ φ θ Ψ cs > sn > θ y x φ > spr w radus R..s s Blc Spr

6 > z y x > > > > > Ψ sn cs sn cs φ φ θ θ θ xampl: θ Suprpsn Sas as Pns n Un Spr

7 Suprpsn Sas as Pns n Un Spr z > xampl: θ π, φ Ψ θ cs > π cs > > φ θ sn > sn > π θ π y x >

8 Suprpsn Sas as Pns n Un Spr z > xampl: θ π/, φ θ Ψ cs > π cs > 4 > > φ θ sn > π sn > 4 > > θ π / y x >

9 Suprpsn Sas as Pns n Un Spr z > xampl: θ π/, φ π/ θ Ψ cs > π cs > 4 > > φ π θ sn > π sn > 4 > > θ π / φ π / > > y x >

10 T b usful fr cmpuan, yu nd prans a cnrl sa f n qub basd n sa f anr. Cnrlld NOT r CNOT: Tw-qub pran a flps scnd qub sa basd n frs qub sa npu sa upu sa,>,>,>,>,>,>,>,> xampl, prfrmng a CNOT pran n α,> β,> γ,> ylds: α,> β,> γ,>

11 Suprpsn and nanglmn ar unbsrvabl n rdnary "macrscpc" bjcs du nracns w r dgrs f frdm and surrundng wrld dsspan and dcrnc w macrscpc s macrscpc? Quanum Mcancs and Quanum Cmpung Quanum nanglmn Scrdngr, 935 Mulpl quanum sysms can xs n nangld supr-psns f sas n wc sa f an ndvdual sysm as n wlldfnd pyscal manng ψ α > β a > b > a > b f ru n macrscpc bjcs dad, lv and lv, dad

12 A classcal cmpur w an n-b mmry can accss sas. Fr xampl, w n bs 4 sas ar,, and. A quanum cmpur can accss suprpsn sas and nangld sas. W n qubs, s gvs f rdr n n sas. xampl: fr n qub w can av: ψ ψ ψ x ψ x ψ y ψ y xampl: fr n qubs w can av 36 prduc sas suc as: ψ ψ ψ ψ xy ψ ψ ψ x plus nangld sas can b wrn as prduc suc as: ψ ψ 3 ψ y ψ 4

13 T xra sas can b usd ackl sm vry dffcul asks: - us Sr's algrm facr larg numbrs quckly and brak RSA ncrypd mssags, - smulang r quanum sysms, - ffcnly sarcng larg daa-bass Grvr s Algrm? Ky Qusn: can a usful quanum cmpur b bul n pracc? Answr: Dfnly mayb. Man xprmnal Callng: All sysms xprnc ns and nrac w r quanum sysms usd wrld, and s vnually dsrys dlca quanum suprpsn sas. Ts s calld dcrnc. Dcrnc s bs undrsd usng dnsy marx. Hr w wll jus ry undrsand w yu can manpula quanum sa f a mul-qub sysm prfrm prans.

14 > z y x > > > > > Ψ sn cs sn cs φ φ θ θ θ xampl: θ Sngl qub cnrl prans as rans n Blc spr

15 Sngl qub cnrl prans as rans n Blc spr z > xampl: θ π, φ θ Ψ cs > π cs > > φ θ sn > sn > π θ π y x > sarng frm > ra abu y-axs by π.. π y -puls.r NOT snc suc a ran wuld als cang > >

16 Sngl qub cnrl prans as rans n Blc spr z > xampl: θ π/, φ θ Ψ cs > π cs > 4 > > φ θ sn > π sn > 4 x > > θ π / > y sarng frm > ra abu y-axs by π/ π/-puls.r... NOT snc w suc rans wuld prduc NOT

17 Sngl qub cnrl prans as rans n Blc spr z > xampl: θ π/, φ π/ θ Ψ cs > π cs > 4 > > φ π θ sn > π sn > 4 > > θ π / φ π / > > y x > sarng frm > > ra abu z-axs by π/. Ts s π Z / r π/ pas ga snc wll ncras pas rm fr any sa by π/

18 Basc Ida fr Drvng a Sysm - Rab Oscllans - Cnsdr a -lvl sysm w nrgy splng Δ. - Cl sysm mpraur T << Δ/k b and wll rlax >. - Apply pwr a prurban cnnuusly a frquncy f Δ/. Δf Δf Δf Apply pwr fr sr m --> Small amplud b n Kp applyng pwr --> vnually sysm pumpd nrly n NOT ga r π-puls Kp applyng pwr --> sysm pumpd back dwn Sysm cycls back and fr bwn and drmnscally a wll-dfnd ra Rab frquncy s by pwr and unng. Sppng pwr a apprpra m can prduc NOT r NOT

19 Tw-lvl Sysm Dynamcs Sa f a sysm dscrbd by wavfuncn Ψ a sasfs m-dpndn Scrdngr s quan ψ Hψ Fr a w-lvl sysm w Hamlnan H a s bng drvn a frquncy w a prurbng nrgy H, w can wr H n marx frm as: H H H ' V cs V cs V *cs V cs wr: nrgy f grund sa, nrgy f xcd sa V csw <H > and wr: α > > Ψ > β

20 Tw-lvl Sysm Dynamcs Plug n Scrdngr s quan: ψ Hψ β α β α cs cs V V Wr as w cupld quans: β α β β α α cs cs V V Farly nasy guss slun f frm: s wll always wrk! B A β α Plug n Scrdngr s quan nc a s says a amplud β b fund n > wll cang basd n amplud α b n >

21 B A V B B V A A cs cs B V A A A cs Fr frs quan, w fnd: Clan ngs up: B V A cs Fr smplcy, l s assum w ar n rsnanc VB A cs xpand s rm

22 B V VB VB A cs Ts rm s cangng vry rapdly and s far frm rsnanc a s can b drppd. rang wav apprxman A V B B V A A V A Assumng A, slun s: cs A V s Rab frquncy ak anr m drvav f s quan and us nd lmna db/d sn B

23 > > Ψ > sn cs sn cs B A β α Tak u an vrall pas facr f xp > > Ψ > sn cs prbably P α P β / π 4π /

24 > > > Ψ sn cs θ θ φ Als nc s s nw n famlar plar crdna frm: wr and θ π φ > > Ψ > sn cs sn cs B A β α Tak u an vrall pas facr f xp > > Ψ > sn cs

25 Rab Oscllan n Blc Spr z > dφ/d θ > > y x > > > T mak NOT ga, sp drvng a π/ Prblm: Sw a s wll NOT any sa!

26 T bavr f a sa n Blc spr s cmplly analgus a magnc mmn prcssng n a magnc fld rnd alng z-axs. Rab Oscllans ar cmplly analgus nuclar magnc rsnanc NMR. In NMR, a sac magnc fld B z s appld and n rsnan rf magnc flds ar appld a frquncy f drv a nuclar spns a rsnanc γb z

The Variance-Covariance Matrix

The Variance-Covariance Matrix Th Varanc-Covaranc Marx Our bggs a so-ar has bn ng a lnar uncon o a s o daa by mnmzng h las squars drncs rom h o h daa wh mnsarch. Whn analyzng non-lnar daa you hav o us a program l Malab as many yps o