If 1 na detects 0i then it leaves b alone 14 ccnot f

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Unformatted text preview: ter |a, bi: Aa,b = a⇤ a ⌦ (b + b⇤ ) + aa⇤ ⌦ 1. You can write this formula Na ⌦ Ab + (1 Na ) ⌦ 1. That is, if Na detects |1i, then it negates b. If 1 Na detects |0i, then it leaves b alone. ¶14. CCNOT: F writes Aab,c for the CCNOT operation applied to lines a, b, and c. Aab,c = 1 + a⇤ ab⇤ b(c + c⇤ 1) (exercise). This formula is more comprehensible in this form: Aab,c = 1 + Na Nb (Ac 1) . ¶15. SWITCH: One of Feynman’s universal computers is based on only two logic gates, Not and Switch (Fig. III.40). If c = |1i, then the “cursor” (locus of control) at p moves to q , but if 182 CHAPTER III. QUANTUM COMPUTATION Feynman 524 O a sMiNoT01tM s =b+ I sN t tN I Fig. 8. C ONTROLLED N OT b y switches, Figure III.41: CNOT implemented by switches. 0/1 annotations on the wires show the a values. [fig. from F85] I n t hese d iagrams, h orizontal o r v ertical l ines will r epresent p rogram a toms. T he s witches a re r epresented b y d iagonal l ines a nd i n b oxes w e'll p ut t he o ther m atrices t hat operate on r egisters s uch a s t he N O T b. T o b e cspecific, it moves to r. f or t his l ittle s ection o f a C O N T R O L L E D N OT, = |0i t he H amiltonian t hinkingnegates s tarting atprocess. It also of i t as c in the s a nd e nding a t t, is g iven b elow: H e(s, t = s ' a Fig. III.40). ¶16. It’s also reversible) (see s + t *a*tM + t *(b + b *) sM + s~va*s nt- l ' a t u q- t ~rs u + C.C ¶17. The switch is a tensor product on |c, p, q, ri: ¶18. ¶19. ¶20. ( The c.c m eans t o a dd t he c omplex conjugate o f all t he p revious t erms.) A lthough t here s eemcpo+ r ⇤ cwo +outes ⇤ q ere pwcr ]. w ould p ossibly q ⇤ t be t ⇤ p r [p⇤ c h + ⇤ hich p roduce a ll k inds o f c omplications c haracteristic o f q uantum m echanics, t his i n ot so. I f t e ntire c omputer s ystem s tarted i n a d efinite s of or (Thes bracketedheexpression is just theis complex conjugatetate fthe first a b y t he t ime t he for reversibility.) Reada the till i n s ome d efinite term from part, required c ursor r eaches s, t he a tom is s factors in each s tate ( although p ossibly d ifferent f rom i ts i nitial s tate d ue t o p revious c omputer right to left: it). T hus o nly o ne o f t he t wo r outes is t aken. T he e xpression o perations on (1) q ⇤ cps:implified b yc oare set,t he S * I unset them utting set S . N . if p and mitting then N t erm and p and t u = q m ay b e ⇤⇤ (2) rOc pn eed p ot b e c oncerned inot set, thehat o ne r oute is set c and r. ne : if n is set and c is n t hat c ase, t unset p and l onger ( two c ursor s ites) t han t he o ther ( one c ursor s ite) f or again t here is n o i nterCNOT: o s cattering ishows CNOTny c ase b y t he i nsertion i nto a cThis is the f erence. N Fig. III.41 s p roduced i n a implemented by switches. hain o f c oupled sites, applied to o f c a, b o f a sequenced f s cursor t he controlled-NOTa n e xtra p iecedata hain andny n umber obyites w ith atoms s, t s ame m utual c oupling b etween s ites ( analogous t o m atching i mpedances i n (= start...
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