Qubit and Quantum Gates

Transcription

1 Quit nd Quntum Gtes Shool on Quntum Dy, Lesson 9:-:, Mrh, 5 Eisuke Ae Deprtment of Applied Physis nd Physio-Informtis, nd REST-JST, Keio University

2 From lssil to quntum Informtion is physil - Rolf Lnduer QUANTUM informtion or quntum INFORMATION? It depends on your kground physis or informtion siene Ultimtely, you need oth At the eginning, it would e etter to keep one perspetive physis here

3 Referenes Quntum omputtion nd Quntum Informtion nd referenes therein, M. A. Nielsen nd I. L. hung, mridge University Press Dy, Lesson - Dy, Lesson Physil Review A 65, 3, N. D. Mermin Dy, Lesson L. M. K. Vndersypen, Ph. D Thesis ville t riv: qunt-ph/593 Dy, Lesson

4 Outline Rules of the gme Quntum it Stte spe Quntum gte Unitry evolution NOT, Y,, Hdmrd H Mesurement Multiple-quit Tensor produt NOT, SWAP, ontrolled-, Toffoli

5 Quntum it For physiists, quntum it quit is synonym for quntum mehnil two-level system g e Superposition

6 Quntum it Vetor nottion for omputtionl sis sttes POSTULATE ψ α β α α β α, β : Proility mplitude Stte spe Hilert spe β : Proilities sum to

7 Unitry evolution POSTULATE The evolution of quit system is desried y unitry trnsformtion suh s ψ t U ψ t Hermitin onjugte: A A T * Hermitin self-djoint: A A Unitry: UU I d * * d * *

8 Unitry evolution ih d ψ onnetion with the Shrödinger eqution dt H ψ H: Hmiltonin of the quit system Hermitin ih t t ψ t exp ψ t U ψ t h Exponentil opertor unitry Any unitry opertor U n e relized in the form U expih where H is some Hermitin opertor For now, tul physil systems tht relize neessry Hmiltonins re NOT our interest

9 Quntum gte ψ t U ψ t Input U Output ψ t ψ t Suessive implementtion Time L ψ ψ U U U 3 U n U ψ U U ψ Time U LU U n ψ

10 Quntum gte Input ψ α β Output U U ψ U α β We hve infinite inputs, ut it suffies to onsider only the omputtionl sis sttes U U Superposition priniple U α U β U U ψ U U

11 NOT gte lssil NOT Input output Quntum NOT The only non-trivil one-it gte in the lssil se or Mtrix representtion

12 Mtrix representtion The first olumn represents the finl stte of The seond olumn represents the finl stte of

13 Puli-, Y, gtes i i Y i Y i Y Y i iy i Y i Y Y Y ], [ ], [ ], [ I Y }, { }, { }, { Y Y Hermitin ommuttion reltions

14 Hdmrd gte, H,,, H H H H H HH Y HYH HH H I Hermitin iruit identities

15 Mesurement gte POSTULATE ψ α β Quntum it lssil it Mthemtil desription Generl mesurement Projetive mesurement POVM with proility α, or with proility β

16 Multiple-quit How do we desrie multiple-quit sttes? omputtionl sis sttes for two-quit sttes my e written s,,, We require them to e orthogonl, so they my e written s Speultion A multiple-quit stte is the tensor produt of the omponent quit systems POSTULATE

17 Tensor produt omputtionl sis set for -quit sttes Mtrix representtion

18 Multiple-quit gtes 3 U U 3 L n M M n n : n -dimensionl vetor U : n y n unitry mtrix

19 Independent gtes H H Y H H Y YH H H YH H YH U 8 y 8 unitry mtrix B B B B B A

20 ontrolled-u gtes ontrol it if Trget it U U U if U n e n ritrry single-quit gte U works only when U is just forml expression

21 NOT gte if if,,, Frequently used

22 NOT gte,,, Never mistke

23 SWAP gte SWAP 3 4 To implement SWAP, we need to. Enode informtion on into nd quit. Erse informtion on from st quit 3. Enode informtion on into st quit 4. Erse informtion on from nd quit

24 ontrolled- gte ontrolled- is nonlol

25 Toffoli Toffoli Toffoli is often referred to s ontrolled-ontrolled-not -NOT

26 Quiz Prove the following iruit identity Also prove the followings H I HH Use the following expressions for quntum gtes H,

27 Answer 3 3 sde implementtion sde ersure

28 Answer H HH onstrutive nd destrutive interferenes

29 Answer H H HH onstrutive nd destrutive interferenes