Quantum manipulation and qubits

Transcription

1 Quantum manipulation and qubits Qubits Quantum information Quantum tlportation Rsonant manipulation Diabatic and adiabatic manipulation Quantum gats Quantiation of a osphson junction Phas qubit Coulomb blockad Spin qubit Qubits Qubit: a two-lvl quantum systm s = α + β, α + β = Lngth fid; global phas not important: A qubit is paramtrid by two angls Visualiation: Bloch sphr α + β Analogy: spin-/ particl Evry singl-qubit transformation is a rotation on th Block sphr γ Rn ( γ ) = p i niσ φ θ y σ Pauli matrics

2 Quantum computation Why can qubits b usful for quantum computation? Boolan logic: bit = stats ( and ) Quantum logic: qubit = dimnsions = infinitly many stats Som algorithms can b mor fficintly ralid with quantum logic: Quantum Fourir transform -> factoring Quantum sarch DiVincno critria for a physical raliation of quantum computr: Physical raliation of qubits Opration of qubits: Quantum information is procssd Rad-out: quantum information can b rad Wak dcohrnc: no rrors during computation Scalability: many qubits can b put togthr Quantum information Quantum information: two subjcts (Alic and Bob) chang quantum information by classical mans Classical information: It is saf to assum that onc stord it rmains thr forvr; can b clond Quantum information: no-cloning thorm Th wav function of th clond stat would b ( )( ) s s = α + β α + β Impossibl du to th fact that Schrödingr quation is linar If A has a qubit and wants to transfr it to B it is impossibl by classical mans

3 Quantum tlportation If A has a qubit and wants to transfr it to B it is impossibl by classical mans Instad, thy prpar in advanc a pair of qubits in th Bll stat B = + ( ) Thn A taks th first qubit and B taks th scond qubit Tlportation protocol: A has th stat α + β Th total stat in th Bll basis of two A s qubits: B = ( ) ( + ) B α + β = ( α + β ) B + ( β + α ) B B = ( ) + ( α β ) B + ( β + α ) B B = ( ) A masurs hr qubit, communicats th rsult to B, B prforms th singl-qubit opraration dpnding on th masurmnt Eampl: if has bn masurd, D dos not hav to do anything Quantum manipulation For N qubits: ˆ i i ij i j H = hσ + U σσ a a ab a b i i< j ab, = y,, Etrnal filds Couplings H = haσ a = h + ihy h Singl qubit: ˆ y h h ih Rotation by puls: switch on h for finit tim τ ˆ ih ˆ στ/ h τ hτ h (): cos ˆ t R = = + iσ sin ψτ ( ) = U ( α) ψ() α = h τ Difficult to rali primntally 3

4 Rsonant manipulation ˆ Ω Ω H = ha() t σ a diag, i t ( h a t ω ) h () t = R (), ω Ω a Schrödingr quation ψ / h() t h() t ihy() t + Ω + ψ + i = t ψ h() t ihy() t / h() t + Ω ψ iωt/ iωt/ W substitut ψ ψ, ψ ψ Thn + + ( ) iωt ψ δω / h() t h() t ihy() t + + ψ + i = t ψ iωt ( h() t + ihy() t ) δω / h() t ψ Rsonant manipulation ( ) iωt ψ δω / h() t h() t ihy() t + + ψ+ i = δω ω t ψ iωt Ω ( h() t + ihy() t ) δω / h() t ψ Rotating wav approimation: W avrag th Hamiltonian with rspct to th rapidly changing filds i t W disrgard ± ω i t and kp ± δω Hˆ RWA A Eignvalus: δω / h () t ih y() t = h () t + ih y() t δω / ± ωr / = δω /4+ h + h Rabi frquncy y h h y Applicability: δω, h, h ω, Ω Now can procd with pulss of, y 4

5 Rsonant manipulation Rabi oscillations: act rsonanc (no dtuning, δω=), initially th qubit is in th stat - ψ + () t i sin ωrt = ψ () t cos π i R R = i Ramsy squnc W apply R, wait for th tim τ, and thn apply R again. Naivly: th two R s cancl ach othr. This is only corrct if thr is no dtuning: δω= Final stat: iδωτ / iδωτ / ( + i + ) / turns - into ( + i + ) / Ramsy frings Diabatic and adiabatic manipulation Hˆ ε T = T ε * Diabatic manipulation: w suddnly chang ε from a larg positiv to a larg ngativ valu Sam rsult as π-puls; th two stats ar not th full spctrum. Adiabatic manipulation: th sam chang slowly ψ () t is (almost) th ignfunction of th instantanous Hamiltonian Finit spd: ε + diabatic, P= Landau-Znr: P + π T = p ε - adiabatic, P= 5

6 Two-qubit manipulation A quantum gat ralis a unitary opration with N qubits Numbr of bits in quals numbr of bits out Th opration is rvrsibl (in contrast to usual computation) Eampl of a two-bit (quantum) gat: CNOT Ngation of th scond (q)bit providd th first is in th stat A thorm: CNOT is nough for arbitrary q. algorithm U CNOT in th basis = },,, { Quantum gats How to rali two-qubit gats using unitary transformations? Mak pulss of th qb-qb coupling Eampl: H σ σ σ σ () () () () int = ( + y y ) in th basis {,,, } Th rsulting two-bit opration is controlld by puls duration R cosγ isinγ ( γ ) =, γ = τ / i sinγ cosγ 6

7 Quantum gats Usually on has to combin (svral) two-bit oprations and (svral) singl-bit rotations Last opration U = R ( π / ) R ( π / ) R ( π) CNOT R ( π /) R ( π /) R ( π /) R ( π /) R ( π /) First opration Charging nrgy Connctd to bulk suprconductor mtal osphson link provids charg transfr suprconducting island gat V g Shifts potntial = inducs charg CgVg Enrgy = osphson nrgy Coulomb nrgy E = E cosϕ CV g + EC N g EC = C 7

8 Quantiation of a osphson junction Rplac classical variabls by oprators Postulat commutation rlations Chck if this rproducs classical quations of motion ϕ ϕ cosϕ E = E ˆ + EC N = H [N ˆ, ˆ ϕ] = i Rstors classical quations of motion dq = ICϕ dt dϕ V = dt Q= CV Q= N; IC = E ; EC = C dn E = ϕ dt dϕ 4E = C N dt Phas qubit Phas (flu) rprsntation: ˆ Q ( ϕ) ( C ( g ) cos ϕ) EΨ = HΨ = E i E Ψ ϕ Currnt-biasd osphson junction E / E >> C 4 I b Z(w) CE, Phas φ is a coordinat osphson potntial E( φ) = E cosφ I φ b nvironmnt, ar usd as qubit stats 8

9 Phas qubit ω = E E Manipulation: by rsonant microwavs with No avrag voltag ω = E E Rad-out: by a rsonant microwav puls with Singl shot masur of P (t) Gats: Capacitiv coupling Coulomb blockad Isolatd island: charg is quantid, Q=n charging nrgy = addition nrgy E = n = E n C C If charging nrgy is not availabl from trnal sourcs, transport is blockd: Coulomb blockad Quantum dot: a dvic whr lctrons ar confind in a visibly discrt lvls 9

10 How many lctrons? Shifts potntial = inducs charg CgVg E CV g g E = EC ( N ) N 4 - CV g g N = CgVg/ CgVg/ Doubl quantum dot (a) C (s) C N N C (s) (b).5 C (g) V (g) C (g) V (g) q / -.5 (,) (c) (,-) (,) (,) (d) (,-) (,) q / (,-) (,) (,) (,-) (,) (,) q / (-,-)(-,) (-,) q / (-,) (-,) q / q /

11 Spin qubit QPC Ptta t al 5 QPC (,) charg configuration: Only singlt stat S (,) charg configuration: Four-fold dgnracy S, T, T, T Split away by magntic fild ˆ H = ε S S + ε S S + ε T T S S + S S ( ) Spin qubit ˆ H = ε S S + ε S S + ε T T S S + S S ( ) S T (,) (,) ε S g Ground stat singlt and triplt coupld by hyprfin intractions ˆ / Ω H = Ω / Manipulation: S and T and th (,) sid Radout: convrting spin into charg by moving quickly to th (,) sid