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Theoretical Realization of Quantum      Gates Using Interacting Endohedral             Fullerene Molecules                ...
Introduction:a. Endohedral Fullerene Molecules (Buckyballs)     N@C60 Buckyball-Ideal Cage
Properties of the N@C60             •The encapsulated Nitrogen                                    atom can be considered a...
Decoherence times:•T1 due to the interactions betweena spin and the surrounding environment• T2 due to the dipolar interac...
Physical system
Physical system:           Two N@C60            Buckyballs       The mutual interaction between the two encased spins is d...
Qubit-encoding two-level system             If we apply a static magnetic field of amplitude B0                     dierec...
Single addressing of each qubit    Current density > 107A/cm2                                                             ...
Theoretical Model
Theoretical model borrowed from NMR quantum computation* ESR techniques allow to induce transitions between the spin state...
SINGLE-SPIN SYSTEM: single-qubit gatesThe state of a ½ spin particle in a static magnetic field B0 directed along the z ax...
When the applied MW-field is resonant with the spin precession frequency, i.e. wm=2w0 ,the Hamiltonian is time-independent...
Two-Spin SystemSingle-qubit gates: can be performed                       Two-qubit gates: naturally accomplishedthrough t...
QUANTUM GATES                       iπ                                                       e 4       0            0  ...
Refocusing: is a set of transformations which allow the removal of            the off-diagonal coupling terms of HDD      ...
p-gate                                                       Circuit representing Gp             π             π          ...
Dynamics of therealistic system
Realistic dynamicsreproduction of theoretical single-qubit and two-qubit quantum gates following the theory               ...
Distant buckyballs: we assume that the distance between the centres of the two                            buckyballs is r=...
B01= B02 =(0.3+3.04x10-5)T,                           ν 0 = 2ω0 / 2π = 8.40 ×109 Hzstatic magnetic field along            ...
•Hamiltonian of two distant buckys subjected to static fields along the z axis                                            ...
Total Hamiltonian (additional MW-field)                  H = H approx + H US (t )                     = −2 g (r )σ z1σ z2 ...
• single-qubit gates: MW-field and the spin to be rotated are in resonance, i.e.                     ωm = 2ω0 î first spin...
•Realization of a p-gate: we need to solve a Schrödinger equation for each of the                            following tra...
• Realization of a CNOT-gate: we need to solve a Schrödinger equation for each of the                              followi...
π            π           π       π                           tout = 3            +3         +         = 1.85µs            ...
Quantum gate fidelity The fidelity quantifies the distance between the realistic evolved state σ  = UσU † and the ideal ev...
Considerations on experimental limitations•Single-qubit rotations: a rotation of spin 1 can be accomplished by centering a...
∆Ω = 8 g (r ) = 1.9 MHz             and     ∆t = t SQ = 32ns  î the bandwidth theorem DWDt@2p is not satisfied Two options...
Conclusions:                 Condition         Dwp>>g(r)               (1)  allows to know exactly the frequency bandwidth...
From (1)îDwp>109HzîNew addressing scheme:We need to investigate alternative designs for addressing each single qubit,     ...
Scalability: Buckyballs can be easily maneuvered:•     buckyballs embedded in a silicon substrate•     Peapod: buckyballs ...
TWO-SPIN SYSTEMTWO-QUBIT GATES: naturally accomplished through the mutual spin dipolar interactionSINGLE-QUBIT GATES: can ...
Since H(t) is time-dependent î Unitary time-evolution                                                               t     ...
Since in the realistic case the dipolar interaction is always     present, we cannot reproduce single-qubit rotations    i...
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Phd thesis- Quantum Computation

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Published papers:
Buckyball quantum computer: realization of a quantum gate , M.S. Garelli and F.V. Kusmartsev, European Physical Journal B, Vol. 48, No. 2, p. 199, (2005)

Fast Quantum Computing with Buckyballs, M.S. Garelli and F.V. Kusmartsev, Proceedings of SPIE, Vol. 6264, 62640A (2006)


Theoretical Realization of Quantum Gates Using Interacting Endohedral Fullerene Molecules:
We have studied a physical system composed of two interacting endohedral fullerene molecules for quantum computational purposes. The mutual interaction between these two molecules is determined by their spin dipolar interaction. The action of static magnetic fields on the whole system allow to encode the qubit in the electron spin of the encased atom.

We herein present a theoretical model which enables us to realize single-qubit and two-qubit gates through the system under consideration. Single-qubit operations can be achieved by applying to the system resonant time-dependent microwave fields. Since the dipolar spin interaction couples the two qubit-encoding spins, two-qubit gates are naturally performed by allowing the system to evolve freely. This theoretical model is applied to two realistic architectures of two interacting endohedrals. In the first realistic system the two molecules are placed at a distance of $1.14 nm$. In the second design the two molecules are separated by a distance of $7 nm$. In the latter case the condition $\Delta\omega_p>>g(r)$ is satisfied, i.e. the difference between the precession frequencies of the two spins is much greater than the dipolar coupling strength. This allows us to adopt a simplified theoretical model for the realization of quantum gates.

The realization of quantum gates for these realistic systems is provided by studying the dynamics of the system. In this extent we have numerically solved a set of Schr{\"o}dinger equations needed for reproducing the respective gate, i.e. phase-gate, $\pi$-gate and CNOT-gate. For each quantum gate reproduced through the realistic system, we have estimated their reliability by calculating their related fidelity.

Finally, we present new ideas regarding architectures of systems composed of endohedral fullerenes, which could allow these systems to become reliable building blocks for the realization of quantum computers.

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Transcript of "Phd thesis- Quantum Computation"

  1. 1. Theoretical Realization of Quantum Gates Using Interacting Endohedral Fullerene Molecules Maria Silvia Garelli (M.S.Garelli@lboro.ac.uk)Department of Physics, Loughborough University, LE11 3TU, U.K.
  2. 2. Introduction:a. Endohedral Fullerene Molecules (Buckyballs) N@C60 Buckyball-Ideal Cage
  3. 3. Properties of the N@C60 •The encapsulated Nitrogen atom can be considered as an independent atom, with all the•Repulsive interaction properties of the free atom.Between the Fullerene •Since the charge is completelycage and the encapsulated screened, the Fullerene cageatom. No charge transfer. does not take part in the interaction process. It can just •The atomic electrons of be considered as a trap for the the encased atom are Nitrogen encased atom. tighter bound than in the free atom. The N atom is stabilized in its ground state.•Nitrogen central site The only Physical quantity of interest position inside the is the spin of the encapsulated atom. fullerene cage. We suppose that the N atom is a ½-spin particle
  4. 4. Decoherence times:•T1 due to the interactions betweena spin and the surrounding environment• T2 due to the dipolar interaction betweenthe qubit encoding spin and the surroundingendohedral spins randomly distributed in thesample• T1 and T2 are both temperature dependent• Their correlation T2 ≅ 2/3 T1 is (N@C60 in CS2)constant over a broad range of temperature• below 160 K, CS2 solvent freezes, leaving regionsof high fullerene concentrations⇒ dramatical increase of the local spin concentration T2=0.25ms⇒ T2 becomes extremely short due to dipolar spin coupling• temperature dependence due to Orbach processesJ. J. L. Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S. A. Lyon, G. A. Briggs, J. Chem. Phys. 124, 014508 (2006).
  5. 5. Physical system
  6. 6. Physical system: Two N@C60 Buckyballs The mutual interaction between the two encased spins is dominated by the dipole-dipole interaction , while the exchange interaction is negligible* r r r r r r ˆ ˆ ˆ ˆ H = g (r )[σ 1 ⊗ σ 2 − 3(σ 1 ⋅ n ) ⊗ (σ 2 ⋅ n )] µ0 µ B 2 where g (r ) = is the dipolar coupling constant 2πr 3 r Hamiltionian of the two-qubit systemBy choosing n ˆ ˆ ˆ ˆ ˆ ˆparellel to the x-axis H = g (r )(σ z1 ⊗ σ z 2 + σ y1 ⊗ σ y2 − 2σ x1 ⊗ σ x2 )*J. C. Greer,Chem. Phys, Lett. 326, 567 (2000); W. Harneit, Phys. Rev. A 65, 032322 (2002); M. Waiblinger, B. Goedde, K. Lips, W. Harneit, P. Jakes,A. Weidinger, K. P. Dinse, AIP Conf. Proc. 544, 195 (2000).
  7. 7. Qubit-encoding two-level system If we apply a static magnetic field of amplitude B0 dierected along the z axis we obtain a two-level system for each spin, due to the splitting of the spin-z componentHamiltonian of a two-qubit system subjected to the spin dipolar mutualinteraction and to the action of static magnetic field along the z direction ˆ ˆ ˆ ˆ ˆ ˆ H = g (r )(σ z1σ z 2 + σ y1σ y2 − 2σ x1σ x2 ) ω0 = µ B B0 1, 2 1, 2 ˆ ˆ −ω01σ z1 − ω02 σ z2whereω and ω are the precession frequencies of spin 1 and spin 2, respectively 01 02
  8. 8. Single addressing of each qubit Current density > 107A/cm2 d = 1µm ρ = 1µm I = 0.3 A With the use of atom chip technology*, two parallel wires carrying a current of the same intensity generate a magnetic field gradient. µ0  1 1  Bg =   x+ ρ +d /2 x−ρ −d /2 +  2π   ïthe two particles are characterized by different resonance frequencies*S. Groth, P. Kruger, S. Wildermuth, R. Folman, T. Fernholz, D. Mahalu, I. Bar-Joseph, J. Schmiedmayer, Appl. Phys. Lett. 85, 14 (2004)
  9. 9. Theoretical Model
  10. 10. Theoretical model borrowed from NMR quantum computation* ESR techniques allow to induce transitions between the spin states by applying microwave fields whose frequency is equal to the precession frequency of the spin. • Single-qubit gates on resonance spin-microwave field interaction • Two-qubit gates naturally existing spin dipolar interaction* M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University press, 2000) L. M. K. Vandersypen, I. L. Chuang, Rev. Mod. Phys. 74, 1037 (2005)
  11. 11. SINGLE-SPIN SYSTEM: single-qubit gatesThe state of a ½ spin particle in a static magnetic field B0 directed along the z axis can rbe manipulated by applying an on resonance MW field,Bm = Bm (cos ωmt + φ , sin ωmt + φ ,0)which rotates in the x-y plane at a frequency wm =2w0 characterized by a phase f andan amplitude Bm H m = − µ0 B0σ z − µ B Bm [cos(ωmt + φ )σ x − sin(ωmt + φ )σ y ]Total Hamiltonian 1 24 14444444 4444444 4 3 spin − static field 2 3 spin − MW fieldConsidering the Schrödinger equation and performing a change of coordinates to a frame rotrotating a frequency wr about the z axis defined by ψ = e −iωrσ z ψ , by choosing wr=w0we obtain the Control Hamiltonian H rot = −ωa [cos[(ωm − 2ω0 )t + φ ]σ x − sin[(ωm − 2ω0 )t + φ ]σ Y ] ωa = µ B Bm
  12. 12. When the applied MW-field is resonant with the spin precession frequency, i.e. wm=2w0 ,the Hamiltonian is time-independent, H = −ωa [cos(φ )σ x − sin(φ )σ Y ] , and its relatedtime evolution can be easily written as follows − iHt iω a t [cos(φ )σ x −sin(φ )σ y ] U (t ) = e =e r r r θσ ⋅n Rotation of an angle q about n axis Rn (θ ) = e r −i 2 •U(t) is a rotation in the x-y plane of an angle q proportional to wat, which is determined by phase f . •Bm (angle of rotation) and f (axis of rotation) can be varied with time. •w0 cannot be varied with time because depends on the amplitude B0 of the static magnetic field πExample: p/2 rotation about the y axis −i σ y 4 U =e it can be realized by choosing f= p/2 and allowing the time evolution for a time t=p/4wa= p /4mBBm
  13. 13. Two-Spin SystemSingle-qubit gates: can be performed Two-qubit gates: naturally accomplishedthrough the selective resonant interaction through the mutual spin dipolar interactionbetween the MW-field and the spinto be transformed Since the dipolar interaction couples the two spins, it naturally realizes two-qubit gates To realize single-qubit gates we need to assume that the spin-dipolar interaction is negligible in comparison with the spin-MW field interaction termASSUMPTION − iHt − i ( H DD + H US ) t − iH US t •HDD dipolar interaction term U (t ) = e =e ≈e •HUS is the interaction between two uncoupled spins and the MW-field The interaction terms between two uncoupled spins and a MW-field dominate the time evolutionï the spin dipolar interaction is negligible ï single-qubit rotation can be performed in good approximation
  14. 14. QUANTUM GATES  iπ  e 4 0 0 0 p/4-phase gate  −i π  realizes a p gate up to a p/2 rotation = 0 0  of both spins about the z axis and 4 U PG e 0  −i π 4  up to a global phase  0 0 e 0  π  i   0 0 0 e4 1 0 0 0 1 0 0 0     0 1 0 0 CNOT-gate 0 1 0 0 p-gate U CNOT = 0 0 1 Gπ =  0 0 1 0  0    0 0  0 0 − 1   0 1 0 
  15. 15. Refocusing: is a set of transformations which allow the removal of the off-diagonal coupling terms of HDD π π − i σ z2 i σ Circuit representing U(t) −iH DD t 2 −iH DD t 2 z2U (t ) = e e e e π π − i σ z2 i σ z2 2 2 = U b (t )e U a (t )e −i 4 g ( r )σ z 1σ z2 t =e π m i σ z2 2 • e is a ±p rotation about the z axis of the second spin • Ua(t) and Ub(t) represent the time evolution when the system is subjected to a static field and to the mutual dipolar interaction only ï they can be interpreted as two-qubit operationsby allowing evolution U(t) for a time t=p/16 g(r), a p/4-phase gate is realized
  16. 16. p-gate Circuit representing Gp π π −i σ z1 4 − i σ z2 4 πGπ = i e e U (t = ) 16 g (r )CNOT-gate Circuit representing CNOT π π π −i σ z1 i σ y2 − i σ y2 2 4 4CNOT = ie e Gπ e
  17. 17. Dynamics of therealistic system
  18. 18. Realistic dynamicsreproduction of theoretical single-qubit and two-qubit quantum gates following the theory previously presentedAssumption e − iHt ≅ e − iHUS t in a realistic system in general is NOT satisfied înumerical solution of the Schrödinger equation The reliability of the realistic system as a candidate for performing quantum gates will be checked from the comparison between the numerical results and the theoretically predicted outcomes and through the study of the fidelity of the quantum gate
  19. 19. Distant buckyballs: we assume that the distance between the centres of the two buckyballs is r=7nm This sut-up can be assembled by encasing buckyballs in a nanotube (peapod)•Buckyball diameter: d@0.7nm•distance between two buckyballsin a nanotube: dist@0.3nm(due to Van der-Waals forces) } We need to place 9 empty buckyballs between the two fullerenes in order to obtain r=7 nm 2 { µ0 µ B g (r ) = = 2.38 × 105 Hz dipolar coupling constant 2πr 3 r=7 nm î Bg1 = 1.87 ×10 −4 T gradient field amplitudes Bg 2 = −1.87 × 10 − 4 T
  20. 20. B01= B02 =(0.3+3.04x10-5)T, ν 0 = 2ω0 / 2π = 8.40 ×109 Hzstatic magnetic field along resonance 1 1 the z direction frequencies ν 0 = 2ω0 / 2π = 8.39 ×109 Hz 2 2 î ∆ω p = ω p1 − ω p2 = 2ω01 − 2ω0 2 = 6.28 ×107 Hz This condition allows us to omit the transverse couplingî Dwp>>g(r) terms in the dipolar Hamiltonianî The mutual dipolar interaction Hamiltonian can be simplified as H approx = g (r )(1 − 3 cos 2 θ )σ z1σ z2 q is the angle between the static magnetic field and the line joining the centres of the buckys H approx = −2 g (r )σ z1σ z2q=0 î
  21. 21. •Hamiltonian of two distant buckys subjected to static fields along the z axis H = H approx + H US = −2 g (r )σ z1σ z2 − ω01σ z1 − ω0 2 σ z2Energy-level diagram for two uncoupled spins (light lines)and for two spins describedby the Hamiltonian presented above (solid lines)
  22. 22. Total Hamiltonian (additional MW-field) H = H approx + H US (t ) = −2 g (r )σ z1σ z2 − ω01σ z1 − ω02 σ z2 − ωa1 [cos(ωm1 t + φ )σ x1 − sin(ωm1 t + φ )σ y1 ] − ωa2 [cos(ωm2 t + φ )σ x2 − sin(ωm2 t + φ )σ y2 ] rot − iω01σ z1t − iω02σ z2 tTotal Hamiltonian in the rotating frame ψ =e e ψ rot H = H approx + H US = −2 g (r )σ z1σ z2 − ωa1 [cos[(ωm1 − 2ω01 )t + φ ]σ x1 − sin[(ωm1 − 2ω01 )t + φ ]σ y1 ] − ωa2 [cos[(ωm2 − 2ω0 2 )t + φ ]σ x2 − sin[(ωm2 − 2ω02 )t + φ ]σ y2 ]
  23. 23. • single-qubit gates: MW-field and the spin to be rotated are in resonance, i.e. ωm = 2ω0 î first spin can be rotated 1 1 ωm = 2ω0 2 2 î second spin can be rotated Typical experimental time exp θ î Bm@1.7mT of a single-qubit rotation* t SQ = ≅ 32ns gµ B Bm • two-qubit gates: naturally realized by the mutual spin dipolar interaction Happrox time-evolution operator if we allow this time-evolution for 2 ig ( r )σ z1σ z2 t U (t ) = e î a time t=p/8g(r)=1.65ms we obtain related to Happrox a controlled p/4 phase gate Happrox is already diagonal î the refocusing procedure is not needed*J.J.L.Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S.A. Lyon, G.A. Briggs,Phys. Rev. A.71, 012332 (2005).
  24. 24. •Realization of a p-gate: we need to solve a Schrödinger equation for each of the following transformations, which define a p-gate 1 0 0 0 π π   −i σ z1 −i σ z2 0 1 0 0 Gπ = i e 4 e 4 U (t = π / 8 g (r )) =  0 0 1 0 •Numerical output matrix   0 0 0 − 1   Up2=Comments :the dipolar interaction influences the perfect reproduction of single-quibit rotationsand subsequently of a p-gate but the time required for performing a single qubit rotationis tSQ=32 ns. The time during which the system is influenced by the spin dipolar interactionis T=2p/g(r)=2.6x10-5s îtSQ<<T during the completion of a single-qubit rotationwe can consider the system as being unaffected by the mutual spin dipolar interactionîwhen performing Single-Qubit rotations, the spin-Mw field term dominates
  25. 25. • Realization of a CNOT-gate: we need to solve a Schrödinger equation for each of the following transformations, which define a CNOT-gate 1 0 0 0 π π π   −i σ z1 i σ y2 2 4 − i σ y2 4 0 1 0 0 CNOT = ie e Gπ e = 0 0 0 1   0 0 1 0 •Numerical output matrix   UCNreal=
  26. 26. π π π π tout = 3 +3 + = 1.85µs 4 µ B Bm 1 4 µ B Bm 8 g (r ) 2 •Operational times: π π π π CNOT tout = 5 +5 + + = 2.05µs 4 µ B Bm1 4 µ B Bm 2 µ B Bm 8 g (r ) 2 1 p/8g(r) determines the order of magnitude of tout•Number of quantum operations T2 T2 n<104 î small number n = π ≅ CNOT ≅ 10 2 allowed before relaxation: tout tout of operationîthe system is not reliable Possibility of increasing T2 two order of magnitude: Proposal: investigation of experiments for the study of relaxation processes of Buckyballs in a nanotube îreduction of dipolar interactions between the encased spin and the randomly distributed spins in the sample The nanotube represents a further shield for the encased spin against the outer environment
  27. 27. Quantum gate fidelity The fidelity quantifies the distance between the realistic evolved state σ = UσU † and the ideal evolved state ψ ideal F(ψ ideal ,σ ) = ideal ψ σψ ideal = ideal ψ U ψ ψ U†ψ ideal Since the starting state is not known in advance, we can consider the minimum fidelity, which minimizes over all possible starting states î F = min F ( ψ ideal ,σ ) c αp-gate: F=0.998 F differs from its ideal value F=1 by of the order of 0.2%(0.8%)CNOT-gate: F=0.991 ïThe realistic transformations are in HIGH ACCORDANCE with the theoretical predictions and the system is highly reliable for reproducing a p-gate through the study of its dynamics
  28. 28. Considerations on experimental limitations•Single-qubit rotations: a rotation of spin 1 can be accomplished by centering a selective MW-pulse at the precession frequency of spin 1, i.e. wm1=2w01, and characterized by a frequency bandwidth which has to cover the range of frequencies 2w01 ±4 g(r) but not overlap the range 2w02 ±4 g(r), which corresponds to the range of frequencies for the excitation of spin 2 Frequency bandwidth difference between the upper and lower values of the range which allow the swap of the selected spin ∆Ω = 2ω01 + 4 g (r ) − (2ω01 − 4 g (r )) = 8 g (r ) î the frequency bandwidth DW depends only on the dipolar coupling constant g(r)
  29. 29. ∆Ω = 8 g (r ) = 1.9 MHz and ∆t = t SQ = 32ns î the bandwidth theorem DWDt@2p is not satisfied Two options: •If tSQ=32ns î DW=1.95x108 Hz The first is preferable because it allows single-qubit rotations in •If DW=1.9 MHz î tSQ=3.3 ms a shorter time The frequency bandwidth depends on g(r). Since tSQ is given, the bandwidth theorem allows us to put a constraint on g(r) and consequently on r, the distance between the two encased particles
  30. 30. Conclusions: Condition Dwp>>g(r) (1) allows to know exactly the frequency bandwidth, i.e. ∆Ω = 8 g (r ) Since Dtª32ns, from the bandwidth theorem DWDtª1, we obtain 8 ∆Ω = 8 g (r ) = 1.96 ×10 Hzwhich implies g(r)=2.45x107Hz and rª1.5nm. This value of r can beobtained by attaching functional groups between the two buckys.In this case The system would be a good candidate as a building block for quantum π T π /4tout ≈ ≅ 1.6 ×10 s ⇒ n = π 24 ≥ 10 4 −8 / computers and would allow the 8 g (r ) tout possibility of applying quantum error correcting codes
  31. 31. From (1)îDwp>109HzîNew addressing scheme:We need to investigate alternative designs for addressing each single qubit, which can allow the achievement of the desirable value of Dwp• Quantum Cellular Automaton with different species of encased particles the two particles have to be characterized by a very different value of the gyromagnetic ratio g •New design for the magnetic field gradient more steep magnetic field gradientFinally: Is it exprimentally possible to realize single-qubit rotations in a time shorter than t=32 ns? T2 n= ≅ 10 4 If so î π( toutCNOT )
  32. 32. Scalability: Buckyballs can be easily maneuvered:• buckyballs embedded in a silicon substrate• Peapod: buckyballs in a nanotube proposal: improved T2 in a peapodReadout: difficulty in the readout of single electron spins. TNT(erbium-doped) fullerene promising candidates for the readoutPromising results of recent experiments:•direct excitation of IONC STATES in TNT’sïopens the opportunity of identifyinguseful readout transitions and coherently and selectively excite these transitions•Application of suitable magnetic fields on TNT samplesïthe observed spectrum splitconfirms that Er3+ ions are Kramer ions. They maintain the two-fold degeneracy in theirquantum states even under complete crystal-field splittingï ENCODING of a QUBITin this pseudo-1/2 spin and EXCITING selective luminecsent transitionsï COULDALLOW THE DETECTION OF INDIVIDUAL SPIN STATES
  33. 33. TWO-SPIN SYSTEMTWO-QUBIT GATES: naturally accomplished through the mutual spin dipolar interactionSINGLE-QUBIT GATES: can be performed through the selective resonant interaction between the MW-field and the spin to be transformed Total Hamiltonian of the two-spin system in the rotating frame H (t ) = H DD + H US = g (r )[cos(2ω01 − 2ω02 )t (σ x1σ x2 + σ y1σ y2 ) − 2σ z1σ z2 ] − ωa1 [cos[(ωm1 − 2ω01 )t + φ ]σ x1 − sin[(ωm1 − 2ω01 )t + φ ]σ y1 ] − ωa2 [cos[(ωm2 − 2ω0 2 )t + φ ]σ x2 − sin[(ωm2 − 2ω02 )t + φ ]σ y2 ] where HDD is the dipolar interaction term and HUS is the interaction between two uncoupled spins and the MW-field
  34. 34. Since H(t) is time-dependent î Unitary time-evolution t U (t , t0 ) = T exp[−i ∫ H (t )dt ] t0 T is the time-ordering operator In order to easily perform unitary transformations, the Hamiltonian has to be time-independent, such that the unitary evolution can be written as U(t)=exp[-iHt]. To cancel the time-dependence in H(t) we chose: • ω0 = ω0 1 2 the precession frequencies of the two spins are equal • ωm1, 2 = 2ω01, 2 resonant MW-fieldASSUMPTION U (t ) = e − iHt = e − i ( H DD + HUS ) t ≈ e − iHUS t The interaction terms between two uncoupled spins and a MW-field dominate the time evolutionï the spin dipolar interaction is negligible ï single-qubit rotation can be performed in good approximation
  35. 35. Since in the realistic case the dipolar interaction is always present, we cannot reproduce single-qubit rotations in perfect agreement with the theoretical predictions. However, the dipolar interaction is essential for performing two-qubit transformations flTwo-qubit gates:can be realized by allowing the system to evolve freely under the action of the mutual spin dipolar interaction.Since the dipolar interaction couples the two spins, it naturally realizes two-qubit gates
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