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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.
Introduction:
a. Endohedral Fullerene Molecules (Buckyballs)




     N@C60 Buckyball-Ideal Cage
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 completely
cage and the encapsulated           screened, the Fullerene cage
atom. 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
Decoherence times:
•T1 due to the interactions between
a spin and the surrounding environment

• T2 due to the dipolar interaction between
the qubit encoding spin and the surrounding
endohedral spins randomly distributed in the
sample
• 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 regions
of 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 processes

J. J. L. Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S. A. Lyon, G. A. Briggs, J. Chem. Phys. 124, 014508 (2006).
Physical system
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 system
By choosing                n                                  ˆ      ˆ       ˆ      ˆ       ˆ      ˆ
parellel 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).
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 component

Hamiltonian of a two-qubit system subjected to the spin dipolar mutual
interaction 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 z2

whereω and ω are the precession frequencies of spin 1 and spin 2, respectively
        01       02
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)
Theoretical Model
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)
SINGLE-SPIN SYSTEM: single-qubit gates
The state of a ½ spin particle in a static magnetic field B0 directed along the z axis can
                                                           r
be 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 and
an 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 field


Considering the Schrödinger equation and performing a change of coordinates to a frame
                                                      rot
rotating a frequency wr about the z axis defined by ψ     = e −iωrσ z ψ , by choosing wr=w0
we 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
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 related
time 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
                                                       π
Example: 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
Two-Spin System
Single-qubit gates: can be performed                       Two-qubit gates: naturally accomplished
through the selective resonant interaction                 through the mutual spin dipolar interaction
between the MW-field and the spin
to 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 term
ASSUMPTION                         − 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
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
                                                                                        
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 z2
U (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 operations
by allowing evolution U(t) for a time t=p/16 g(r), a p/4-phase gate is realized
p-gate                                                       Circuit representing Gp



             π             π
           −i σ z1
             4
                         − i σ z2
                            4
                                                π
Gπ = 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                      4
CNOT = ie                   e          Gπ e
Dynamics of the
realistic system
Realistic dynamics
reproduction of theoretical single-qubit and two-qubit quantum gates following the theory
                                    previously presented

Assumption                      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
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 buckyballs
in 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
B01= B02 =(0.3+3.04x10-5)T,                           ν 0 = 2ω0 / 2π = 8.40 ×109 Hz
static 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 coupling
î Dwp>>g(r)                           terms in the dipolar Hamiltonian


î     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 z2
q=0     î
•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 z2




Energy-level diagram for two uncoupled spins (light lines)and for two spins described
by the Hamiltonian presented above (solid lines)
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 t
Total 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 ]
• 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).
•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 rotations
and subsequently of a p-gate but the time required for performing a single qubit rotation
is tSQ=32 ns. The time during which the system is influenced by the spin dipolar interaction
is T=2p/g(r)=2.6x10-5s îtSQ<<T during the completion of a single-qubit rotation
we can consider the system as being unaffected by the mutual spin dipolar interaction
îwhen performing Single-Qubit rotations, the spin-Mw field term dominates
• 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=
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
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 σ'ψ     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         α
p-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
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)
∆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
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 Hz
which implies g(r)=2.45x107Hz and rª1.5nm. This value of r can be
obtained 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 /4
tout    ≈          ≅ 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
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 gradient
Finally:
         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 )
Scalability: Buckyballs can be easily maneuvered:
•     buckyballs embedded in a silicon substrate
•     Peapod: buckyballs in a nanotube
    proposal: improved T2 in a peapod


Readout: difficulty in the readout of single electron spins.
    TNT(erbium-doped) fullerene promising candidates for the readout
Promising results of recent experiments:
•direct excitation of IONC STATES in TNT’sïopens the opportunity of identifying
useful readout transitions and coherently and selectively excite these transitions
•Application of suitable magnetic fields on TNT samplesïthe observed spectrum split
confirms that Er3+ ions are Kramer ions. They maintain the two-fold degeneracy in their
quantum states even under complete crystal-field splittingï ENCODING of a QUBIT
in this pseudo-1/2 spin and EXCITING selective luminecsent transitionsï COULD
ALLOW THE DETECTION OF INDIVIDUAL SPIN STATES
TWO-SPIN SYSTEM

TWO-QUBIT GATES: naturally accomplished through the mutual spin dipolar interaction

SINGLE-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
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-field

ASSUMPTION               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
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

                            fl
Two-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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Phd thesis- Quantum Computation

  • 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. Introduction: a. Endohedral Fullerene Molecules (Buckyballs) N@C60 Buckyball-Ideal Cage
  • 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 completely cage and the encapsulated screened, the Fullerene cage atom. 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. Decoherence times: •T1 due to the interactions between a spin and the surrounding environment • T2 due to the dipolar interaction between the qubit encoding spin and the surrounding endohedral spins randomly distributed in the sample • 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 regions of 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 processes J. J. L. Morton, A. M. Tyryshkin, A. Ardavan, K. Porfyrakis, S. A. Lyon, G. A. Briggs, J. Chem. Phys. 124, 014508 (2006).
  • 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 system By choosing n ˆ ˆ ˆ ˆ ˆ ˆ parellel 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. 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 component Hamiltonian of a two-qubit system subjected to the spin dipolar mutual interaction 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 z2 whereω and ω are the precession frequencies of spin 1 and spin 2, respectively 01 02
  • 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)
  • 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. SINGLE-SPIN SYSTEM: single-qubit gates The state of a ½ spin particle in a static magnetic field B0 directed along the z axis can r be 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 and an 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 field Considering the Schrödinger equation and performing a change of coordinates to a frame rot rotating a frequency wr about the z axis defined by ψ = e −iωrσ z ψ , by choosing wr=w0 we 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. 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 related time 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 π Example: 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. Two-Spin System Single-qubit gates: can be performed Two-qubit gates: naturally accomplished through the selective resonant interaction through the mutual spin dipolar interaction between the MW-field and the spin to 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 term ASSUMPTION − 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. 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. 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 z2 U (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 operations by allowing evolution U(t) for a time t=p/16 g(r), a p/4-phase gate is realized
  • 16. p-gate Circuit representing Gp π π −i σ z1 4 − i σ z2 4 π Gπ = 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 4 CNOT = ie e Gπ e
  • 18. Realistic dynamics reproduction of theoretical single-qubit and two-qubit quantum gates following the theory previously presented Assumption 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. 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 buckyballs in 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. B01= B02 =(0.3+3.04x10-5)T, ν 0 = 2ω0 / 2π = 8.40 ×109 Hz static 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 coupling î Dwp>>g(r) terms in the dipolar Hamiltonian î 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 z2 q=0 î
  • 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 z2 Energy-level diagram for two uncoupled spins (light lines)and for two spins described by the Hamiltonian presented above (solid lines)
  • 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 t Total 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. • 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. •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 rotations and subsequently of a p-gate but the time required for performing a single qubit rotation is tSQ=32 ns. The time during which the system is influenced by the spin dipolar interaction is T=2p/g(r)=2.6x10-5s îtSQ<<T during the completion of a single-qubit rotation we can consider the system as being unaffected by the mutual spin dipolar interaction îwhen performing Single-Qubit rotations, the spin-Mw field term dominates
  • 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. 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. 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 σ'ψ 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 α p-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. 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. ∆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. 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 Hz which implies g(r)=2.45x107Hz and rª1.5nm. This value of r can be obtained 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 /4 tout ≈ ≅ 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. 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 gradient Finally: 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. Scalability: Buckyballs can be easily maneuvered: • buckyballs embedded in a silicon substrate • Peapod: buckyballs in a nanotube proposal: improved T2 in a peapod Readout: difficulty in the readout of single electron spins. TNT(erbium-doped) fullerene promising candidates for the readout Promising results of recent experiments: •direct excitation of IONC STATES in TNT’sïopens the opportunity of identifying useful readout transitions and coherently and selectively excite these transitions •Application of suitable magnetic fields on TNT samplesïthe observed spectrum split confirms that Er3+ ions are Kramer ions. They maintain the two-fold degeneracy in their quantum states even under complete crystal-field splittingï ENCODING of a QUBIT in this pseudo-1/2 spin and EXCITING selective luminecsent transitionsï COULD ALLOW THE DETECTION OF INDIVIDUAL SPIN STATES
  • 33. TWO-SPIN SYSTEM TWO-QUBIT GATES: naturally accomplished through the mutual spin dipolar interaction SINGLE-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. 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-field ASSUMPTION 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. 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 fl Two-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