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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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.
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