Maria Silvia Garelli
Department of Physics, Loughborough University, LE11 3TU, U.K.
Theoretical Realization of Quantum
Gates Using Interacting Endohedral
Fullerene Molecules
Introduction:Introduction:
a.a. Endohedral FullereneEndohedral Fullerene Molecules Molecules ((BuckyballsBuckyballs))
N@C60 Buckyball-Ideal Cage
•Repulsive interaction
Between the Fullerene
cage and the encapsulated atom. No charge transfer.
Properties of the N@C60Properties of the N@C60
•The atomic electrons of
the encased atom are
tighter bound than in
the free atom. The N atom is stabilized in its ground state.
•Nitrogen central site
position inside the
fullerene cage.
•Since the charge is completely screened, the Fullerene cage does not take part in the interaction process. It can just be considered as a trap for the Nitrogen encased atom.
•The encapsulated Nitrogen atom can be considered as an independent atom, with all the properties of the free atom.
The only Physical quantity of interest
is the spin of the encapsulated atom.
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
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 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).
≅(N@C60 in CS2)
⇒⇒
T2=0.25ms
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*
*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).
)]ˆ()ˆ(3ˆˆ)[( 2121 nnrgHrrrrrr
⋅⊗⋅−⊗= σσσσ
3
2
0
2)(
rrg B
π
µµ=where is the dipolar coupling constant
By choosing
parellel to the x-axisnr
)ˆˆ2ˆˆˆˆ)((21211 2 xxyyzzrgH σσσσσσ ⊗−⊗+⊗=
Hamiltionian of the two-qubit 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
Qubit-encoding
two-level system
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
2211
21211
ˆˆ
)ˆˆ2ˆˆˆˆ)((
00
2
zz
xxyyzzrgH
σωσω
σσσσσσ
−−
−+=
where10ω
20ωand are the precession frequencies of spin 1 and spin 2, respectively
2,12,1 00 BBµω =
With the use of atom chip technology*,
two parallel wires carrying a current
of the same intensity generate
a magnetic field gradient.
Single addressing of each qubit
*S. Groth, P. Kruger, S. Wildermuth, R. Folman, T. Fernholz, D. Mahalu, I. Bar-Joseph, J. Schmiedmayer, Appl. Phys. Lett. 85, 14 (2004)
AI
m
md
3.0
1
1
=
=
=
µρ
µ
ïthe two particles are characterized by different
resonance frequencies
Current density > 107A/cm2
−−+
++=
2/
1
2/
1
2
0
dxdxBg
ρρπ
µ
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.
* 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-qubit gates
on resonance spin-microwave field interaction
• Two-qubit gates
naturally existing spin dipolar interaction
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
be manipulated by applying an on resonance MW field,
which rotates in the x-y plane at a frequency wm =2w0 characterized by a phase f and
an amplitude Bm
4444444 34444444 2143421fieldMWspin
ymxmmB
fieldstaticspin
zm ttBBH
−−
+−+−−= ])sin()[cos(00 σφωσφωµσµTotal Hamiltonian
Considering the Schrödinger equation and performing a change of coordinates to a frame
rotating a frequency wr about the z axis defined by , by choosing wr=w0
we obtain the Control Hamiltonian
ψψ σω zrirote
−=
]])2sin[(])2[cos[( 00 Ymxma
rotttH σφωωσφωωω +−−+−−=
mBa Bµω =
)0,sin,(cos φωφω ++= ttBB mmmm
r
When the applied MW-field is resonant with the spin precession frequency, i.e. wm=2w0 ,
the Hamiltonian is time-independent, , and its related
time evolution can be easily written as follows ])sin()[cos( YxaH σφσφω −−=
])sin()[cos()( yxatiiHt
eetUσφσφω −− ==
•U(t) is a rotation in the x-y plane of an angle qproportional 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
Example: p/2 rotation about the y axisyi
eUσ
π
4−
=it can be realized by choosing f= p/2 and allowing the time evolution for a time
t=p/4wa= p /4mBBm
2)(n
i
n eR
rr
r
⋅−
=σθ
θRotation of an angle q about axisn
r
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
ASSUMPTION tiHtHHiiHt USUSDD eeetU−+−− ≈==
)()(
The interaction terms between two uncoupled spins and a MW-field
dominate the time evolutionï the spin dipolar interaction is negligible ïsingle-qubit rotation can be performed in good approximation
•HDD dipolar interaction term
•HUS is the interaction between
two uncoupled spins and the MW-field
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
QUANTUM GATES
p/4-phase gate
=−
−
4
4
4
4
000
000
000
000
π
π
π
π
i
i
i
i
PG
e
e
e
e
U
realizes a p gate up to a p/2 rotation
of both spins about the z axis and
up to a global phase
p-gate
−
=
1000
0100
0010
0001
πG CNOT-gate
=
0100
1000
0010
0001
CNOTU
Refocusing: is a set of transformations which allow the removal of
the off-diagonal coupling terms of HDD
trgi
i
a
i
b
itiH
itiH
zz
zz
zDD
zDD
e
etUetU
eeeetU
21
22
22
)(4
22
22
)()(
)(
σσ
σπ
σπ
σπ
σπ
−
−
−−
−
=
=
=
• 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
22zi
eσ
πm
by allowing evolution U(t) for a time t=p/16 g(r), a p/4-phase gate is realized
Circuit representing U(t)
p-gate
))(16
(21 44
rgtUeeiG
zz ii πσπ
σπ
π ==−−
Circuit representing Gp
CNOT-gate
221 442yyz iii
eGeieCNOTσ
π
π
σπ
σπ
−−
=
Circuit representing CNOT
Dynamics of the
realistic system
Realistic dynamics
reproduction of theoretical single-qubit and two-qubit quantum gates following the theory
previously presented
Assumption tiHiHt USee−− ≅
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: [email protected]
•distance between two buckyballs
in a nanotube: [email protected]
(due to Van der-Waals forces) We need to place 9 empty buckyballs between
the two fullerenes in order to obtain r=7 nm
r=7 nm îHz
rrg B 5
3
2
0 1038.22
)( ×==π
µµ
TB
TB
g
g
4
4
1087.1
1087.1
2
1
−
−
×−=
×=dipolar coupling constant
gradient field amplitudes
Hz
Hz
9
00
9
00
1039.82/2
1040.82/2
22
11
×==
×==
πων
πωνresonance
frequencies
î
Hzppp
7
00 1028.6222121
×=−=−=∆ ωωωωωî
Dwp>>g(r)This condition allows us to omit the transverse coupling
terms in the dipolar Hamiltonian
î The mutual dipolar interaction
Hamiltonian can be simplified as21
)cos31)(( 2
zzapprox rgH σσθ−=
q is the angle between the static magnetic field
and the line joining the centres of the buckys
q=0 î 21)(2 zzapprox rgH σσ−=
B01= B02 =(0.3+3.04x10-5)T,
static magnetic field along
the z direction
•Hamiltonian of two distant buckys subjected to static fields along the z axis
Energy-level diagram for two uncoupled spins (light lines)and for two spins described
by the Hamiltonian presented above (solid lines)
221121 00)(2 zzzz
USapprox
rg
HHH
σωσωσσ −−−=
+=
Total Hamiltonian in the rotating frame
]])2sin[(])2[cos[(
]])2sin[(])2[cos[(
)(2
2222222
1111111
21
00
00
ymxma
ymxma
zz
rot
USapprox
tt
tt
rg
HHH
σφωωσφωωω
σφωωσφωωω
σσ
+−−+−−
+−−+−−
−=
+=
ψψσωσω titirot
zz ee 220110 −−=
])sin()[cos(
])sin()[cos(
)(2
)(
22222
11111
221121 00
ymxma
ymxma
zzzz
USapprox
tt
tt
rg
tHHH
σφωσφωω
σφωσφωω
σωσωσσ
+−+−
+−+−
−−−=
+=
Total Hamiltonian (additional MW-field)
• single-qubit gates: MW-field and the spin to be rotated are in resonance, i.e.
î first spin can be rotated
î second spin can be rotated
11 02ωω =m
22 02ωω =m
• two-qubit gates: naturally realized by the mutual spin dipolar interaction Happrox
time-evolution operator
related to Happrox
trig zzetU 21)(2
)(σσ
=if we allow this time-evolution for
î a time t=p/8g(r)=1.65ms we obtain
a controlled p/4 phase gate
Happrox is already diagonal î the refocusing procedure is not needed
Typical experimental time
of a single-qubit rotation* nsBg
tmB
SQ 32exp ≅=µ
*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
−
===−−
1000
0100
0010
0001
))(8/(21 44 rgtUeeiG
zz ii
πσ
πσ
π
π
•Numerical output matrix
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
Comments :
Up2=
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
==−−
0100
1000
0010
0001
221 442yyz iii
eGeieCNOTσ
π
π
σπ
σπ
•Numerical output matrix
UCNreal=
•Operational times:s
rgBBBt
srgBB
t
mBmBmB
CNOT
out
mBmB
out
µπ
µ
π
µ
π
µ
π
µπ
µ
π
µ
ππ
05.2)(824
54
5
85.1)(84
34
3
121
21
=+++=
=++=
•Number of quantum operations
allowed before relaxation:
222 10≅≅=CNOT
outout t
T
t
Tn
πn<104 î small number
of operationîthe system
is not reliable
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
Possibility of increasing T2 two order of magnitude:
p/8g(r) determines the order of magnitude of tout
UUσσ =' †The fidelity quantifies the distance between the realistic evolved state
and the ideal evolved stateideal
ψ
Since the starting state is not known in advance, we can consider the
minimum fidelity, which minimizes over all possible starting states
)',(min σψα
idealcF=Fî
idealidealidealidealidealUUF ψψψψψσψσψ == ')',(
Quantum gate fidelity
p-gate: F=0.998
CNOT-gate: F=0.991
F differs from its ideal value F=1by of the order of 0.2%(0.8%)
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 bandwidthdifference between the upper and lower values
of the range which allow the swap of the selected spin
)(8))(42()(4211 00 rgrgrg =−−+=∆Ω ωω
î the frequency bandwidth DW depends only on the dipolar coupling constant g(r)
MHzrg 9.1)(8 ==∆Ω nstt SQ 32==∆and
î the bandwidth theorem DWDt@2p is not satisfied
Two options:
•If tSQ=32ns î DW=1.95x108 Hz
•If DW=1.9 MHz î tSQ=3.3 ms
The first is preferable because it
allows single-qubit rotations in
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
Condition Dwp>>g(r)allows to know exactly the frequency bandwidth, i.e.
)(8 rg=∆ΩSince Dtª32ns, from the bandwidth theorem DWDtª1, we obtain
Hzrg81096.1)(8 ×==∆Ω
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
(1)
Conclusions:
4
4/
284/ 10106.1)(8
≥=⇒×≅≈ −
π
π π
out
outt
Tns
rgt
The system would be a good candidate
as a building block for quantum
computers and would allow the
possibility of applying quantum
error correcting codes
• Quantum Cellular Automaton with different species of encased particlesthe 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
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
Is it exprimentally possible to
realize single-qubit rotations in
a time shorter than t=32 ns?
Finally:
If so î4
)(
2 10≅=CNOT
outt
Tn
π
Readout: difficulty in the readout of single electron spins.
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
Scalability: Buckyballs can be easily maneuvered:
• buckyballs embedded in a silicon substrate
• Peapod: buckyballs in a nanotube
TNT(erbium-doped) fullerene promising candidates for the readout
proposal: improved T2 in a peapod
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
]])2sin[(])2[cos[(
]])2sin[(])2[cos[(
]2)()22)[cos((
)(
2222222
1111111
21212121
00
00
00
ymxma
ymxma
zzyyxx
USDD
tt
tt
trg
HHtH
σφωωσφωωω
σφωωσφωωω
σσσσσσωω
+−−+−−
+−−+−−
−+−=
+=
Total Hamiltonian of the two-spin system in the rotating frame
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
]')'(exp[),(
0
0 ∫−=
t
t
dttHiTttU
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:
21 00 ωω =• the precession frequencies of the two spins are equal
• resonant MW-field2,12,1 02ωω =m
ASSUMPTIONtiHtHHiiHt USUSDD eeetU
−+−− ≈==)(
)(
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
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
fl
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