Maria Silvia Garelli

(M.S.Garelli@lboro.ac.uk)

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: 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

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

θ î Bm@1.7mT

*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