Journal of Quantum Informatio n Science, 2011, 1, 121-126
doi:10.4236/jqis.2011.13017 Published Online December 2011 (http://www.SciRP.org/journal/jqis)
121
On the Theory of Topological Computation in the Lowest
Landau Level of QHE
Dipti Banerjee
Department of Physi cs , Vidyasagar College for Women, Kolkata, India
Received August 28, 2011; revised November 20, 2011; accepted November 28, 2011
Abstract
We have studied the formation of Hall-qubit in lowest Landau level of (LLL) Quantum Hall effect due to the
Aharonov-Bohm oscillation of quasiparticles. The spin echo method plays the key role in the topological
entanglement of qubits. The proper ratio of fluxes for maximally entangling qubits has also been pointed out.
The generation of higher Quantum Hall state may be possible with the help of quantum teleportation.
Keywords: Spin Echo, Aharonov-Bohm Phase
1. Introduction
Entanglement is one of the basic aspects of Quantum
mechanics that exhibits the peculiar correlations between
two physically distant parts of the total systems. The
Geometric phase such as Berry phase (BP) [1] and Aha-
ronov-Bohm phase [2] play an important role in Quan-
tum mechanics. The effect of Berry phase on the entan-
gled quantum system is less known. The quantum gates
that convey topological transformation is known as
topological gates. These gates are advantageous for their
immunity and resistive power for local disturbances.
They do not depend on the overall time evolution nor on
small deformations on the control parameter. This indi-
cates that a quantum mechanical state would carry its
memory during its spatial variation and the influence of
Berry phase on an entangled state could be linked up
with the local observations of spins. Few attempts have
been made connecting the Berry phase with entangle-
ment of spin-1/2 particles resulting the outcome of Geo-
metric/Holonomic quantum computation [3].
The transport of information through quasi-particles
exhibiting long-range non-abelian Aharonov-Bohm in-
teractions can yield similar topological quantum compu-
tation [4]. Kitaev [5] pointed out topological quantum
computer as a device in which quantum numbers carried
by quasi-particles residing on 2DES have long range
Aharonov-Bohm interactions with one another. There
exist a strong quantum correlation between the quasi-
particles interacting by A-B effect extending out over
large distances [6].
The interwinding of the quasi-particles trajectories in
the course of time evolution of the qubits realizes con-
trolled-phase transformations with nontrivial phase val-
ues. One of the remarkable discoveries of recent decades
is the infinite range A-B interactions observed in Frac-
tional Quantum Hall effect [7]. The electrons in the
Quantum Hall systems are so highly frustrated that the
ground state is an extremely entangled state. The quan-
tum entanglement of particle having nontrivial Berry
phase is associated with the transport of a charge around
a flux which is equivalent to the Aharonov-Bohm phase
in the analogy with the gauge theory. A similar reflection
is seen in FQHE when one quasi-particle/composite par-
ticle goes around another encircling an area A. The total
phase associated with this path is given by [8]


*
0
2πBA2 enc pN   (1)
where enc is the number of composite fermion inside
the loop. The first term on the right hand side is usual A-
B phase and the second term is the contribution from the
vortices bound to composite fermions indicating that
each enclosed composite fermion effectively reduces the
flux by flux quanta. These particles of a nontrivial
condensed matter state obey fractional statistics and
Arovas et al. [9] pointed out that the exchange of parti-
cles over a half loop producing phase factor
N
2p

1
πi
e



.
In the entanglement of two spin-1/2 particles in pres-
ence of magnetic field through the spin echo method BP
plays an important role [10]. We have studied in pres-
ence of Rabi oscillation the rotation of qubit through BP

D. BANERJEE
122
[11]. A reflection of this idea is seen to study the forma-
tion of Hall qubit rotation recently in 1

 the ground
state in the light of quantum entanglement [12]. Here we
aim at understanding the rotation of Hall qubits having
A-B interactions in the background of topological com-
putation.
2. Formation of Hall-Qubit from Berry
Phase
One single qubit can be sufficiently constructed [3] using
the two well known quantum gates—Hadamard gate (H)
and Phase gate as follows




2π2
0
cos 0sin1
i
HH
e





 
 (2)
With the use of quantum gates and elementary qubits
0 and 1 the spinors for up or down polarization can
be written as
2
sin0cos 1
22
i
e



 






(3)
and
2
cos 0 sin1
22
i
e



 





(4)
This above qubit representing a spinor acquires the
geometric phase [11] over a closed path.
,,dit t



 (5)

diA



 (6)
d
eff
Lt

 (7)

π1cos


  (8)
which is a solid angle subtended about the quantization
axis. For the conjugate state the Berry phase over the
closed path becomes

π1cos



 (9)
Here the angle

measures the deviation of the
magnetic flux line from the Z-axis. The fermionic or the
antifermionic nature of the two spinors (up/down) can be
identified by the maximum value of topological phase
π

  at an angle π2

. For 0

 we get the
minimum value of 0

 and at π

 no extra effect
of phase is realized.
In the spin echo method, this Berry phase can be
fruitfully [13,14] isolated in the construction of two qubit
through rotation of one qubit (spin 1/2) in the vicinity of
another. Incorporating the spin-echo for half period we
find the antisymmetric Bell’s state after one cycle


t


,



12 12
1ee
2
ii
t







(10)
And symmetric state becomes



12 12
1ee
2
ii
t




 
(11)
where



. It may be noted that for

1.

1c os
2


This phase plays the key role through the

factor,
in the measurement of entanglement by the concurrence
“C” of an entangled state. For a two qubit state
 

  the concurrence [4] is given by
2C

  (12)
When 1C

the entanglement is maximum and dis-
entanglement for 0C

. The value of 0 
 implies
disentanglement for 0


. For π

 there is maxi-
mum deviation of flux line yielding 1 
 as a signa-
ture of maximum entanglement.
Splitting up these above two Equations (11) and (12)
into the symmetric and antisymmetric states [12] and
rearranging we have
00
cos sini



 

(13)
0
sin cosi



 
0

(14)
The doublet acquiring the matrix Berry phase

as
rotated from 0t

to t


.
0 


  

 

 

(15)
cossin cos2
sin cos
i
i





 


(16)
This non-abelian matrix Berry phase is developed
from the abelian Berry phase


. For 0

 there is
symmetric rotation of states, but for π

 the return is
antisymmetric and the respective values of = I and –I
(where I = identity matrix).

There is a deep analogy between FQHE and superflu-
idity [15]. The ground state of anti-ferromagnetic Hei-
senberg model on a lattice introduce frustration giving
rise to the resonating valence bond (RVB) states corre-
sponding spin singlets where two nearest-neighbor bonds
are allowed to resonate among themselves. The RVB
state is a coherent superposition of spin singlet pairs and

D. BANERJEE 123
can be written as

11
RVB,, nn kk ij ijij 

(17)
in which


1
,2 ijij ij



is a spin singlet
pair (valance bond) between sites i and j. This RVB state
support fractionalized excitation of spin 1/2 spinon [4,5].
The topological order is closely related to the coherent
motion of fractionalised spin excitation in RVB back-
ground. It is suggested that RVB states [6] is a basis of
fault tolerant topological quantum computation. Since
these spin singlet states forming a RVB gas is equivalent
to fractional quantum Hall fluid, its description through
quantum computation will be of ample interest.
The Quantum Hall effect can be considered on a 3D
anisotropic space having the N-particle wave-function of
parent Hall states


m
mij ji
Nuv uv

 
 (18)
Represent the one qubit in the language of quantum
information. In the Jain's formalism [16], m

 the hier-
archical FQHE incompressible state for Landau filling
factor

2
11
pn mn
qnm n


 

1
becomes
   
12
11
mm
m
n
zzz z


1n (19)
where odd for making the state anti-
symmetric and = integer, specifying Lanadu level in
QHE. The state is the QHE state at the lowest
landau level n and filling factor
m

m
z


1
n
1

1z

. States of
the above form are grouped into a family depending on
the values of . Any FQHE state can be expressed in
terms of the IQHE ground state [17].
m
Recently we have identified [12] this ground state
as the Hall qubit, the basic building block for
constructing any other IQHE/FQHE state formed when
two nearest neighbor bonds are allowed to resonate
among themselves. It is an extremely entangled state
visualized by the formation of singlet state between a
pair of spinors.

1z

,th ij



1
01
10
j
ijjiii
j
u
zuvuv uvv


  




(20)
01
10
i





j
(21)
A singlet state is a two qubit developed as one qubit
sin e
2
cos 2
i
u
v






  
 
 

through the spin echo method where Berry’s topological
phase dominates in acquiring the Hall qubit with the de-
scription of a two component up spinor.
In this present work we have focused on Quantum
Hall effect where the nature of the state will be only an-
tisymmetric. Hence the Equation (17) reduces to
0 



The Hall qubit


1z has resemblance with


.
In the lowest Landau level 1m



would develop
similar non-abelian Berry phase . This is visualizing
the spin conflict during parallel transport leading to ma-
trix Berry phase. Over a closed period t

 the QHE
state


1z at 1


filling factor will acquire the
matrix Berry phase.
 
10
e
H
i
z




1
z
(22)
Here e
H
i

is the non-abelian matrix Berry phase
iij
H
j
ij










(23)
where i

and
j

are the BPs for the ith and jth spinor
as seen in Equation (16) and the off-diagonal BP ij

arises due to local frustration in the spin system.
All the above explanation is restricted for lowest Lan-
dau level 1


, but concentrating on the other parent
state 1m


where m

odd integers, the Berry
phase of a qubit is πim


[12] that is associated with
the two qubit Hall state


π
2
1π
11
2
0e
1
e0 2
im
im
z











(24)
through the process of quantum entanglement between
two one qubit where the reflection of spin echo is visual-
ized.
3. Hall-Qubit Formation in QHE through
Aharonov-Bohm Phase
In the composite fermion theory of Quantum Hall effect
the qubits are equivalent to the fluxes attached with the
charged particles. When an electron is attached with a
magnetic flux, its statistics changes and it is transformed
into a boson. These bosons condense to form cluster
which is coupled with the residual fermion or boson
(composed of two fermions). Indeed the residual boson
or fermion will undergo a statistical interaction tied to a
geometric Berry phase effect that winds the phase of the
particles as it encircles the vortices. Indeed as two vor-
tices cannot be brought very close to each other, there
will be a hard core repulsion in the system which ac-

rotates in the vicinity of another

D. BANERJEE
124
counts for the incompressibility of the Quantum Hall
fluid.
These non-commuting fluxes have their own interest-
ing Aharonov-Bohm interactions. As the quasi-particles
encircle another in their way of topological transport, the
Aharonov-Bohm type statistical phase is developed. Fo-
llowing a generalization of Pauli exclusion principle,
Haldane [18] pointed out that the quasi-particles carrying
flux


and charge q

orbiting around another object
carrying flux


and charge q

has the relative statis-
tical phase



 
expexp πiig

g


  

(25)
where gq



 . With this view we have recently
shown that when two non-identical composite fermions
residing in two consecutive Landau levels in FQHE en-
circle each other, the relative Aharonov-Bohm (AB) type
phase is developed. As the quasi-particles advance to-
wards the edge of FQHE in a similar circular way, the
developed current [19] should have a connection with
this AB type phase through Berry's topological phase.
These A-B interactions are the key source of forming
two qubit Hall states identified as Hall qubit. Hence
movement of Hall qubits would develop the A-B phase.
In the physics of spin echo instead of Berry phase the
incorporation of Aharonov-Bohm phase would be more
appropriate as the rotation of qubits are equivalent to the
rotation of fluxes around charges. If eis

be the Aharo-
nov-Bohm phase between the two qubits, for half period
we find the antisymmetric Bell’s state after one cycle

t

,


12 12
1ee
2
is is
t



 

(26)
Similar consideration for symmetric states


12 12
1ee
2
is is
t



   (27)
Splitting up these states and rearranging the symmetric
and antisymmetric parts we have the doublet acquiring
the matrix form of Aharonov-Bohm phase as rotated
from to t
0 t

.
0 


  






(28)
where
cossin cos 2
sin cos
ss
s
ss
i
i





 


(29)
This topological matrix phase is developed from
the Aharonov-Bohm phase

s

as one qubit rotates
around another. The qubits in QHE are quantized spinor
having flux attached with charge. Their entanglement is
equivalent to spin type echo where the topological phase
dominates due to Aharonov-Bohm oscillation between
them. This compel to change the Berry phase of the
singlet state as in Equation (24) by the relative A-B type
phase
s

.


10
eis z



1
z
(30)
This Hall qubit can be visualized in terms of entan-
glement of two oscillating qubits.


2
1z
ee
11
2
0e 1
e0
2
is
is












1
(31)
To form the singlet state between the qubits under A-B
interactions in the spin echo method, the essential condi-
tion for antisymmetric QHE states are visualized by
πis i 

 .
In the formation of Hall qubit through the entangling
of qubits in the different Landau level the A-B phase
plays the key role in the spin type echo method. If the
rotating qubits are in the same Landau level, the A-B
phase changes to statistical phase. Considering the inter-
action [20] between two identical qubits in the same
Landau level for the composite particles filling factor
th
n
2eff
n


 the statistical phase becomes
π
exp 2
sn
i   (32)
where for 2, 4, 6n

 the change of statistics will be
fermionic visualized by the phase exp π
si

. On the
other hand for 1, 3,5n

 the statistics will be bosonic
exp π2
si


. It may be noted that for fractional filling
factors the final statistics will be fermionic through
proper combinations. With this view, the entanglement
of two one qubits in the same Landau Level, the Hall
qubit


1z will be formed when the exchange phase
is πi



π
2
1π
11
2
0e
1
e0
2
i
i
z













(33)
Whenever the interaction takes place between dis-
similar qubits in different Landau level the rotation of
one against another develops the Aharonov-Bohm type
phase that does not express their statistics. We assumed
the transfer of the composite particle [19] from the inner
edge in the Landau level having filling factor n
th
n

picking up even integral


2m of flux 1

through the
bulk of QH system and forming a new composite particle
in the Landau level in the outer edge. The fill-

1n

th

D. BANERJEE 125
ing factor of the effective particle becomes 1
eff eff
n



.
The monopole strength eff

of the state can
be considered as
2
1
mn

1
2
eff n
m

 . (34)
Encircling of the composite particle in the inner edge
having flux n

with charge n around the composite
particle in the outer edge having corresponding
eff
q
flux

would develop a relative AB type phase

π
exp 2

s
neffeff n
iqq

  (35)
In more simplified way it becomes
1
π1
exp 22
sn
inm





 




(36)
Since the concurrence 1C

indicate the maximum
entanglement and for disentanglement the value of mini-
mum concurrence is , we can establish a relation-
ship between the fluxes of the entangling qubits on the
Hall surface. The maximum entanglement between the
two quasi-particle results a relation between the respec-
tive two fluxes 1
0C

and n

in terms of parent filling
factor and Landau level .
m n
1
π1
expexp π
22 n
inm









i (37)
This gives a ratio between the entangling fluxes n

and 1

in order to form the singlet pairs through AB
oscillations in Quantum Hall effect.
1
2
23
nm
n

  (38)
It may be noted that for maximum entanglement
11

 results
2
23
nm
n
 

and for minimum entanglement both 1

and n

be-
comes zero.
The physics behind the formation of higher Hall states
take place through the entanglement of Hall qubits

1 in the lowest landau level. Here the A-B phase
or statistical phase plays the key role in the process of
spin echo with the essential condition .
The outcome of entanglement of two Hall qubits is
z
1
π
ee
is i 

  


π
1
π
1
1
0e
e0
0
0
i
i
zz
z
z




 






where




1ijij
zuvvu. On the similar manner we
realize that the entanglement of two

z  gives rise
to the state formed by the square of Hall qubits.
  

2
1
2
1
01 0
10 0
z
zzz z
 





 

 
(40)
In order to maintain the antisymmetric nature of the
Hall state the power of the Hall qubit should be odd. This
is possible only when two asymmetric Hall qubits (one
even power with another odd power) entangle under
topological interactions
  

3
1
3
1
01 0
10 0
z
zz
z



 
 

 
(41)
Forming a Hall state in the parent Landau filling factor
oddm


.
We like to conclude mentioning that the hierarchical
FQHE states are formed through the process of quantum
teleportation. If there are three entities defined by 1, 2,
and 3, then transportation of 1 to 3 through 2 will be
123
123
 (42)

13
12 3
11
2


  (43)
Similar reflection of quantum teleportation [21] in
FQHE motivate us to write

11
22 13
111 1
11
2
mm m
nn
n


2
1
m

  (44)
A hierarchical FQHE state whose extensive study
through the entanglement of Hall qubits maintaining the
antisymmetric nature of the exchange phase is to be done
in future.
4. Conclusions
1
z
(39)
In this paper we have studied the Physics behind the Hall
qubit formed by the entanglement of two qubits where
one is rotating in the field of the other with Aharonov-
Bohm (AB) phase. The image of spin echo between the
entangling composite fermion/qubit has been reflected in
the field of Quantum Hall effect. Topological quantum
computation has been executed considering the Hall
qubit at 1


as a building block for the formation of
other higher IQHE/FQHE states at different filling fac-
tors. With the condition of concurrence for maximum
entanglement, a proper ratio between the fluxes of the
entangling qubits has been evaluated. At the end, we
have mentioned that the states in hierarchies of FQHE
can be studied in the light of quantum teleportation

D. BANERJEE
126
whose extensive study will be of ample interest in future.
5. Acknowledgements
This work has been partly supported by The Abdus Sa-
lam International Center for Theoretical Physics, Trieste,
Italy during the visit as Regular associate.
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