Topologically protected quantum states and quantum computing in Josephson junctions arrays
We review recent results on a new class of Josephson arrays which have non-trivial topology and exhibit a novel quantum states at low temperatures. One of these states is characterized by long range order in a two Cooper pair condensate and by a discrete topological order parameter. The second st...
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irk-123456789-1198392017-06-11T03:03:38Z Topologically protected quantum states and quantum computing in Josephson junctions arrays Ioffe, L.B. Feigel`man, M.V. Douçot, B. Сверхпроводимость и мезоскопические структуры We review recent results on a new class of Josephson arrays which have non-trivial topology and exhibit a novel quantum states at low temperatures. One of these states is characterized by long range order in a two Cooper pair condensate and by a discrete topological order parameter. The second state is insulating and can be considered as a result of evolution of the former state due to Bose-condensation of usual superconductive vortices with a flux quantum 0. Quantum phase transition between these two states is controlled by variation of external magnetic field. Both the superconductive and insulating states are characterized by the presence of 2K-degenerate ground states, with K being the number of topologically different cycles existing in the plane of the array. This degeneracy is «protected» from the external perturbations (and noise) by the topological order parameter and spectral gap. We show that in ideal conditions the low order effect of the external perturbations on this degeneracy is exactly zero and that deviations from ideality lead to only exponentially small effects of perturbations. We argue that this system provides a physical implementation of an ideal quantum computer with a built in error correction. A number of relatively simple «echo-like» experiments possible on small-size arrays are discussed. 2004 Article Topologically protected quantum states and quantum computing in Josephson junctions arrays / L.B. Ioffe, M.V. Feigel`man, B. Douçot // Физика низких температур. — 2004. — Т. 30, № 7-8. — С. 841-855. — Бібліогр.: 36 назв. — англ. 0132-6414 PACS: 74.50.+r, 85.25.–j http://dspace.nbuv.gov.ua/handle/123456789/119839 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Сверхпроводимость и мезоскопические структуры Сверхпроводимость и мезоскопические структуры |
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Сверхпроводимость и мезоскопические структуры Сверхпроводимость и мезоскопические структуры Ioffe, L.B. Feigel`man, M.V. Douçot, B. Topologically protected quantum states and quantum computing in Josephson junctions arrays Физика низких температур |
| description |
We review recent results on a new class of Josephson arrays which have non-trivial topology
and exhibit a novel quantum states at low temperatures. One of these states is characterized by
long range order in a two Cooper pair condensate and by a discrete topological order parameter.
The second state is insulating and can be considered as a result of evolution of the former state due
to Bose-condensation of usual superconductive vortices with a flux quantum 0. Quantum phase
transition between these two states is controlled by variation of external magnetic field. Both the
superconductive and insulating states are characterized by the presence of 2K-degenerate ground
states, with K being the number of topologically different cycles existing in the plane of the array.
This degeneracy is «protected» from the external perturbations (and noise) by the topological order
parameter and spectral gap. We show that in ideal conditions the low order effect of the external
perturbations on this degeneracy is exactly zero and that deviations from ideality lead to only
exponentially small effects of perturbations. We argue that this system provides a physical implementation
of an ideal quantum computer with a built in error correction. A number of relatively
simple «echo-like» experiments possible on small-size arrays are discussed. |
| format |
Article |
| author |
Ioffe, L.B. Feigel`man, M.V. Douçot, B. |
| author_facet |
Ioffe, L.B. Feigel`man, M.V. Douçot, B. |
| author_sort |
Ioffe, L.B. |
| title |
Topologically protected quantum states and quantum computing in Josephson junctions arrays |
| title_short |
Topologically protected quantum states and quantum computing in Josephson junctions arrays |
| title_full |
Topologically protected quantum states and quantum computing in Josephson junctions arrays |
| title_fullStr |
Topologically protected quantum states and quantum computing in Josephson junctions arrays |
| title_full_unstemmed |
Topologically protected quantum states and quantum computing in Josephson junctions arrays |
| title_sort |
topologically protected quantum states and quantum computing in josephson junctions arrays |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2004 |
| topic_facet |
Сверхпроводимость и мезоскопические структуры |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119839 |
| citation_txt |
Topologically protected quantum states and quantum computing in Josephson junctions arrays / L.B. Ioffe, M.V. Feigel`man, B. Douçot // Физика низких температур. — 2004. — Т. 30, № 7-8. — С. 841-855. — Бібліогр.: 36 назв. — англ. |
| series |
Физика низких температур |
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AT ioffelb topologicallyprotectedquantumstatesandquantumcomputinginjosephsonjunctionsarrays AT feigelmanmv topologicallyprotectedquantumstatesandquantumcomputinginjosephsonjunctionsarrays AT doucotb topologicallyprotectedquantumstatesandquantumcomputinginjosephsonjunctionsarrays |
| first_indexed |
2025-07-08T16:45:32Z |
| last_indexed |
2025-07-08T16:45:32Z |
| _version_ |
1837097952248594432 |
| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8, p. 841–855
Topologically protected quantum states and quantum
computing in Josephson junctions arrays
L.B. Ioffe
Center for Materials Theory, Department of Physics and Astronomy, Rutgers University
136 Frelinghuysen Rd., Piscataway NJ 08854, USA
Landau Institute for Theoretical Physics, 2 Kosygina Str., Moscow 117940, Russia
M.V. Feigel’man
Landau Institute for Theoretical Physics, 2 Kosygina Str., Moscow 117940, Russia
E-mail: feigel@landau.ac.ru
B. Douçot
Laboratoire de Physique Théorique et Hautes’ Energies, CNRS UMR 7589,
Universités Paris 6 et 7, 4 Place Jussieu, Paris 75252 Cedex 05, France
Received January 5, 2004
We review recent results on a new class of Josephson arrays which have non-trivial topology
and exhibit a novel quantum states at low temperatures. One of these states is characterized by
long range order in a two Cooper pair condensate and by a discrete topological order parameter.
The second state is insulating and can be considered as a result of evolution of the former state due
to Bose-condensation of usual superconductive vortices with a flux quantum �0. Quantum phase
transition between these two states is controlled by variation of external magnetic field. Both the
superconductive and insulating states are characterized by the presence of 2K-degenerate ground
states, with K being the number of topologically different cycles existing in the plane of the array.
This degeneracy is «protected» from the external perturbations (and noise) by the topological or-
der parameter and spectral gap. We show that in ideal conditions the low order effect of the exter-
nal perturbations on this degeneracy is exactly zero and that deviations from ideality lead to only
exponentially small effects of perturbations. We argue that this system provides a physical imple-
mentation of an ideal quantum computer with a built in error correction. A number of relatively
simple «echo-like» experiments possible on small-size arrays are discussed.
PACS: 74.50.+r, 85.25.–j
1. Introduction
Quantum computing [1,2] is in principle a very
powerful technique for solving classic «hard» prob-
lems such as factorizing large numbers [3] or sorting
large lists [4]. The remarkable discovery of quantum
error correction algorithms [5] shows that there is no
problem of principle involved in building a function-
ing quantum computer. However, implementation
still seems dauntingly difficult: the essential ingredi-
ent of a quantum computer is a quantum system with
2K (with K �� 100) quantum states which are degen-
erate (or nearly so) in the absence of external pertur-
bations and are insensitive to the «random» fluctua-
tions which exist in every real system, but which may
be manipulated by controlled external fields with er-
rors less than10 6� . Moreover, the standard schemes of
error-corrections (assuming error rate of order 10 6� )
require very big system sizes, K � 104–106, to correct
the errors (i.e., the total number of all qubits is by
factor of 100–1000 exceeds the number of qubits
needed to perform computational algorithm in the
«ideal» conditions of no errors). If frequency of errors
could be reduced by orderes of magnitude, the condi-
tions for residual errow-corrections would becomes
© L.B. Ioffe, M.V. Feigel’man, and B. Douçot, 2004
much less stringent, and the total size of the system,
K, would be much smaller [6].
Insensitivity to random fluctuations means that
any coupling to the external environment neither in-
duces transitions among these 2K states nor changes
the phase of one state with respect another. Mathe-
matically, this means that one requires a system whose
Hilbert space contains a 2K -dimensional subspace
(called «the protected subspace» [7–9]) within which
any local operator �O has (to a high accuracy) only
state-independent diagonal matrix elements:
n O m O o Lmn| � | (exp ( ))� � �0� ,
where L is a parameter such as the system size that
can be made as large as desired. It has been very diffi-
cult to design a system which meets these criteria.
Many physical systems (for example, spin glasses
[10]) exhibit exponentially many distinct states so
that the off-diagonal matrix elements of all physical
operators between these states are exponentially
small. In such systems the longitudinal relaxation of
a superposition of these states is very slow. The ab-
sence of the transverse relaxation which is due to the
different diagonal matrix elements O Omm nn� is a
different matter: it is highly non-trivial requirement
that is not satisfied by usual physical systems (such as
spin glasses) and which puts systems satisfying it in a
completely new class.
One very attractive possibility, proposed in an im-
portant paper by Kitaev [7] and developed further in
[11] involves a protected subspace [8,9] created by a
topological degeneracy of the ground state. Typically
such degeneracy happens if the system has a conserva-
tion law such as the conservation of the parity of the
number of «particles» along some long contour, and
the absence of any local order parameter. Physically,
it is clear that two states that differ only by the parity
of some big number that cannot be obtained from any
local measurement are very similar to each other. Car-
toon example of this idea can be presented as follows.
Consider two locally flat surfaces, one with a topol-
ogy of simple cylinder, whereas another one is the
Moebius stripe, and imagine an observer moving on
one of these surfaces. Clearly, the only way to decide
on which surface the observer is located is to walk the
whole loop around the stripe and find himself at the
same point (then the surface is a cylinder) or on the
other side of the surface (if it is Moebius stripe).
The model proposed in [7] has been shown to ex-
hibit many properties of the ideal quantum computer;
however before now no robust and practical imple-
mentation was known. In a recent paper we and others
proposed a Josephson junction network which is an
implementation of a similar model with protected de-
generacy and which is possible (although difficult) to
build in the laboratory [12].
In this paper we review our recent results on fur-
ther development of ideas of Ref. 12. We propose a
new Josephson junction network that has a number of
practical advantages (shorter account of this approach
can be found in [13,14]). This network operates in a
phase regime (i.e., when Josephson energy is larger
than the charging energy), which reduces undesired
effects of parasitic stray charges. All Josephson junc-
tions in this array are similar which should simplify
the fabrication process. This system has 2K -degenerate
ground states «protected» to even higher extent than
in [12]: matrix elements of local operators scale as �L,
where � 01. is a measure of non-ideality of the sys-
tem’s fabrication (e.g., the spread of critical currents
of different Josephson junctions and geomertical areas
of different elementary cells in the network). The new
array does not require a fine tuning of its parameters
into a narrow region. The relevant degrees of freedom
of this new array are described by the model analogous
to the one proposed in Ref. 7.
In physical terms, the array we propose may exist
in two different phases: i) topological superconductor,
that is a superfluid of 4e composite objects, and ii)
topological insulator that is a superfluid of super-
conductive vortices with a flux quantum �0 2� hc/ e.
The topological degeneracy of the ground state
i) arises because 2e excitations have a gap. Indeed, in
such system with the geometry of an annulus, one ex-
tra Cooper pair injected at the inner boundary can
never escape it; on the other hand, it is clear that two
states differing by the parity of the number of Cooper
pairs at the boundary are practically indistinguishable
by a local measurement. In the ground state ii) the
lowest excitation is a «half-vortex» (i.e., vortex with
a flux �0 2/ ), and topological double-degeneracy ap-
pears due to the possibility to put a half-vortex inside
the opening, without paying any energy.
Below we first describe the physical array, with the
«topological superconductor» state, identify its rele-
vant low energy degrees of freedom and the mathemat-
ical model which describes their dynamics. We then
show how the protected states appear in this model,
derive the parameters of the model and identify vari-
ous corrections appearing in a real physical system and
their effects. Then we discuss generalization of this ar-
ray that is needed to obtain in a controllable way a
second phase of «topologically insulator». Finally, we
discuss how one can manipulate quantum states in a
putative quantum computer based upon those arrays,
and the physical properties expected in a small arrays
of this type.
842 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8
L.B. Ioffe, M.V. Feigel’man, and B. Douçot
We remark that the properties of the excitations
and topological order parameter exhibited by the sys-
tem we propose here are in many respects similar to
the properties of the ring exchange and frustrated
magnets models discussed recently in [8,9,15–25].
2. Array 1: Topological superconductor
The basic building block of the lattice is a rhombus
made of four Josephson junctions with each side of the
rhombus containing one Josephson contact, these
rhombi form a hexagonal lattice as shown in Fig. 1.
We denote the centers of the hexagons by letters
a b, ... and the individual rhombi by ( ), ( )...ab cd ,
because each rhombus is one-to-one correspondence
with the link ( )ab between the sites of the triangular
lattice dual to the hexagonal lattice. The lattice is
placed in a uniform magnetic field so that the flux
through each rhombus is �0 2/ . The geometry is cho-
sen in such a way that the flux, � s through each
David's star is a half-integer multiple of �0: � s �
� �( )n /s 1 2 0� *. Finally, globally the lattice con-
tains a number, K, of big openings (the size of the
opening is much larger than the lattice constant, a lat-
tice with K � 1 is shown in Fig. 1,a). The dimension of
the protected space will be shown to be equal 2K . The
system is characterized by the Josephson energy,
E / e IJ c� ( )� 2 , of each contact and by the capaci-
tance matrix of the islands (vertices of the lattice).
We shall assume (as is usually the case) that the ca-
pacitance matrix is dominated by the capacitances of
individual junctions, we write the charging energy as
E e / CC � 2 2 . The «phase regime» of the network men-
tioned above implies that E EJ C� . The whole system
is described by the Lagrangian
L � � � � �
1
16
2
E
E a
Cij
i j J i j ij
( )
( � � ) cos ( )� � � � ,
(1)
where � i are the phases of individual islands and aij
are chosen to produce the correct magnetic fluxes.
The Lagrangian (1) contains only gauge invariant
phase differences, � � �ij i j ija� � � , so it will be
convenient sometimes to treat them as indepen-
dent variables satisfying the constraint �
� ij �
� �2 20� �� �
/ n, where the sum is taken over
closed loop
and n is arbitrary integer.
As will become clear below, it is crucial that the
degrees of freedom at the boundary have dynamics
identical to those in the bulk. To ensure this one needs
to add additional superconducting wires and
Josephson junctions at the boundary. There are a few
ways to do this, two examples are shown in Fig. 1,a
and Fig. 1,b: type I boundary (entire length of bound-
aries in Fig. 1,a, parts AB,CD of Fig. 1,b) and type II
Topologically protected quantum states and quantum computing in Josephson junctions arrays
Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 843
�S
�0 /2
�m
a b J
A
B
C
D
Fig. 1. Examples of the proposed Josephson junction ar-
ray. Thick lines show superconductive wires, each wire
contains one Josephson junction as shown in detailed view
of one hexagon. The width of each rhombi is such that the
ratio of the area of David's star to the area of one rhombi
is odd integer. The array is put in magnetic field such that
the flux through each elementary rhombus and through
each David's star (inscribed in each hexagon) is half inte-
ger. Thin lines show the effective bonds formed by the ele-
mentary rhombi. The Josephson coupling provided by
these bonds is �-periodic. Array with one opening, gener-
ally the effective number of qubits, K is equal to the num-
ber of openings. The choice of boundary condition shown
here makes superconducting phase unique along the entire
length of the outer (inner) boundary, the state of the en-
tire boundary is described by a single degree of freedom.
The topological order parameter controls the phase differ-
ence between inner and outer boundaries. Each boundary
includes one rhombus to allow experiments with flux pen-
etration; magnetic flux through the opening is assumed to
be ( )1 2 20/ m /� � with any integer m (a). With this choice
of boundary circuits the phase is unique only inside the
sectors AB and CD of the boundary; the topological de-
gree of freedom controls the difference between the phases
of these boundaries. This allows a simpler setup of the ex-
perimental test for the signatures of the ground state de-
scribed in the text, e.g., by a SQUID interference experi-
ment sketched here that involves a measuring loop with flux
�m and a very weak junction J balancing the array (b).
* The flux � s can be also chosen so that it is an integer multiple of �0: this would not change significantly the final
results but would change intermediate arguments and make them longer, so for clarity we discuss in detail only the
half-integer case here. Note, however, that the main quantitative effect of this alternative choice of the flux is beneficial:
it would push up the phase transition line separating the topological and superconducting phases shown in Fig. 3 for
half integer case.
boundary (BC, AD). For both types of boundaries one
needs to include in each boundary loop the flux which
is equal to 1
4 0Zb� , where Zb is coordination number
of the dual triangular lattice site. For instance, for the
four coordinated boundary sites one needs to enclose
the integer flux in these contours. In type I boundary
the entire boundary corresponds to one degree of free-
dom (phase at some point) while type II boundary
includes many rhombi so it contains many degrees of
freedom.
Note that each (inner and outer) boundary shown
in Fig. 1,a contains one rhombus; we included it to al-
low flux to enter and exit through the boundary when
it is energetically favorable.
3. Ground state, excitations and topological
order
In order to identify the relevant degrees of freedom
in this highly frustrated system we consider first an in-
dividual rhombus. As a function of the gauge invari-
ant phase difference between the far ends of the rhom-
bus the potential energy is
U E / /ij J ij ij( ) (| cos ( )| | sin ( )| ).� � �� � �2 2 2 (2)
This energy has two equivalent minima, at
� �ij /� � 2 which can be used to construct elemen-
tary unprotected qubit, see [26]. In each of these
states the phase changes by � �/4 in each junction
clockwise around the rhombus. We denote these
states as � and � respectively. In the limit of large
Josephson energy the space of low energy states of the
full lattice is described by these binary degrees of
freedom, the set of operators acting on these states is
given by Pauli matrices � ab
x y z, , . We now combine
these rhombi into hexagons forming the lattice shown
in Fig. 1. This gives another condition: the sum of
phase differences around the hexagon should equal to
the flux � s through each David's star inscribed in
this hexagon. The choice � �ij /� 2 is consistent with
flux � s that is equal to a half integer number of flux
quanta. This state minimizes the potential energy (2)
of the system. This is, however, not the only choice.
Although flipping the phase of one rhombus changes
the phase flux around the star by � and thus is pro-
hibited, flipping two, four and six rhombi is allowed;
generally the low energy configurations of U( )� sat-
isfy the constraint
�Pa ab
z
b
� ��� 1 , (3)
where the product runs over all neighbors, b, of site a.
The number of (classical) states satisfying the con-
straint (3) is still huge: the corresponding configura-
tional entropy is extensive (proportional to the num-
ber of sites). We now consider the charging energy of
the contacts, which results in the quantum dynamics
of the system. We show that it reduces this degener-
acy to a much smaller number 2K . The dynamics of
the individual rhombus is described by a simple
Hamiltonian H t x� ~� but the dynamics of a rhombus
844 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8
L.B. Ioffe, M.V. Feigel’man, and B. Douçot
� � '
a
b
Fig. 2. Location of the discrete degrees of freedom respon-
sible for the dynamics of the Josephson junction array
shown in Fig. 1. The spin degrees of freedom describing
the state of the elementary rhombi are located on the
bonds of the triangular lattice (shown in thick lines)
while the constraints are defined on the sites of this lat-
tice. The dashed line indicates the boundary condition im-
posed by a physical circuitry shown in Fig. 1,a. Contours
� and �� are used in the construction of topological order
parameter and excitations (a). The lattice with K � 3
openings, the ground state of Josephson junction array on
this lattice is 2 8k � fold degenerate (b).
embedded in the array is different because individual
flips are not compatible with the constraint (3). The
simplest dynamics compatible with (3) contains flips
of three rhombi belonging to the elementary triangle,
( , , )a b c , �
( )Q abc ab
x
bc
x
ca
x� � � � and therefore the simplest
quantum Hamiltonian operating on the subspace de-
fined by (3) is
H r Q abc
abc
� �
( )
( )
. (4)
We discuss the derivation of the coefficient r in this
Hamiltonian and the correction terms and their ef-
fects below but first we solve the simplified model
(3), (4) and show that its ground state is «protected»
in the sense described above and that excitations are
separated by the gap*.
Clearly, it is very important that the constraint is
imposed on all sites, including boundaries. Evidently,
some boundary hexagons are only partially complete
but the constraint should be still imposed on the corre-
sponding sites of the corresponding triangular lattice.
This is ensured by additional superconducting wires
that close the boundary hexagons in Fig. 1.
We note that constraint operators commute not
only with the full Hamiltonian but also with individ-
ual �
( )Q abc : [ � , � ]( )P Qa abc � 0. The Hamiltonian (4) with-
out constraint has an obvious ground state, 0 , in
which � ab
x � 1 for all rhombi. This ground state, how-
ever, violates the constraint. This can be fixed noting
that since operators �Pa commute with the Hamil-
tonian, any state obtained from 0 by acting on it by
�Pa is also a ground state. We can now construct a true
ground state satisfying the constraint by
G
Pa
a
�
��1
2
0
�
. (5)
Here ( � )1 2� P /a is a projector onto the subspace
satisfying the constraint at site a and preserving the
normalization.
Obviously, the Hamiltonian (4) commutes with
any product of �Pa which is equal to the product of � ab
z
operators around a set of closed loops. These integrals
of motion are fixed by the constraint. However, for a
topologically non-trivial system there appear a num-
ber of other integrals of motion. For a system with K
openings a product of � ab
z operators along contour, �
that begins at one opening and ends at another (or at
the outer boundary, see Fig. 2)
Tq ab
z
q
^
( )
� ��
�
(6)
commutes with Hamiltonian and is not fixed by the
constraint. Physically these operators count the par-
ity of «up» rhombi along such contour. The presence
of these operators results in the degeneracy of the
ground state. Note that multiplying such operator by
an appropriate �Pa gives similar operator defined on
the shifted contour so all topologically equivalent
contours give one new integral of motion. Further,
multiplying two operators defined along the contours
beginning at the same (e.g., outer) boundary and
ending in different openings, A, B is equivalent to the
operator defined on the contour leading from A to B,
so the independent operators can be defined (e.g.) by
the set of contours that begin at one opening and ends
at the outer boundary. The state G constructed
above is not an eigenstate of these operators but this
can be fixed defining
G
c T
Gf
q q
q
�
�
�
1
2
�
, (7)
where cq � �1 is the eigenvalue of Tq
^
operator defined
on contour � q. Equation (7) is the final expression
for the ground state eigenfunctions.
Construction of the excitations is similar to the
construction of the ground state. First, one notices
that since all operators �Qabc commute with each other
and with the constraints, any state of the system can
be characterized by the eigenvalues (Qabc � �1) of
�Qabc. The lowest excited state correspond to only one
Qabc being �1. Notice that a simple flip of one
rhombus (by operator �
( )ab
z somewhere in the system
changes the sign of two Qabc eigenvalues corres-
ponding to two triangles to which it belongs. To
change only one Qabc one needs to consider a con-
tinuous string of these flip operators starting from the
boundary: ( ) ( )abc v abc� 0 with v abc cd
z
( ) ( )
,� �
�
�
�
where the product is over all rhombi ( )cd that belong
to the path �� that begins at the boundary and ends at
( )abc (see Fig. 2, which shows one such path). This
operator changes the sign of only one Qabc, the one
that corresponds to the «last» triangle. This const-
ruction does not satisfy the constraint, so we have to
apply the same «fix» as for the ground state const-
ruction above
Topologically protected quantum states and quantum computing in Josephson junctions arrays
Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 845
* In a rotated basis � �x z� , � �z x� this model is reduced to a special case of Z2 lattice gauge theory [27,28] which
contains only magnetic term in the Hamiltonian with the constraint (3) playing a role of a gauge invariance condition.
v
c T P
vabc
q q
q
a
a
abc( )
^
( )
�
�
� �� �
1
2
1
2
0 (8)
to get the final expression for the lowest energy exci-
tations. The energy of each excitation is 2r. Note that
a single flip excitation at a rhombus ( )ab can be
viewed as a combination of two elementary excita-
tions located at the centers of the triangles to which
rhombus ( )ab belongs and has twice their energy.
Generally, all excited states of the model (4) can be
characterized as a number of elementary excitations
(8), so they give exact quasiparticle basis. Note that
creation of a quasiparticle at one boundary and mov-
ing it to another is equivalent to the �Tq operator, so
this process acts as �q
z in the space of the 2K degener-
ate ground states. As will be shown below, in the
physical system of Josephson junctions these excita-
tions carry charge 2e so that �q
z process is equivalent
to the charge 2e transfer from one boundary to an-
other.
Consider now the matrix elements,
O G O G�� � �� | � | of a local operator, �O, between two
ground states, e.g., of an operator that is composed of
a small number of � ab. To evaluate this matrix
element we first project a general operator onto the
space that satisfies the constraint: � �O O� P P , where
P � �� ( � )1 2P /a
a
. The new (projected) operator is
also local, it has the same matrix elements between
ground states but it commutes with all �Pa . Since it is
local it can be represented as a product of � z and �Q
operators which implies that it also commutes with
all Tq
^
. Thus, its matrix elements between different
states are exactly zero. Further, using the fact that it
commutes with �Pa and Tq
^
we write the difference
between its diagonal elements evaluated between the
states that differ by a parity over contour q as
O O
P
T Oi
i
q� �� �
��0
1
2
0
�
�^
. (9)
This equation can be viewed as a sum of products of
� z operators. Clearly to get a non-zero contribution
each � z should enter even number of times. Each �P
contains a closed loop of six � z operators, so any
product of these terms is also a collection of a closed
loops of � z . In contrast to it, operator �Tq contains a
product of � z operators along the loop �, so the prod-
uct of them contains a string of � z operators along
the contour that is topologically equivalent to �.
Thus, one gets a non-zero O O� �� only for the opera-
tors �O which contain a string of � z operators along
the loop that is topologically equivalent to � which is
impossible for a local operator. Thus, we conclude
that for this model all non-diagonal matrix elements
of a local operator are exactly zero while all diagonal
are exactly equal.
4. Effect of physical perturbations
We now come back to the original physical system
described by the Lagrangian (1) and derive the param-
eters of the model (4) and discuss the most important
corrections to it and their effect. We begin with the
derivation. In the limit of small charging energy the
flip of three rhombi occurs by a virtual process in
which the phase, � i at one (6-coordinated) island, i,
changes by �. In the quasiclassical limit the phase dif-
ferences on the individual junctions are � �ind � � /4;
the leading quantum process changes the phase on one
junction by 3 2�/ and on others by � �/2 changing the
phase across the rhombus � � �� � . The phase differ-
ences, �, satisfy the constraint that the sum of them
over the closed loops remain 2 0�( )n /s� � � . The
simplest such process preserves the symmetry of the
lattice, and changes simultaneously the phase differ-
ences on the three rhombi containing island i keeping
all other phases constant. The action for such process
is three times the action of elementary transitions of
individual rhombi, S0:
r E E S S E /EJ C J C� � �3 4 1 4
0 03 161/ / exp( ), . .
(10)
In the alternative process the phase differences be-
tween i and other islands change in turn, via high en-
ergy intermediate state in which one phase difference
has changed while others remained close to their orig-
inal values. The estimate for this action shows that it
is larger than 3 0S , so (10) gives the dominating con-
tribution. There are in fact many processes that con-
tribute to this transition: the phase of island i can
change by �� and, in addition, in each rombus one
can choose arbitrary the junction in which the phase
changes by �3 2�/ ; the amplitude of all these pro-
cesses should be added. This does not change the re-
sult qualitatively unless these amplitudes exactly can-
cel each other, which happens only if the charge of
the island is exactly half-integer (because phase and
charge are conjugate the amplitude difference of the
processes that are different by 2� is exp ( )2�iq ). We
assume that in a generic case this cancellation does
not occur. External electrical fields (created by, e.g.,
stray charges) might induce non-integer charges
on each island which would lead to a randomness in
the phase and amplitude of r. The phase of r can
be eliminated by a proper gauge transformation
846 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8
L.B. Ioffe, M.V. Feigel’man, and B. Douçot
� � �
ab ab ab
iexp ( )� and has no effect at all. The
amplitude variations result in a position dependent
quasiparticle energy.
We now consider the corrections to the model (4).
One important source of corrections is the difference
of the actual magnetic flux through each rhombus
from the ideal value �0 2/ . If this difference is small it
leads to the bias of «up» versus «down» states, their
energy difference becomes
2 2 2
0
� � �
��
�
d
JE . (11)
Similarly, the difference of the actual flux through
David's star and the difference in the Josephson ener-
gies of individual contacts leads to the interaction be-
tween «up» states:
� � � �H V Vab
ab
ab
z
ab cd
ab cd
ab
z
cd
z
1 � �
( )
( ),( )
( ),( )
, (12)
where Vab � � for uniform field deviating slightly
from the ideal value and V ab cd( ),( ) � 0 for rhombi be-
longing to the same hexagon. Consider now the effect
of perturbations described by �H1, Eq. (12). These
terms commute with the constraint but do not com-
mute with the main term, H, so the ground state is no
longer G� . In other words, these terms create excita-
tions (8) and give them kinetic energy. In the leading
order of the perturbation theory the ground state be-
comes G / r G
ab
z
ab
i� ��
( )
( )
( )
� 4 � . Qualitatively, it
corresponds to the appearance of virtual pairs of
quasiparticles in the ground state. The density of
these quasiparticles is � /r. As long as these
quasiparticles do not form a topologically non-trivial
string all previous conclusions remain valid. How-
ever, there is a nonzero amplitude to form such a
string, it is now exponential in the system size. With
exponential accuracy this amplitude is ( )� / r L2
which leads to an energy splitting of the two ground
state levels and the matrix elements of typical local op-
erators of the same order
E E O O / r L
� � � �� �� � ( )� 2 .
The physical meaning of the v abc( ) excitations be-
comes more clear if one consider the effect of the addi-
tion of one � z operator to the end of the string defin-
ing the quasiparticle: it results in the charge transfer
of 2e across this last rhombus. To prove this, note that
the wave function of a superconductor corresponding
to the state which is symmetric combination of � and
� is periodic with period � and thus corresponds to
charge which is multiple of 4e while the antisymmetric
corresponds to charge ( )2 1 2n e� . The action of one � z
� z induces the transition between these states and
thus transfers the charge 2e. Thus, these excitations
carry charge 2e. Note that continuous degrees of free-
dom are characterized by the long range order in
cos ( )2� and thus correspond to the condensation of
pairs of Cooper pairs. In other words, this system
superconducts with elementary charge 4e and has a
gap, 2r, to the excitations carrying charge 2e. Similar
pairing of Cooper pairs was shown to occur in a chain
of rhombi in a recent paper [29]; formation of a classi-
cal superconductive state with effective charge 6e in
frustrated Kagome wire network was predicted in
[30].
The model (4) completely ignores the processes
that violate the constraint at each hexagon. Such pro-
cesses might violate the conservation of the topologi-
cal invariants �Tq and thus are important for a long
time dynamics of the ground state manifold. In order
to consider these processes we need to go back to the
full description involving the continuous supercon-
ducting phases � i . Since potential energy (2) is peri-
odic in � it is convenient to separate the degrees of
freedom into continuous part (defined modulus �) and
discrete parts. Continuous parts have a long range or-
der: � � � �cos ( )2 2 10� � r . The elementary excita-
tions of the continuous degrees of freedom are har-
monic oscillations and vortices. The harmonic
oscillations interact with discrete degrees of freedom
only through the local currents that they generate.
Further the potential (2) is very close to the qua-
dratic, so we conclude that they are practically de-
coupled from the rest of the system. In contrast to this
vortices have an important effect. By construction,
the elementary vortex carries flux � in this problem.
Consider the structure of these vortices in a greater de-
tail. The superconducting phase should change by 0 or
2� when one moves around a closed loop. In a half vor-
tex this is achieved if the gradual change by � is com-
pensated (or augmented) by a discrete change by � on
a string of rhombi which costs no energy. Thus, from
the viewpoint of discrete degrees of freedom the posi-
tion of the vortex is the hexagon where constraint (3)
is violated. The energy of the vortex is found from the
usual arguments
E R
E
R c cv
J( ) (ln ), . ;� � �
�
4 6
12 (13)
it is logarithmic in the vortex size, R. The process
that changes the topological invariant �Tq is the one in
which one half vortex completes a circle around an
opening. The amplitude of such process is exponen-
tially small: (~ ( ))t/E Dv
�, where ~t is the amplitude
to flip one rhombus and � is the length of the shortest
path around the opening. In the quasiclassical limit
the amplitude ~t can be estimated analogously to (10):
Topologically protected quantum states and quantum computing in Josephson junctions arrays
Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 847
~ exp ( )t E E SJ Q� � 0 . The half vortices would ap-
pear in a realistic system if the flux through each
hexagon is systematically different from the ideal half
integer value. The presence of free vortices destroys
topological invariants, so a realistic system should ei-
ther be not too large (so that deviations of the total
flux do not induce free vortices) or these vortices
should be localized in prepared traps (e.g., David's
stars with fluxes slightly larger or smaller than � s).
If the absence of half vortices the model is equivalent
to the Kitaev model [7] placed on triangular lattice in
the limit of the infinite energy of the excitation vio-
lating the constraints.
Quantitatively, the expression for the parameters
of the model (4) becomes exact only if E EJ C�� .
One expects, however, that the qualitative conclu-
sions remain the same and the formulas derived above
provide reliable estimates of the scales even for
E EJ C� , provided that charging energy is not so
large as to result in a phase transition to a different
phase. One expects this transition to occur at
E EC J
* � � with � � 1 which exact value can be re-
liably determined only from numerical simulations.
Practically, since the perturbations induced by flux
deviations from �0 are proportional to ��� �0) !
! ( )E /rJ and r becomes exponentially small at small
EC, the optimal choice of the parameters for the phy-
sical system is E EC C� * . We show the schematics of
the phase diagram in Fig. 3. We assume here that tran-
sition to insulating phase is direct, another alternative
is the intermediate phase in which the energy of the
vortex becomes finite instead of being logarithmic. If
this phase indeed exists it is likely to have properties
more similar to the one discussed in [7] (in the next
Section we consider generalized JJ array, where such
am intermediate phase does exist). The «topological»
phase is stable in a significant part of the phase dia-
gram. The phase transition between «topological su-
perconductor» and usual superconductor belongs to
the class of quantum spin-1 2/ 2D Ising model on a
hexagonal lattice, placed in a transverse field:
H s s r sj
x
ij
i
x
i
z
i
Ising � � �
�
( )
. (14)
Here i j, denote sites of hexagonal lattice, eigenvalue
of operator si
z measures parity of Cooper pairs num-
ber ni on the ith island: n si i
z( ) ( )mod 2 11
2� � ; pa-
rameters r and � are defined in Eqs. (10), (11). As
long as the ratio " � r/ � is larger than some critical
value " c, ground state obeys even number of Cooper
pairs on each island, which corresponds to our «topo-
logical superconductor» phase. The values of " c for
square, triangular and cubic lattices was found via
quantum Monte—Carlo simulations [31]; in particu-
lar, " c
triang � �4 6 0 3. . , and " c
square � �2 7 0 3. . . There is
no available data for a hexagonal lattice; based upon
above-cited results one could estimate " c
hex � �2 0 5. .
Furthermore, since the vortex excitations have log-
arithmic energy, we expect that this phase survives at
finite temperatures as well. In the thermodynamic
limit, at T � 0 one gets a finite density of 2e carrying
excitations (n r/Tv � �exp ( )2 ) but the vortices re-
main absent as long as temperature is below BKT-like
depairing transition for half-vortices.
5. Aray 2: Topological insulator
Generally, increasing the charging energy in a
Josephson junction array makes it an insulator. This
transition is due to an increase of phase fluctuations in
the original array and the resulting appearance of free
vortices that form a superfluid of their own. The new
situation arises in topological superconductor because
it allows half-vortices. Two scenarios are now possi-
ble. The «conventional» scenario would involve con-
densation of half-vortices since they are conjugate to
4e charges. In this case we get an insulator with ele-
mentary excitations carrying charge 4e. An alternative
is condensation of full vortices (pairs of half vortices)
with a finite gap to half vortices. In this case the ele-
mentary excitations are charge 2e objects. Similar
fractionalization was discussed in the context of high
Tc superconductors in [19,32] and in the context of
spin or quantum dimer systems in [22,33–36,]. Such
insulator acquires interesting topological properties
on a lattice with holes because each hole leads to a
new binary degree of freedom which describes pres-
ence or absence of half vortex. The energies of these
848 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8
L.B. Ioffe, M.V. Feigel’man, and B. Douçot
0.2
0 �
SC
SCT
Ins
�� �d 0/
E /EC J
Fig. 3. Schematic of the phase diagram for half integer � s
at low temperatures: ��d is the deviation of the magnetic
flux through each rhombus from its ideal value. SC stands
for usual superconducting phase, SCT for the phase with
cos ( )2� long range order of the continuous degrees of free-
dom and discrete topological order parameter discussed ex-
tensively in the bulk of the paper. The SCT phase and SC
phase are separated by 2D quantum Ising phase transition.
states are equal up to corrections which vanish expo-
nentially with the size of the holes. These states can-
not be distinguished by local measurements and have
all properties expected for a topological insulator.
They can be measured, however, if the system is adia-
batically brought into the superconductive state by
changing some controlling parameter. Here we pro-
pose a modification of the «topologically supercon-
ductive» array that provides such control parameter
and, at the same time, allows us to solve the model
and compute the properties of the topological insula-
tor. The key idea of this modification is to allow full
(of flux �0) vortices move with large amplitude be-
tween plackets of the hexagonal lattice, so that they
lower their energy due to delocalization, and eventu-
ally Bose-condense, while half-vortices are still kept
(almost) localized.
Consider array shown in Fig. 4 that contains
rhombi with junctions characterized by Josephson and
charging energies E EJ C� and weak junctions with
� �J C CE�� �� . Each rhombus encloses half of a flux
quantum leading to an exact degeneracy between the
two states of opposite chirality of the circulating cur-
rent [13,29]. This degeneracy is a consequence of the
symmetry operation which combines the reflection
about the long diagonal of the rhombus and a gauge
transformation needed to compensate the change of
the flux � �0 02 2/ /� � . This gauge transformation
changes the phase difference along the diagonal by �.
This Z2 symmetry implies the conservation of the par-
ity of the number of pairs at each site of the hexagonal
lattice and is the origin of the Cooper pair binding.
We assume that each elementary hexagon contains ex-
actly k such junctions: in case each link contains one
weak junction k � 6, but generally it can take any
value k # 1. As will be shown below, the important
condition is the number of weak junctions that one
needs to cross in the elementary loop. Qualitatively, a
value k # 1 ensures that it costs a little to put vortex in
any hexagon.
For the general arguments that follow below the
actual construction of the weak links is not important,
however, practically it is difficult to vary the ratio of
the capacitance to the Josephson energy so weaker
Josephson contact usually implies larger Coulomb en-
ergy. This can be avoided if weak contact is made from
Josephson junction loop frustrated by magnetic field.
The charging energy of this system is half the charging
energy of the individual junction while the effective
Josephson junction strength is �J E� 2 0 0��� $� � ,
where E0 is the Josephson energy of each contact and
�� � �� � 0 2/ is the difference of the flux from half
flux quanta. This construction also allows to control
the system under varying the magnetic field.
In these conditions the whole array is insulating.
Assume that �J sets the lowest energy scale in this
problem (the exact condition will be discussed be-
low). The state of the array is controlled by discrete
variables uij � 0 1, which describe the chiral state of
each rhombus and by continuous phases � ij that spec-
ify the state of each weak link (here and below i j,
denote the sites of hexagonal lattice). If Josephson
coupling �J % 0, different islands are completely de-
coupled and potential energy does not depend on dis-
crete variables uij . For small �J we can evaluate its ef-
fect in the perturbation theory:
Topologically protected quantum states and quantum computing in Josephson junctions arrays
Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 849
�
A
B
Fig. 4. Schematics of the array. The main figure: Global
structure of the array. Discrete variables controlling the
low energy properties are defined on the links of the hex-
agonal lattice. Generally, the lattice might have K big
holes, here we show K � 1 example. Zoom in: Each inner
bond of the lattice contains a rhombus made out of four
Josephson junctions; some bonds also contain an effective
weaker link made of two Josephson junctions so that each
hexagon of the lattice contains k � 3 such links. The flux
through each rhombus is half-flux quanta, �0 2/ , the flux
through a loop constituting a weak link is close to half
flux quanta � � �� �0 2/ � . The boundary of the lattice
contains rhombi and weak links so that each boundary
plaquette has the same number, k, of weak links as the
bulk hexagon.
V u W u W
k
kij
k
J
J
C
k
( ) cos ( ),
!
� � �
&
'
((
)
*
++
�
�
hex
�
�
�8
1
.
(15)
This potential energy lowers the relative energy of
classical configurations of uij that satisfy the con-
straint u
hex
% 0 (mod 2) but it does not prohibit con-
figurations with u
hex
% 1 (mod 2).
Consider now the dynamics of discrete variables.
Generally, two types of tunneling processes are possi-
ble. In the first type the phase changes by � across
each of the three rhombi that have a common site. This
is the same process that gives the leading contribution
to the dynamics of the superconducting array [13], its
amplitude is given (in the quasiclassical approxima-
tion) by Eq. (10) above. In the second type of process
the phase changes across one rhombus and across one
weak junction. Because the potential energy of the
weak junction is assumed to be very small the main ef-
fect of the weak junction is to change the kinetic en-
ergy. The total kinetic energy for this process is the
sum of the terms due to the phase across the rhombi
and across the weak link. Assuming that these phase
variations are equal and opposite in sign, the former is
about EC
�1 2� while the latter �C
�1 2� , so the effective
charging energy of this process is ~ ( )E EC C C� �� � �1 1 1
� .
For �C CE�� this charging energy is small and such
process is suppressed. Thus, in these conditions the
dominating process is the simultaneous flip of three
rhombi as in the superconducting case. In the follow-
ing we restrict ourselves to this case. Further, we shall
assume that r W�� so that in the leading order one
can neglect the potential energy compared to the ki-
netic energy corresponding to the flip of three rhombi.
As W is increased by turning on �J the continuous
phase � ij orders and the transition into the supercon-
ducting state happens at � �J C� . At larger �J , W be-
comes �J and with a further increase of �J , for �J r��
vortices completely disappear from the low energy
spectrum and the array becomes equivalent to the one
studied in [13].
The low energy states are the ones that minimize
the kinetic energy corresponding to simultaneous flip
processes:
H rT ij
x
j ii
� � �
�
( )
. (16)
Here j i( ) denote the nearest neighbors of site i, � ij
x is
the operator that flips discrete variables uij and r is
given by (10). The states minimizing this energy sat-
isfy the gauge invariance condition
� ij
x
j i( )
� �, , . (17)
The Hilbert space of states that satisfy the condition
(17) is still huge. If all weaker terms in the
Hamiltonian are neglected all states that satisfy (17)
are degenerate. These states can be visualized in terms
of half vortices positioned on the sites of the dual lat-
tice, a b, . Indeed, a convenient way to describe differ-
ent states that satisfy (17) is to note that operator
� ij
x
j i( )� does not change the value of uijhex
for
second type of tunneling processes. Thus, one can fix
the values of u vij ahex
� on all hexagonal pla-
quettes, a and impose the constraint (17). In physical
terms the binary values vi � 0 1, describe the positions
of half-vortices on dual (triangular) lattice. This de-
generacy between different states is lifted when the
subdominant terms are taken into account. The main
contribution to the potential energy of these half-vor-
tices comes from (15), it is simply proportional to
their number. The dynamics of these vortices is due to
the processes in which only one rhombus changes its
state and the corresponding flip of the phase accross
the weak junction. The amplitude of this process is
~ exp ( ), .~ ~
~r E E S S
E
E
J
/
C
/ J
C
� � �3 4 1 4
0 0 161 .
In this approximation the effective Hamiltonian
controlling these vortices becomes (cf., Eq. (14)):
H r Wv a
x
ab
b
x
a
z
a
� � �
~
( )
� � � , (18)
where operators � a act in the usual way on the states
with/without half-vortices at plaquette a and the
first sum runs over adjacent plaquettes ( )ab . This
Hamiltonian describes an Ising model in a transverse
field. For small W/r c
~ � �" 1 its ground state is «dis-
ordered»: � z � 0 but � x � 0 while for W/r c
~ � "
it is «ordered»: � z � 0, � x � 0. The critical value
of transverse field is known from extensive numerical
simulations [31]: " c � �4 6 0 3. . for triangular lattice.
The «disordered» state corresponds to the liquid of
half-vortices, while in the «ordered» state the density
of free half-vortices vanishes, i.e., the ground state
contains even number of half-vortices so the total
vorticity of the system is zero. To prove this we start
from the state � which is the ground state at
~r/W � 0 and consider the effect of ~
( )r b
x
ab a
x� �
in
perturbation theory. Higher energy states are sepa-
rated from the ground state by the gap W so each or-
der is finite. Further in each order operator � �a
x
b
x cre-
ates two more half-vortices proving that the total
number of half-vortices remains even in each order.
850 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8
L.B. Ioffe, M.V. Feigel’man, and B. Douçot
The states with odd number of half-vortices have a
gap -(~ )r/W which remains non-zero for W/r c
~ � " .
In terms of the original discrete variable defined on
the rhombi the Hamiltonian (18) becomes
H r Wu ij
x
ij
ij
z
j ii
� � �
�
~
( ) ( )
� � , (19)
where �-operators act on the state of each rhombus.
This Hamiltonian commutes with the constraint (17)
and is in fact the simplest Hamiltonian of the lattice
Z2 gauge theory. The disordered regime corresponds
to a confined phase of this Z2 gauge theory, leading
to elementary 4e charge excitations and the ordered
regime to the deconfined phase.
Consider now the system with non-trivial topology,
e.g., a hole. In this case the set of variables va is not
sufficient to determine uniquely the state of the sys-
tem, one has to supplement it by the variable
v uabL0 �
, where sum is taken over a closed contour
L that goes around the hole. Physically, it describes
the presence/absence of the half-vortex in the hole.
The effective Hamiltonian of this additional variable
has only kinetic part because presence or absence of
half vortex in a hole which has l weak links in its per-
imeter gives potential energy W c /J J C
l
0 � � � �( )
which is exponentially small for l �� 1. The kinetic
part is similar to other variables: H r a
x
a I
x
0 0� �
.
~ ,� �
it describes a process in which half-vortex jumps from
the hole into the inner boundary, I, of the system. In
the state with � z � 0 this process increases the en-
ergy of the system by ~ (~ / )W r W ( ~ ( )W W0 � and
~ ( )W c" � 0). In the state with � x � 0 it costs noth-
ing. Thus, the process in which half-vortex jumps
from the hole into the system and another half-vortex
exits into the outside region appears in the second or-
der of the perturbation theory, the amplitude of this
process is t r gv abi I j O�
. .
~
,
2 , where sum is per-
formed over all sites of the inner (I) and outer (O)
boundaries and
g
H Eab a
x
b
x�
�
� �
1
0
has a physical meaning of the half vortex tunneling
amplitude from inner to outer boundaries. At small
~r/W, deep in the insulating phase, we can estimate
gab using the perturbation expansion in ~r/W: the
leading contribution appears in | |a b� th order of the
perturbation theory, thus g r/Wab
a b/ �(~ )| | . Thus
for small ~r/W the tunneling amplitude of the half
vortex is exponentially small in the distance, L, from
the outer to the inner boundary; we expect that it re-
mains exponentially small for all ~r/W c� " . For
~r/W c� " this amplitude is of the order of ~r /W2 and
therefore is significant.
In a different language, in the system with a hole
we can construct a topological invariant P � � �� ij
x
(contour � is shown in Fig. 4) which can take values
�1 . Note that now the contour goes via triangular lat-
tice sites (where vortices are defined), whereas in the
first (superconductive array) version the correspond-
ing path was drawn via sites of basic hexagonal lat-
tice. The same arguments as used for the supercon-
ducting array show that any dynamics consistent with
constraint (17) preserves P . Thus, formally, the prop-
erties of the topological insulator are very similar to
the properties of the topological superconductor dis-
cussed in [12], if one replaces the words Cooper pair
by half-vortex and vice versa. We summarize this du-
ality in the table.
Note that at small ~r/W � 0 the ground state of the
Hamiltonian (19) satisfies the condition (17) and
minimizes the second term in (19), i.e., satisfies the
condition � ij
z
j i( )� �, , ; it can be explicitly writ-
ten as
0
1
2
1ins � � �� � �
i
ij
z
j i kl
kl
( ) .
( )
� (20)
Topologically protected quantum states and quantum computing in Josephson junctions arrays
Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 851
Table
The typical properties of topological superconductor and insulator
State Topological superconductor Topological insulator
Ground state Condensate of 4e charges Condensate of 2� phase vortices
Fluxons Gapful, charge 2e Gapful, � phase vortices
Pseudocharges Half fluxes with energy � � EJ Llog Charge 2e with � �� �2 0r C L c/c� min [log ,log ( )]*
Ground state degeneracy Charge on the inner boundary mod 4e Number of � vortices inside the hole mod 2
Ground state splitting ( )( )�� �0 EJ/r L (~ )r/W L
C o m m e n t: * here � is the number � 1, c is the capacitance of a weak link and c0 is the self-capacitance of an island.
This state is a linear superposition of the degenerate
states with P � 1 and P � �1:
0
1
2ins ins ins� � � �( ) (21)
and it coincides with the ground state G of discrete
variables in the superconducting array (cf., Eq.(5)).
The orthogonal superposition of P � �1 states,
1
1
2ins ins ins� � � �( ) , (22)
corresponds to the half-vortex inside the hole. The en-
ergy difference between the above two states E E1 0�
is exponentially small in the insulating state of the
array, whereas it is large in the superconductive state.
6. Quantum manipulations
We now discuss the manipulation of the protected
states formed in this system. We start from super-
conductive version of array.
First, we note that here the topological invariant
Tq
^
has a simple physical meaning — it measures the
total phase difference (modulus 2�) between the inner
and outer boundaries. In an array with even number of
rhombi between internal and outer boundaries, the
state with eigenvalue cq � 1 has phase difference 0,
whereas the state with eigenvalue cq � �1 has phase
difference �. It means that measuring this phase differ-
ence measures the state of the qubit in the same basis
where Tq
^
is diagonal. For the following discussion we
define a set of Pauli matrices � q
x y z, , acting in the 2 2!
qubit space, such that � q
z
qT% .
Introducing a weak coupling between these bound-
aries by a very weak Josephson circuit (characterized
by a small energy eJ) would change the phase of these
states in a controllable manner, e.g., in a unitary
transformation
U ie tz
J q
z� exp ( ) .� (23)
The transformation coupling two qubits can be ob-
tained if one introduces a weak Josephson circuit that
connects two different inner boundaries (correspond-
ing to different qubits). Namely, it will produce oper-
ation
U ie tp q
z
J q
z
p
z
, exp ( )� � � . (24)
Analogously, the virtual process involving half vortex
motion around the opening gives the tunneling ampli-
tude, �t between topological sectors, e.g., unitary
transformation U itx
t q
x� exp ( )� � . This tunneling
can be controlled by magnetic field if the system is
prepared with some number of vortices that are
pinned in the idle state in a special plaquette, where
the flux is integer. The slow (adiabatic) change of
this flux to towards a normal (half-integer) value
would release the vortex and result in the transitions
between topological sectors with � t t/D� ~ 2.
These operations are analogous to usual operations
on a qubit and are prone to usual source of errors. This
system, however, allows another type of operation
that are naturally discrete. As we show above the
transmission of the elementary quasiparticle across the
system changes its state by � q
z . This implies that a dis-
crete process of one pair transfer across the system is
equivalent to the � q
z transformation. Similarly, a con-
trolled process in which a vortex is moved around a
hole results in a discrete � q
x transformation. More-
over, this system allows one to make discrete transfor-
mations such as � x z, . Consider, for instance, a pro-
cess in which, by changing the total magnetic flux
through the system, one half vortex is placed in a cen-
ter of the system shown in Fig. 1,b and then released.
It can escape through the left or through the right
boundary, in one case the state does not change, in an-
other it changes by � x . The amplitudes add resulting
in the operation ( )1 2� i /x� . Analogously, using the
electrostatic gate(s) to pump one charge 2e from one
boundary to the island in the center of the system and
then releasing it results in a ( )1 2� i /z� transforma-
tion. This type of processes allow a straightforward
generalization for the array with many holes: there ex-
tra half vortex or charge should be placed at equal dis-
tances from the inner and outer boundaries.
The degenerate ground states in the insulating ar-
ray can be manipulated in the same way as in the su-
perconductor, up to duality (half-vortex � Coo-
per-pair and vice verse). As mentioned above, these
states 0 ins
and 1 ins
correspond to the absence or
presence of the half-vortex inside the hole. We define
by ~ , ,� x y z Pauli operators acting in the space spanned
by 0 ins
and 1 ins
. The adiabatic change of local mag-
netic field that drags one half vortex across the sys-
tem, flips the state of the system, providing us with
the implementation of the operator ~� x acting on the
state of the qubit. Analogously, motion of elementary
charge 2e around the hole changes by � the relative
phase of the states 0 ins
and 1 ins
providing us with
the operator ~� z . Operators ~ ,� x z can be realized in a
way similar to that described for superconductive ar-
ray. Rotation by an arbitrary angle U ix x� exp ( ~ )��
that is an analog of the operator (23) can achieved by
modifying (during time t) parameter r in a way to pro-
duce non-negligible amplitude of half-vortex tunnel-
ling A across the system: � � At. In the same way
852 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8
L.B. Ioffe, M.V. Feigel’man, and B. Douçot
two-qubit entanglement operation can be realized,
which is an analog of operator (24)):
~ exp ( ~ ~ ) ,,U itAp q
x
pq p
x
q
x� � � (25)
in this case Apq is the half-vortex tunnelling ampli-
tude between holes p and q.
7. Physical properties of small arrays
7.1. Superconducting array
Even without these applications for quantum com-
putation the physical properties of this array are re-
markable: it exhibits a long range order in the square
of the usual superconducting order parameter:
cos ( ( ))2 10� �� �r without the usual order:
cos ( )� �0 0� �r ; the charge transferred through
the system is quantized in the units of 4e. This can be
tested in a interference experiment sketched in
Fig. 1,b, as a function of external flux, �m the
supercurrent through the loop should be periodic with
half the usual period. This simpler array can be also
used for a kind of «spin-echo» experiment: applying
two consecutive operations ( )1 2� i /x� described
above should give again a unique classical state while
applying only one of them should result in a quantum
superposition of two states with equal weight.
The echo experiment can be used to measure the
decoherence time in this system. Generally one distin-
guishes processes that flip the classical states and the
ones that change their relative phases. In NMR litera-
ture the former are referred to as longitudinal relax-
ation and the latter as transverse one. The transverse
relaxation occurs when a vortex is created and then
moved around the opening by an external noise. As-
suming a thermal noise, we estimate the rate of this
process � 0
� � �1 ~ exp ( ( ) )t E L /TV . Similarly, the
transfer of a quasiparticle from the outer to the inner
boundary changes the relative phase of the two states,
leading to a longitudinal relaxation. This rate is
proportinal to the density of quasiparticles,
� | |
� � �1 2R r/Texp ( ). The coefficient R depends on
the details of the physical system. In an ideal system
with some nonzero uniform value of � (defined above
(12)) quasiparticles are delocalized and R L� � / 2.
Random deviations of fluxes � r from half-integer
value produce randomness in �, in which case one ex-
pects Anderson localization of quasiparticles due to
off-diagonal disorder, with localization length of the
order of lattice spacing, thus R cL� �� exp ( ) with
c � 1, and � is the typical value of �. Stray charges in-
duce randomness in the values of r, i.e., add some dia-
gonal disorder. When the random part of r, �r becomes
larger than � the localization becomes stronger:
R r L� � �( / )� , where �r is the typical value of �r.
Upon a further increase of stray charge field there ap-
pear rare sites where ri is much smaller than an aver-
age value. Such site acts as an additional openings in
the system. If the density of these sites is significant,
the effective length that controls the decoherence be-
comes the distance between these sites. For typical
E L EV J( ) � � 2 K the transverse relaxation time
reaches seconds for T � 01. K while realistic � /r � 01.
imply that due to a quasiparticle localization in a ran-
dom case the longitudinal relaxation reaches the same
scale for systems of size L � 10; note that temperature
T has to be only somewhat lower than the excitation
gap, 2r, in order to make the longitudinal rate low.
Most properties of the array are only weakly sensi-
tive to the effect of stray charges: as discussed above,
they result in a position-dependent quasiparticle po-
tential energy which has very little effect because
these quasiparticles had no kinetic energy and were lo-
calized anyway. A direct effect of stray charges on the
topologically protected subspace can be also physi-
cally described as a effect of the electrostatic potential
on the states with even and odd charges at the inner
boundary; since the absolute value of the charge fluc-
tuates strongly, this effect is exponentially weak.
7.2. Insulating array
The signature of the topological insulator is the
persistence of the trapped half flux inside the central
hole (see Fig. 4) which can be observed by cycling
magnetic field so as to drive the system back and forth
between insulating and superconducting states. This
trapping is especially striking in the insulator. Experi-
mentally, this can be revealed by driving slowly the
array into a superconducting state and then measuring
the phase difference between opposite points such as A
and B in Fig. 4. In the state with a half vortex the
phase difference is � �/ n2 � while it is �n in the other
state. The �n contribution is due to the usual vortices
that get trapped in a big hole. This slow transforma-
tion can be achieved by changing the strength of weak
links using the external magnetic field as a control pa-
rameter. The precise nature of the superconductive
state is not essential because phase difference � be-
tween points A and B can be interpreted as due to a
full vortex trapped in a hole in a conventional super-
conductor or due to a � periodicity in a topological one
which makes no essential difference. These flux trap-
ping experiments are similar to the ones proposed for
high-Tc cuprates [18,19] with a number of important
differences: the trapped flux is half of �0, the cycling
does not involve temperature (avoiding problems with
excitations) and the final state can be either conven-
tional or topological superconductor.
Topologically protected quantum states and quantum computing in Josephson junctions arrays
Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 853
8. Conclusion
We have shown that a Josephson junction array of
a special types (shown in Fig. 1 and Fig. 4) have de-
generate ground states described by a topological or-
der parameter. The manifold of these states is pro-
tected in the sense that local perturbations have
exponentially weak effect on their relative phases and
transition amplitudes. The simpler array Fig. 1 pos-
sesses superconductive state with topological order.
Its minor modification shown in Fig. 4 may be
brought into both superconductive and insulating
«topological» phases in a controllable manner*. Both
versions of the array are expected to demonstrate very
long coherence time in «spin-echo» — type experi-
ments and to be promising basic elements for scalable
quantum computers.
The main building block of the array is the rhom-
bus which has two (almost) degenerate states, in the
array discussed here these rhombi are assembled into
hexagons but we expect that lattices in which these
rhombi form other structures would have similar prop-
erties. However, the dynamics of these arrays is de-
scribed by quartic (or higher) order spin exchange
terms which have larger barrier in a quasiclassical re-
gime implying that their parameter r is much smaller
than in for the array considered here. This makes them
more difficult to built in the interesting regime.
Acknowledgements
We are grateful to G. Blatter, D. Ivanov, S. Kor-
shunov, A. Larkin, A. Millis, B. Pannetier, E. Serret,
and L. Shchur for the discussions and useful com-
ments, and to ENS (Paris), to LPTMS, Orsay, and to
LSI, Ecole Polytechnique for their hospitality which
allowed this work to be completed.
We acknowledge the support by NSF grant
4-21262. MF was supported by SCOPES program of
Switzerland, Dutch Organization for Fundamental re-
search (NWO), RFBR grant 01-02-17759, the pro-
gram «Quantum Macrophysics» of Russian Academy
of Science and the program «Physics of Quantum
Computing» of Russian Ministry of Industry, Science
and Technology.
1. A. Steane, Rep. Prog. Phys. 61, 117 (1998).
2. A. Ekert and R. Jozsa, Rev. Mod. Phys. 68, 733
(1996).
3. P.W. Shor, in: Proceedings of the 35th Symposium on
the Foundations of Computer Science, Los Alamitos,
CA:IEEE Press (1994).
4. L. Grover, in: Proceedings of the Annual ACM Sym-
posium on the Theory of Computing, ACM Press, New
York (1996).
5. P.W. Shor, Phys. Rev. A52, 2493 (1995).
6. J. Preskill, Proc. Roy. Soc. A454, 385 (1998).
7. A.Yu. Kitaev, Preprint, http://xxx.lanl.gov/abs/quant
— ph/9707021 (1997); I. Kant, in: Critique of Pure
Reason (1781).
8. X.-G. Wen and Q. Niu, Phys. Rev. B41, 9377 (1990).
9. X.-G. Wen, Phys. Rev. B44, 2664 (1991).
10. M. Mezard, G. Parisi, and M. Virasoro, Spin Glass
Theory and Beyond, Word Scientific, Singapore (1997).
11. S.B. Bravyi and A. Kitaev, Preprint, http://xxx.
lanl.gov/abs/quant-ph/0003137; M.H. Freedman, A.
Kitaev, and Z. Wang, Preprint, http://xxx.lanl.gov/
abs/quant — ph/0001071.
12. L.B. Ioffe, M.V. Feigel’man, A. Ioselevich, D. Iva-
nov, M. Troyer, and G. Blatter, Nature 415, 503
(2002).
13. L.B. Ioffe and M.V. Feigel’man, Phys. Rev. B66,
224503 (2002).
14. B. Douçot, M.V. Feigel’man, and L.B. Ioffe, Phys.
Rev. Lett. 90, 107003 (2003).
15. N. Read and B. Chakraborty, Phys. Rev. B40, 7133
(1989).
16. N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991).
17. S. Kivelson, Phys. Rev. B39, 259 (1989).
18. T. Senthil and M.P.A. Fisher, Phys. Rev. Lett. 86,
292 (2001).
19. T. Senthil and M.P.A. Fisher, Phys. Rev. B63,
134521 (2001).
20. L. Balents, S.M. Girvin, and M.P.A. Fisher, Preprint,
http://xxx.lanl.gov/abs/cond-mat/0110005.
21. A. Paramekanti, L. Balents, and M.P.A. Fisher, Pre-
print, http://xxx.lanl.gov/abs/cond-mat/0203171.
22. G. Misguich, D. Serban, and V. Pasquier, Preprint,
http://xxx.lanl.gov/abs/cond-mat/0204428.
23. O. Motrunich and T. Senthil, Preprint, http://xxx.
lanl.gov/abs/cond-mat/0205170.
24. P. Fendley, R. Moessner, and S.L. Sondhi, Preprint,
http://xxx.lanl.gov/abs/cond-mat/0206159.
25. A.S. Ioselevich, D.A. Ivanov, and M.V. Feigel’man,
Preprint, http://xxx.lanl.gov/abs/cond-mat/0206451.
26. G. Blatter, V.B. Geshkenbein, and L.B. Ioffe, Phys.
Rev. B63, 174511 (2001).
27. F. Wegner, J. Math. Phys. 12, 2259 (1971).
28. R. Balian, J.M. Droufle, and C. Itzykson, Phys. Rev.
D11, 2098 (1975).
29. B. Douçot and J. Vidal, Preprint, http://xxx.
lanl.gov/abs/cond-mat/0202115.
30. K. Park and D.A. Huse, Phys. Rev. B64, 134522
(2001).
31. R.F. Bishop, D.J.J. Farrel, and M.L. Ristig, Int. J.
Mod. Phys. B14, 1517 (2000).
854 Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8
L.B. Ioffe, M.V. Feigel’man, and B. Douçot
* It is possible that array shown in Fig. 1 also has topological insulator phase but we cannot prove it.
32. T. Senthil and M.P.A. Fisher, Phys. Rev. B62, 7850
(2000).
33. N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773
(1991).
34. L. Balents, M.P.A. Fisher, and S.M. Girvin, Phys.
Rev. B65, 224412 (2002).
35. T. Senthil and O. Motrunich, Preprint, http://xxx.
lanl.gov/abs/cond-mat/0201320.
36. R. Moessner, S.L. Sondhi, and E. Fradkin, Phys. Rev.
B65, 24504 (2002).
Note added in proof
The approach to the constructing of topologically
stable superconductive qubits and anomalous super-
conducting states with Cooper-pair paring, that is re-
viewed in the present article, has been further de-
veloped in very recent papers: B. Douçot, M.V.
Feigel’man, L.B. Ioffe, and A.S. Ioselevich, Protected
Qubits and Chern Simons Theories in Josephson-
Junction Arrays, cond-mat/0403712; I.V. Protopo-
pov and M.V. Feigel’man, Theory of 4e Versus 2e
Supercurrent in Frustrated Josephson-Junction
Rhombi Chain, cond-mat/0405170.
Topologically protected quantum states and quantum computing in Josephson junctions arrays
Fizika Nizkikh Temperatur, 2004, v. 30, Nos. 7/8 855
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