Symmetryprotected phases for measurementbased quantum computation
ABSTRACT Ground states of spin lattices can serve as a resource for measurementbased
quantum computation. Ideally, the ability to perform quantum gates via
measurements on such states would be insensitive to small variations in the
Hamiltonian. Here, we describe a class of symmetryprotected topological orders
in onedimensional systems, any one of which ensures the perfect operation of
the identity gate. As a result, measurementbased quantum gates can be a robust
property of an entire phase in a quantum spin lattice, when protected by an
appropriate symmetry.

 SourceAvailable from: de.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: We define generalized cluster states based on finite group algebras in analogy to the generalization of the toric code to the Kitaev quantum double models. We do this by showing a general correspondence between systems with CSS structure and finite group algebras, and applying this to the cluster states to derive their generalization. We then investigate properties of these states including their PEPS representations and global symmetries. Following this, we explore the relationship between the generalized cluster states and the Kitaev quantum double models and sketch a protocol for universal adiabatic topological cluster state quantum computation that makes use of the generalized cluster states.08/2014;  SourceAvailable from: Leon Loveridge[Show abstract] [Hide abstract]
ABSTRACT: We report first steps towards elucidating the relationship between contextuality, measurementbased quantum computation (MBQC) and the nonclassical logic of a topos associated with the computation. We show that in a class of MBQC, classical universality \emph{requires} nonclassical logic, which is "consumed" during the course of the computation, therefore pinpointing another potential quantum computational resource.08/2014;
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Symmetryprotected phases for measurementbased quantum computation
Dominic V. Else,1Ilai Schwarz,1,2Stephen D. Bartlett,1and Andrew C. Doherty1
1Centre for Engineered Quantum Systems, School of Physics,
The University of Sydney, Sydney, NSW 2006, Australia
2Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
Ground states of spin lattices can serve as a resource for measurementbased quantum compu
tation. Ideally, the ability to perform quantum gates via measurements on such states would be
insensitive to small variations in the Hamiltonian. Here, we describe a class of symmetryprotected
topological orders in onedimensional systems, any one of which ensures the perfect operation of the
identity gate. As a result, measurementbased quantum gates can be a robust property of an entire
phase in a quantum spin lattice, when protected by an appropriate symmetry.
Quantum computation exploits quantum entangle
ment to achieve computational speedups. However, cre
ating entanglement between particles in a sufficiently
controlled way to allow for quantum computation has
proved a major technical challenge.
approach is measurementbased quantum computation
(MBQC) [1, 2], where universal quantum computation is
achieved by means of nonentangling operations (namely,
singleparticle measurements) on an already entangled
resource state. The resource state need not be prepared
coherently; instead, one could imagine constructing inter
actions between neighboring spins on a lattice, governed
by a gapped Hamiltonian whose ground state is a uni
versal resource state for MBQC [3–5]. For this approach
to be robust, the capability of ground states to serve as
a resource for MBQC would have be insensitive to small
variations in the Hamiltonian, like a form of quantum
order [3].
One potential
In this Letter, we draw an explicit connection between
MBQC and a type of quantum order called symmetry
protected topological order (SPTO) [6–8].
we will describe a class of quantum phases in which
the perfect operation of the identity gate in MBQC can
be derived directly from the presence of SPTO, and
consequently this perfect operation is a robust prop
erty which is maintained throughout the entire phase.
Our results will be expressed in the context of one
dimensional systems. Such systems are not expected to
allow for universal MBQC, but the ground states of cer
tain 1D spin chains can be used as quantum computa
tional wires [9], meaning, loosely, that through single
particle measurements one can propagate a logical qubit
down the chain while applying singlequbit unitaries.
Later, we will also explain how our results can be ap
plied to higherdimensional systems (which can allow for
universal MBQC) by considering them as ‘quasi1D’.
Specifically,
A wellknown example of a onedimensional system
whose ground state can serve as a quantum computa
tional wire is the AffleckKennedyLiebTasaki (AKLT)
antiferromagnetic spin1 chain [10, 11]. This system lies
in a quantum phase, called the Haldane phase, charac
terized by SPTO and protected by a Z2× Z2 rotation
symmetry [12, 13], so that no symmetryrespecting path
of local Hamiltonians can interpolate between the Hal
dane phase and a product state without crossing a phase
transition.The perfect operation of the identity gate
throughout the Haldane phase has been noticed before
in various guises [14, 15], as well as the strictly weaker
condition of diverging localizable entanglement length
[16]. Our purpose in this Letter will be to exhibit this
property as a direct manifestation of SPTO. As a result,
we can apply our technique more generally to a whole
class of quantum phases characterized by SPTO, includ
ing phases containing the 1D cluster state; qudit cluster
states [17]; and cluster states in higher dimensions. In
addition, we show that gates other than the identity are
not expected to exhibit similar robustness, explaining the
numerical observations in Ref. [15].
Symmetryprotection of the identity gate in correlation
space.—The connection between SPTO and MBQC will
be expressed through the correlation space picture of [18],
which is a particularly natural way to formulate MBQC
on 1D resource states. This picture assumes a resource
state Ψ? that can be represented as a matrixproduct
state (MPS),
?
Ψ? =
k1,...,kN
?RA[kN]A[kN−1]···A[k1]L?
× k1,...,kN?, (1)
where each A[kj], kj= 1,...,d is a linear operator acting
on a Ddimensional vector space (known as the correla
tion space), L? and ?R are states in correlation space,
and d is the dimension of the Hilbert space of each spin.
Here we are assuming translational invariance, for nota
tional simplicity only. When a projective measurement
is performed on the first spin, with outcome ψ?, the ef
fect is to remove the first spin from the chain and induce
an evolution L? → A[ψ]L? on the correlation system,
where we use the notation A[ψ] =?
it for the special case of the Haldane phase. One sys
tem within this phase is the spin1 AKLT chain, for
which the ground state has an exact MPS representa
tion of the form (1), with D = 2. Expressed in the basis
kA[k]?ψk?.
As an introduction to our result, we will first state
arXiv:1201.4877v2 [quantph] 6 Feb 2012
Page 2
2
{x?,y?,z?}, where α? is the zero eigenstate of the spin
1 operator Sαfor α = x,y,z, we have AAKLT[α] = σα,
where σαare the Pauli spin operators. Thus, the AKLT
state has the particular property that there exists a basis,
namely the {x?,y?,z?} basis, such that measurements
in this basis induce an identity evolution (up to Pauli
byproducts) on the correlation system. Additionally, by
measuring in a basis corresponding to a rotated set of
axes, it is possible to execute any singlequbit rotation in
correlation space (up to Pauli byproducts) [11]. There
fore, the AKLT state can be said to act as a quantum
computational wire.
We will now extend our correlationspace analysis be
yond the AKLT chain to other ground states within the
Haldane phase. We confine our discussion to states that
can be exactly represented as an MPS with a bond dimen
sion D that is independent of the system size. Because
arbitrary gapped ground states can be approximated by
MPS [19], we expect that our discussion will apply also
to arbitrary systems in the Haldane phase.
The Haldane phase containing the AKLT chain is pro
tected by the Z2× Z2symmetry generated by the π ro
tations about three orthogonal axes. The action of this
symmetry on a spin1 chain can be written as a tensor
product [u(g)]⊗N, where N is the number of spins, and
u(g) is the appropriate singlespin rotation operator for
each group element g in the symmetry group G = Z2×Z2.
We therefore refer to it as an onsite symmetry.
In general, the invariance of a ground state under such
an onsite symmetry leads to symmetry constraints on
the MPS tensor A[·] used to construct the state’s MPS
representation [7, 8, 20, 21]; we will exploit these con
straints to prove our result. Specifically, under an injec
tivity assumption which we expect to be satisfied in a
gapped phase, we have [7, 8, 20]
V (g)†A[ψ?]V (g) = β(g)A[u(g)†ψ?],
where V (g) is some projective representation of G acting
on the correlation system, and β(g) is a onedimensional
linear representation of G. Now, in general V (g) can
be decomposed as a tensor sum of irreducible projective
representations as V (g) =?
state in the Haldane phase, it is a consequence of Lemma
2 below that only one irrep?V (g) (of dimension 2) appears
V (g) =?V (g) ⊗ Ijunk.
correlation system into a protected subsystem [on which
V (g) acts irreducibly as?V (g)] and a junk subsystem (on
multaneous eigenstates of all the elements u(g). By an
argument involving Schur’s Lemma (given in greater gen
erality in Theorem 1), it follows that the tensor A appear
ing in the MPS representation of the ground state must
(2)
JVJ(g) ⊗ ImJ, where mJ
is the multiplicity of the irrep J in V . For any ground
in this decomposition, so that
(3)
That is, we have a tensor product decomposition of the
which V (g) acts trivially). The states x?,y?,z? are si
take the form
A[α] = σα⊗ Ajunk[α],
for some set of operators Ajunk[α] acting on the junk
subsystem. Recall that A[α] is the evolution induced on
the correlation system when a projective measurement
results in the outcome α?. Thus, Eq. (4) shows that the
ability to induce an identity evolution in the protected
subsystem (up to Pauli byproducts, dependent on the
measurement outcome but independent of the resource
state) by measuring in the {x?,y?,z?} basis is dictated
by the symmetry properties of the MPS tensor; it is a
property not just of the AKLT state, but rather of the
entire Haldane phase.
Another state which can serve as a quantum com
putational wire is the 1D cluster state, which is the
ground state on a row of qubits of the local Hamil
tonian H = −?
it lies within a symmetryprotected phase with respect
to a Z2× Z2 symmetry [22], in this case generated by
?
sites. The simultaneous eigenstate of the onsite sym
metry representation is then {++?,+−?,−+?,−−?},
where ±? =
product basis, so that blocking sites does not change the
singlequbit nature of the measurements). Identical to
the AKLT case above, we again find that the ability to
perform the identity gate by measuring in the appropri
ate basis is maintained throughout the phase. Similar
results hold for the generalization of the cluster state to
ddimensional particles [17], for which the relevant sym
metry group is Zd× Zd.
General statement of the result.—We will now give the
statement and proof of our result in a general setting.
We consider a ground state that is invariant under an
onsite symmetry [u(g)]⊗N, where u(g) is a representa
tion of some symmetry group G. We assume the ground
state has an MPS representation satisfying the symme
try condition (2), and we absorb β(g) into u(g) so that
β(g) = 1. A projective representation V (g) is character
ized by its factor system ω, such that
α = x,y,z,(4)
iZi−1XiZi+1. Like the AKLT state,
the cluster state has an exact MPS representation, and
ievenXi and?
ioddXi. We can treat this symmetry
as onsite provided that we group pairs of qubits into
1
√2(0? ± 1?) (we emphasize that this is a
V (g)V (h) = ω(g,h)V (gh).(5)
An equivalence class of factor systems related by rephas
ing of the operators V (g) is called a cohomology class,
and we denote the cohomology class containing a given
factor system ω as [ω].It was argued in Refs. [7, 8]
that each cohomology class of G corresponds to a distinct
symmetryprotected phase. For example, in the case of
the MPS AAKLT[α] = σα for the AKLT state, where
G = Z2× Z2 = {1,x,y,z}, it can be verified that Eq.
(2) is satisfied with the Pauli projective representation
V (1) = I and V (α) = σα for α = x,y,z. This corre
sponds to a nontrivial cohomology class [not containing
Page 3
3
the trivial factor system ω(g,h) = 1], so that the AKLT
chain lies in a nontrivial symmetryprotected phase.
We now relate the symmetry condition (2), which
holds throughout the entire symmetryprotected phase,
to the operation of gates in the correlationspace pic
ture. We consider the case where the symmetry group
G is a finite abelian group. For simplicity, we will fo
cus on the case where the cohomology class [ω] charac
terizing the symmetryprotected phase is of a particular
type. (An analogous result holds for all nontrivial coho
mology classes, but the structure of correlation space is
more involved in that case.) In particular, we consider
the case where the factor systems contained in [ω] are
maximally noncommutative, meaning that the subgroup
G(ω) = {g ∈ Gω(g,h) = ω(h,g) ∀ h ∈ G} is trivial.
(Note, this condition does not depend on the choice of
the representative ω.) Under these conditions, our main
result can be stated as follows:
Theorem 1. Consider a symmetryprotected phase
characterized by a finite abelian symmetry group and
a maximally noncommutative cohomology class [ω].
Then for any MPS in this phase, there exists a de
composition of the correlation system into protected
and junk subsystems, and a site basis {i?}, such
that measuring in the basis {i?} leads to an identity
gate evolution on the protected subsystem up to an
outcomedependent byproduct Bi. That is to say,
the MPS tensor A has the decomposition
A[i] = Bi⊗ Ajunk[i]. (6)
The byproduct operators Biare unitary and are el
ements of a finite group. Furthermore, they are the
same for all possible MPS in the symmetryprotected
phase.
For example, the factor system for the Pauli projective
representation of Z2×Z2is maximally noncommutative,
and Eq. (4) is a special case of Eq. (6).
Proof of Theorem 1.— We will make use of the fol
lowing consequences of maximal noncommutativity of a
factor system:
Lemma 1. Let ω be a maximally noncommutative fac
tor system of a finite abelian group G. For every linear
character χ of G, there exists an element hχ∈ G such
that, for any projective representation V (g) with factor
system ω,
V (hχ)V (g) = χ(g)V (g)V (hχ), (7)
Proof. We define a homomorphism ϕ : G → G∗, where
G∗is the group of linear characters of G, according to
[ϕ(h)](g) = ω(h,g)ω(g,h)−1. (That ϕ(g) ∈ G∗for all g,
and ϕ is a homomorphism, follows from the associativ
ity condition satisfied by ω, e.g. see Lemma 7.1 in [23]).
Because the kernel of ϕ is G(ω), which is trivial by as
sumption, and G = G∗ for finite abelian groups, it
follows that ϕ is invertible. We then set hχ= ϕ−1(χ). It
can be checked that this satisfies Eq. (7).
Lemma 2. For each maximally noncommutative fac
tor system ω of a finite abelian group G, there exists a
unique (up to unitary equivalence) irreducible projective
representation?V (g) with factor system ω. The dimen
Proof. See [24, 25].
sion of this irreducible representation is
?G.
For an MPS tensor A satisfying the symmetry condi
tion (2), Lemma 2 implies that there exists a tensor prod
uct decomposition of the correlation system into a pro
tected and a junk subsystem such that V (g) acts within
the protected subsystem as?V (g) as in Eq. (3).
ment basis {i?} to be the simultaneous eigenbasis {i?} of
the elements u(g), such that such that u(g)i? = χi(g)i?,
where each χiis a linear representation of G. Expressed
in the basis {i?}, Eq. (2) then becomes
V (g)†A[i]V (g) = χi(g)A[i].
Now we can prove Theorem 1. We choose the measure
(8)
Making use of Eq. (7), we find that
V (g)?V (hχi)†A[i]?=?V (hχi)†A[i]?V (g).
We can now conclude by Schur’s Lemma that
(9)
A[i] =?V (hχi) ⊗ Ajunk[i](10)
for some operators Ajunk[i]. Therefore Theorem 1 holds
with Bi=?V (hχi).
the identity gate, which involves measuring in the simul
taneous eigenbasis of the operators u(g), is symmetry
protected. We will now see that nontrivial gates (i.e.
those involving measurement in a different basis) are not
symmetryprotected.
For example, let us consider a measurement that on the
exact AKLT state would correspond to a rotation by an
angle 2θ about the z axis (up to Pauli byproducts). One
of the possible measurement outcomes is θ? ≡ cosθx?+
sinθy?. Then from the decomposition (4) of the MPS
tensor A for a generic state in the Haldane phase, we
find that
Nontrivial gates.—In Theorem 1, we have proven that
A[θ] = (cosθ)σx⊗Ajunk[x]+(sinθ)σy⊗Ajunk[y].
If Ajunk[x] = Ajunk[y] (as for the exact AKLT state) then
this implies
(11)
A[θ] = [(cosθ)σx+ (sinθ)σy] ⊗ Ajunk[x],(12)
Page 4
4
and the evolution on the protected subsystem is the
same as it would be for the exact AKLT state. How
ever, there is no symmetry constraint that guarantees
Ajunk[x] = Ajunk[y] (because any choice whatsoever for
Ajunkin Eq. (4) gives rise to an MPS satisfying the sym
metry constraints). Therefore, the evolution induced by
measurements in this basis is not fixed by the symmetry;
similar arguments apply to all nontrivial gates.
The preceding discussion of nontrivial gates applies
to systems with only the Z2× Z2 rotation symmetry,
and larger symmetry groups will lead to stronger con
straints on the MPS tensor. In particular, one might ex
pect that for the AKLT state, imposing the full SO(3)
rotation symmetry would lead to all gates being pro
tected, because all gates are achieved by measuring in
the basis {x??,y??,z??} for some rotated orthogonal set
of axes x?,y?,z?. This would indeed be true if only
the spin1/2 projective representation V1/2(g) of SO(3)
appeared in the irrep decomposition of V (g), so that
V (g) = V1/2(g)⊗I. However, all the halfinteger spin rep
resentations of SO(3) have the same cohomology class,
so this will not hold in general. Indeed, the numerical
results of [15] show reduced performance of nontrivial
gates. This should be contrasted with the protocol of
[26], where a logical qubit is encoded into an explicitly
spin1/2 edge mode and particles are adiabatically decou
pled from the chain before being measured. In that case
it was found that all gates operate perfectly throughout
the Haldane phase so long as the full rotational symmetry
is maintained.
Initialization and readout.—Apart from performing
unitary gates in correlation space, the other essential in
gredient for MBQC is the ability to initialize and read
out the state of the correlation system.
verified (in the same way as for nontrivial gates) that
the usual procedures for doing this in the cluster or
AKLT states are not symmetryprotected. However, a
symmetryprotected readout can be achieved through
out the Haldane phase by terminating a finite chain of
spin1’s with a spin1/2, as in [15].
Higherdimensionalsystems.—The
symmetryprotectedtopological
been extended to higherdimensional systems [27, 28],
and we speculate that our results could be generalized
in this context.However, if we consider a ‘quasi1D’
system whose extent in all but one dimension is finite
(but could be set arbitrarily large), then the results of
this Letter can be applied directly.
For example, a 2D cluster model of extent 2N in the
vertical direction (with periodic boundary conditions in
that direction) has a (Z2×Z2)×Nsymmetry, as depicted
in Figure 1. This symmetry is represented in correlation
space by a tensor product of N copies of the Pauli repre
sentation; this is a maximally noncommutative projec
tive representation of the symmetry group. By Lemma
2, the protected subsystem has dimension 2N, which cor
It is easily
notion
has
of
orderrecently
X
Z
Z
X
Z
Z
X
Z
Z
X
Z
Z
X
Z
Z
X
Z
Z
······
FIG. 1.
the 2D cluster state. The other generators can be obtained
from this one by a displacement by 1 horizontally and/or an
even number vertically. The circles represent qubits in the
2D square lattice.
One generator of the (Z2 × Z2)×Nsymmetry in
responds to the capacity for N qubits to be propagated
in the horizontal direction by measuring each ‘site’ (here
a pair of adjacent columns) in a simultaneous eigenba
sis of the symmetry. It can be checked that there exists
such an eigenbasis which is also a product basis over the
qubits making up the site, so that this propagation can
be achieved by singlequbit measurements. These prop
erties are consequences of the symmetry and thus remain
true throughout the symmetryprotected phase.
Conclusion.—In summary, we have identified a class of
symmetryprotected topological orders, each of which en
sures the perfect operation of the identity gate in MBQC
throughout an entire symmetryprotected phase. Such
connections between MBQC and quantum order can be
expected to lead to a greater understanding of the poten
tial for singleparticle measurements on ground states of
quantum spin systems to be a robust form of quantum
computation.
By contrast, we have shown that the perfect operation
of nontrivial gates is a property only of specific systems
within such a phase, contrary to some previous hopes
[3]. However, we have not given a complete characteriza
tion of the operation of nontrivial gates away from these
points, and it is possible that their performance could be
made arbitrarily good by a suitable choice of adaptive
measurement protocol, as in [15].
We acknowledge discussions with A. Miyake, and sup
port from the ARC via the Centre of Excellence in
Engineered Quantum Systems (EQuS), project num
ber CE110001013. I.S. acknowledges support from
the AustraliaIsrael Scientific Exchange Foundation
(AISEF).
[1] Robert Raussendorf and Hans J. Briegel, “A one
way quantum computer,” Phys. Rev. Lett., 86, 5188
(2001); Robert Raussendorf, Daniel E. Browne,
Hans J. Briegel, “Measurementbased quantum computa
tion on cluster states,” Phys. Rev. A, 68, 022312 (2003),
arXiv:quantph/0301052.
and
Page 5
5
[2] H. J. Briegel, D. E. Browne, W. Dur, R. Raussendorf,
and M. Van den Nest, “Measurementbased quantum
computation,” Nat. Phys., 5, 19 (2009), arXiv:0910.1116.
[3] Andrew C. Doherty and Stephen D. Bartlett, “Identify
ing phases of quantum manybody systems that are uni
versal for quantum computation,” Phys. Rev. Lett., 103,
020506 (2009), arXiv:0802.4314.
[4] Xie Chen, Bei Zeng, ZhengCheng Gu, Beni Yoshida,
and Isaac L. Chuang, “Gapped twobody hamiltonian
whose unique ground state is universal for oneway quan
tum computation,” Phys. Rev. Lett., 102, 220501 (2009),
arXiv:0812.4067.
[5] Akimasa Miyake, “Quantum computational capability of
a 2D valence bond solid phase,” Ann. Phys., 326, 1656
(2011), arXiv:1009.3491;
and Robert Raussendorf, “AffleckKennedyLiebTasaki
state on a honeycomb lattice is a universal quantum
computational resource,” Phys. Rev. Lett., 106, 070501
(2011), arXiv:1102.5064.
[6] ZhengChengGu and
entanglementfiltering renormalization
symmetryprotected topological order,” Phys. Rev. B,
80, 155131 (2009), arXiv:0903.1069.
[7] Xie Chen, ZhengCheng Gu,
“Classification of gapped symmetric phases in one
dimensional spin systems,” Phys. Rev. B, 83, 035107
(2011), arXiv:1008.3745.
[8] Norbert Schuch, David P´ erezGarc´ ıa, and Ignacio Cirac,
“Classifying quantum phases using matrix product states
and projected entangled pair states,” Phys. Rev. B, 84,
165139 (2011), arXiv:1010.3732.
[9] D. Gross and J. Eisert, “Quantum computational webs,”
Phys. Rev. A, 82, 040303 (2010), arXiv:0810.2542.
[10] Ian Affleck, Tom Kennedy, Elliott H. Lieb,
Tasaki, “Rigorous results on valencebond ground states
in antiferromagnets,” Phys. Rev. Lett., 59, 799 (1987);
I. Affleck, T. Kennedy, E.H. Lieb, and H. Tasaki, “Va
lence bond ground states in isotropic quantum antiferro
magnets,” Commun. Math. Phys., 115, 477 (1988).
[11] Gavin K. Brennen and Akimasa Miyake, “Measurement
based quantum computer in the gapped ground state of
a twobody hamiltonian,” Phys. Rev. Lett., 101, 010502
(2008), arXiv:0803.1478.
[12] Frank Pollmann, Erez Berg, Ari M. Turner, and Masaki
Oshikawa, “Symmetry protection of topological order
in onedimensional quantum spin systems,”
arXiv:0909.4059.
[13] Frank Pollmann, Ari M. Turner, Erez Berg,
Masaki Oshikawa, “Entanglement spectrum of a topolog
ical phase in one dimension,” Phys. Rev. B, 81, 064439
(2010), arXiv:0910.1811.
[14] J. P. Barjaktarevic, R. H. McKenzie, J. Links, and G. J.
Milburn, “Measurementbased teleportation along quan
tum spin chains,” Phys. Rev. Lett., 95, 230501 (2005),
TzuChieh Wei, Ian Affleck,
XiaoGangWen,
approach
“Tensor
and
and XiaoGang Wen,
and Hal
(2009),
and
arXiv:quantph/0501180.
[15] Stephen D. Bartlett,
Miyake, and Joseph M. Renes, “Quantum computational
renormalization in the Haldane phase,” Phys. Rev. Lett.,
105, 110502 (2010), arXiv:1004.4906.
[16] L.C. Venuti and M. Roncaglia, “Analytic relations be
tween localizable entanglement and string correlations
in spin systems,” Phys. Rev. Lett., 94, 207207 (2005),
arXiv:condmat/0503021.
[17] D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, “Quantum
computation based on dlevel cluster state,” Phys. Rev.
A, 68, 062303 (2003), arXiv:quantph/0304054.
[18] D. Gross, J. Eisert, N. Schuch,
“Measurementbased quantum computation beyond the
oneway model,” Phys. Rev. A, 76, 052315 (2007),
arXiv:0706.3401.
[19] F. Verstraete and J. I. Cirac, “Matrix product states rep
resent ground states faithfully,” Phys. Rev. B, 73, 094423
(2006), arXiv:condmat/0505140; M. B. Hastings, “An
area law for onedimensional quantum systems,” Journal
of Statistical Mechanics: Theory and Experiment, 2007,
P08024 (2007), arXiv:0705.2024.
[20] D. P´ erezGarc´ ıa, M. M. Wolf, M. Sanz, F. Verstraete,
and J. I. Cirac, “String order and symmetries in quan
tum spin lattices,” Phys. Rev. Lett., 100, 167202 (2008),
arXiv:0802.0447.
[21] Sukhwinder Singh, Robert N. C. Pfeifer, and Guifr´ e Vi
dal, “Tensor network decompositions in the presence of
a global symmetry,” Phys. Rev. A, 82, 050301 (2010),
arXiv:0907.2994.
[22] W. Son, L. Amico, R. Fazio, A. Hamma, S. Pascazio,
and V. Vedral, “Quantum phase transition between clus
ter and antiferromagnetic states,” Europhys. Lett., 95,
50001 (2011), arXiv:1103.0251.
[23] Adam Kleppner, “Multipliers on abelian groups,” Math
ematische Annalen, 158, 11 (1965).
[24] R. Frucht, “¨Uber die darstellung endlicher abelscher
gruppen durch kollineationen,” Journal f¨ ur die reine und
angewandte Mathematik, 1932, 16 (1932).
[25] Ya. G. Berkovich and E. M. Zhmud?, Characters of Finite
Groups, Vol. 1 (American Mathematical Society, Provi
dence, Rhode Island, 1998).
[26] Akimasa Miyake, “Quantum computation on the edge
of a symmetryprotected topological order,” Phys. Rev.
Lett., 105, 040501 (2010), arXiv:1003.4662.
[27] Xie Chen, ZhengXin Liu, and XiaoGang Wen, “Two
dimensional symmetryprotected topological orders and
their protected gapless edge excitations,” Phys. Rev. B,
84, 235141 (2011), arXiv:1106.4752.
[28] Xie Chen, ZhengCheng Gu, ZhengXin Liu, and Xiao
Gang Wen, “Symmetry protected topological orders and
the cohomology class of their symmetry group,” (2011),
arXiv:1106.4772.
Gavin K. Brennen, Akimasa
and D. PerezGarcia,