Page 1

Symmetry-protected phases for measurement-based 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 measurement-based 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 symmetry-protected

topological orders in one-dimensional systems, any one of which ensures the perfect operation of the

identity gate. As a result, measurement-based 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 measurement-based quantum computation

(MBQC) [1, 2], where universal quantum computation is

achieved by means of non-entangling operations (namely,

single-particle 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 1-D 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 single-qubit unitaries.

Later, we will also explain how our results can be ap-

plied to higher-dimensional systems (which can allow for

universal MBQC) by considering them as ‘quasi-1D’.

Specifically,

A well-known example of a one-dimensional system

whose ground state can serve as a quantum computa-

tional wire is the Affleck-Kennedy-Lieb-Tasaki (AKLT)

antiferromagnetic spin-1 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 symmetry-respecting 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 1-D 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].

Symmetry-protection 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 1-D resource states. This picture assumes a resource

state |Ψ? that can be represented as a matrix-product

state (MPS),

?

|Ψ? =

k1,...,kN

?R|A[kN]A[kN−1]···A[k1]|L?

× |k1,...,kN?, (1)

where each A[kj], kj= 1,...,d is a linear operator acting

on a D-dimensional 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 spin-1 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]?ψ|k?.

As an introduction to our result, we will first state

arXiv:1201.4877v2 [quant-ph] 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

by-products) on the correlation system. Additionally, by

measuring in a basis corresponding to a rotated set of

axes, it is possible to execute any single-qubit rotation in

correlation space (up to Pauli by-products) [11]. There-

fore, the AKLT state can be said to act as a quantum

computational wire.

We will now extend our correlation-space 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 spin-1 chain can be written as a tensor

product [u(g)]⊗N, where N is the number of spins, and

u(g) is the appropriate single-spin rotation operator for

each group element g in the symmetry group G = Z2×Z2.

We therefore refer to it as an on-site symmetry.

In general, the invariance of a ground state under such

an on-site 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 one-dimensional

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 1-D cluster state, which is the

ground state on a row of qubits of the local Hamil-

tonian H = −?

it lies within a symmetry-protected phase with respect

to a Z2× Z2 symmetry [22], in this case generated by

?

sites. The simultaneous eigenstate of the on-site sym-

metry representation is then {|++?,|+−?,|−+?,|−−?},

where |±? =

product basis, so that blocking sites does not change the

single-qubit 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

d-dimensional 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

on-site 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 on-site 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

symmetry-protected 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 symmetry-protected phase.

We now relate the symmetry condition (2), which

holds throughout the entire symmetry-protected phase,

to the operation of gates in the correlation-space 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 symmetry-protected phase is of a particular

type. (An analogous result holds for all non-trivial 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 non-commutative, meaning that the subgroup

G(ω) = {g ∈ G|ω(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 symmetry-protected phase

characterized by a finite abelian symmetry group and

a maximally non-commutative 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

outcome-dependent 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 symmetry-protected

phase.

For example, the factor system for the Pauli projective

representation of Z2×Z2is maximally non-commutative,

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 non-commutativity of a

factor system:

Lemma 1. Let ω be a maximally non-commutative 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 non-commutative 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 non-trivial gates (i.e.

those involving measurement in a different basis) are not

symmetry-protected.

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? ≡ cosθ|x?+

sinθ|y?. Then from the decomposition (4) of the MPS

tensor A for a generic state in the Haldane phase, we

find that

Non-trivial 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 non-trivial gates.

The preceding discussion of non-trivial 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 spin-1/2 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 half-integer 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 non-trivial

gates. This should be contrasted with the protocol of

[26], where a logical qubit is encoded into an explicitly

spin-1/2 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 non-trivial gates) that

the usual procedures for doing this in the cluster or

AKLT states are not symmetry-protected. However, a

symmetry-protected readout can be achieved through-

out the Haldane phase by terminating a finite chain of

spin-1’s with a spin-1/2, as in [15].

Higher-dimensional systems.—The

symmetry-protectedtopological

been extended to higher-dimensional systems [27, 28],

and we speculate that our results could be generalized

in this context.However, if we consider a ‘quasi-1D’

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 2-D 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 non-commutative 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 2-D 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

2-D 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 single-qubit measurements. These prop-

erties are consequences of the symmetry and thus remain

true throughout the symmetry-protected phase.

Conclusion.—In summary, we have identified a class of

symmetry-protected topological orders, each of which en-

sures the perfect operation of the identity gate in MBQC

throughout an entire symmetry-protected phase. Such

connections between MBQC and quantum order can be

expected to lead to a greater understanding of the poten-

tial for single-particle 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 non-trivial 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 non-trivial 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 Australia-Israel 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, “Measurement-based quantum computa-

tion on cluster states,” Phys. Rev. A, 68, 022312 (2003),

arXiv:quant-ph/0301052.

and

Page 5

5

[2] H. J. Briegel, D. E. Browne, W. Dur, R. Raussendorf,

and M. Van den Nest, “Measurement-based quantum

computation,” Nat. Phys., 5, 19 (2009), arXiv:0910.1116.

[3] Andrew C. Doherty and Stephen D. Bartlett, “Identify-

ing phases of quantum many-body systems that are uni-

versal for quantum computation,” Phys. Rev. Lett., 103,

020506 (2009), arXiv:0802.4314.

[4] Xie Chen, Bei Zeng, Zheng-Cheng Gu, Beni Yoshida,

and Isaac L. Chuang, “Gapped two-body hamiltonian

whose unique ground state is universal for one-way 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, “Affleck-Kennedy-Lieb-Tasaki

state on a honeycomb lattice is a universal quantum

computational resource,” Phys. Rev. Lett., 106, 070501

(2011), arXiv:1102.5064.

[6] Zheng-ChengGu and

entanglement-filtering renormalization

symmetry-protected topological order,” Phys. Rev. B,

80, 155131 (2009), arXiv:0903.1069.

[7] Xie Chen, Zheng-Cheng 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´ erez-Garc´ 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 valence-bond 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 two-body 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 one-dimensional 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, “Measurement-based teleportation along quan-

tum spin chains,” Phys. Rev. Lett., 95, 230501 (2005),

Tzu-Chieh Wei, Ian Affleck,

Xiao-Gang Wen,

approach

“Tensor-

and

and Xiao-Gang Wen,

and Hal

(2009),

and

arXiv:quant-ph/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:cond-mat/0503021.

[17] D. L. Zhou, B. Zeng, Z. Xu, and C. P. Sun, “Quantum

computation based on d-level cluster state,” Phys. Rev.

A, 68, 062303 (2003), arXiv:quant-ph/0304054.

[18] D. Gross, J. Eisert, N. Schuch,

“Measurement-based quantum computation beyond the

one-way 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:cond-mat/0505140; M. B. Hastings, “An

area law for one-dimensional quantum systems,” Journal

of Statistical Mechanics: Theory and Experiment, 2007,

P08024 (2007), arXiv:0705.2024.

[20] D. P´ erez-Garc´ 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 symmetry-protected topological order,” Phys. Rev.

Lett., 105, 040501 (2010), arXiv:1003.4662.

[27] Xie Chen, Zheng-Xin Liu, and Xiao-Gang Wen, “Two-

dimensional symmetry-protected topological orders and

their protected gapless edge excitations,” Phys. Rev. B,

84, 235141 (2011), arXiv:1106.4752.

[28] Xie Chen, Zheng-Cheng Gu, Zheng-Xin 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. Perez-Garcia,