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arXiv:0802.4314v3 [quant-ph] 16 Jul 2009

Identifying phases of quantum many-body systems that are universal for quantum

computation

Andrew C. Doherty1and Stephen D. Bartlett2

1School of Physical Sciences, The University of Queensland, St Lucia, Queensland 4072, Australia

2School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia

(Dated: 21 June 2009)

Quantum computation can proceed solely through single-qubit measurements on an appropriate

quantum state, such as the ground state of an interacting many-body system. We investigate a

simple spin-lattice system based on the cluster-state model, and by using nonlocal correlation func-

tions that quantify the fidelity of quantum gates performed between distant qubits, we demonstrate

that it possesses a quantum (zero-temperature) phase transition between a disordered phase and an

ordered “cluster phase” in which it is possible to perform a universal set of quantum gates.

Measurement-based quantum computation (MBQC) is

a fundamentally new approach to quantum computing.

MBQC proceeds by using only local adaptive measure-

ments on single qubits. No entangling operations are

required; all entanglement for the computation is sup-

plied by a fixed initial resource state on a lattice of

qubits. The canonical example of such a resource state

is the so-called cluster state [1, 2]. Although a handful

of other universal resources have recently been identi-

fied [3, 4, 5, 6], there currently exists very little under-

standing of precisely which properties of quantum states

allow for universal MBQC. For example, given a state

that is slightly perturbed from the cluster state, it is not

currently known how to determine if it is a universal re-

source. New theoretical tools are required to identify the

properties of potential resource states that allow for uni-

versal MBQC.

A useful perspective to approach this problem is to

view the resource state for MBQC as the ground state of

a strongly-coupled quantum many-body system. With

this perspective, we propose that the ability to perform

MBQC is a type of quantum order – one which can be

identified using appropriate correlation functions as or-

der parameters. We show that a natural choice for such

correlation functions are the expectation values of non-

local strings of operators that can be identified with mea-

surement sequences for performing quantum logic gates

within MBQC. One way of understanding MBQC is that,

by means of a set of local measurements, it is possible to

prepare the resource states required for gate teleporta-

tion [2, 7, 8] between distant components of the many-

body system. The performance of the MBQC scheme

can be characterized by calculating the fidelity of the pre-

pared resource state with the ideal one [9]. This fidelity

will depend on a set of non-local correlation functions as a

result of the many local measurements that are required

to prepare the resource state. (Because the fidelity of

the identity gate is quantified by the ability to prepare

an entangled state between two distant qubits using local

measurements, it is closely related to the much-studied

property of localizable entanglement [10].) We show that,

for the cluster state implementation of MBQC, the spe-

cific correlation functions corresponding to any gate can

be calculated, and we investigate a specific model where

the fidelities of a gate set indeed serve as order parame-

ters identifying a cluster phase. This result suggests the

existence of spin systems that possess a phase for which

any state is a universal resource for MBQC. These meth-

ods provide new tools for identifying properties of quan-

tum many-body systems that are required for MBQC.

Consider the following model system. The cluster state

on a lattice L is defined as the unique +1 eigenstate of a

set of stabilizer operators Kµ= Xµ?

(Zµ) is the Pauli X (Z) operator at site µ and where

ν ∼ µ denotes that ν is connected to µ by a bond in

the lattice L. The Hamiltonian H = −?

cluster state as its unique ground state [11]. Although the

terms in this Hamiltonian are many-body interactions, it

can be realized as the effective low-energy theory of a

Hamiltonian consisting only of two-body terms [12].

As a model system to consider how robust is this

Hamiltonian in the presence of local perturbations, we

supplement it with a local field term,

ν∼µZν, where Xµ

µ∈LKµhas the

H(B) = −?

µ∈L(Kµ+ BXµ), (1)

representing a local transverse field with magnitude

B.We refer to a lattice with this Hamiltonian as

the transverse-field cluster model (TFCM), and we will

demonstrate the existence of a single zero-temperature

phase transition in the ground state of such models

on both a 1-D line and a 2-D square lattice, separat-

ing a disordered phase from a “cluster phase”. Rather

than solving these models explicitly, we explore duality

transformations that relate these models to others with

well-understood phases and order parameters. We then

demonstrate that the order parameters of these mod-

els, mapped back to the TFCM, are precisely equiva-

lent to the correlation functions in the cluster state that

quantify the fidelity of the identity gate (i.e., teleporta-

tion) in MBQC. That is, the ability to perform the iden-

tity gate over a long range serves as an order parameter

for this phase; similar results hold for other single-qubit

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gates as well. In addition, in two dimensions, we per-

form a similar analysis of the two-qubit CSIGN gate,

exp(iπ|1??1| ⊗ |1??1|), which together with our single-

qubit gates yields a universal gate set for MBQC. (In

contrast, the case with a local longitudinal field instead

of a transverse one was investigated in [13]; this model

demonstrates no such phase but nevertheless can still al-

low for MBQC for some range of parameters.)

General properties of the transverse field cluster model.

We first present some general properties of the TFCM

that are valid in any dimension and on many lattices,

before investigating one- and two-dimensional models in

detail. An immediate observation is that this model is

self-dual. The canonical transformation of Pauli opera-

tors given by applying the CSIGN operation between all

neighbouring pairs of qubits takes Kµ↔ Xµ, and thus

the Hamiltonian (1) transforms as H(B) → BH(1/B).

This self-duality ensures that, if this model has a single

quantum phase transition in the range B > 0, then it

must occur at B = 1.

Also, consider lattices which are bipartite, meaning we

can divide the sites into two subsets Lrand Lb, labeled

red and blue, such that the neighbours of any site are all

of the other colour. With this colouring, the Hamiltonian

(1) can be written as the sum of two commuting terms,

H = Hr+ Hb, where

Hr= −?

µ∈LbKµ− B?

µ∈LrXµ, (2)

with Hb consisting of the remaining terms. In the fol-

lowing, we present mappings of Hr(equivalently, Hb) in

one and two dimensions to known models, which allows

us to identify the phases and relevant order parameters.

One dimension. Consider the TFCM on a 1-D lattice

with fixed boundary conditions – a line. A state of a 1-D

lattice cannot serve as a universal resource for MBQC;

however, it will be illustrative to consider this model as

a prelude for studying higher dimensions. The Hamilto-

nian (1) on a line with boundary terms is

H(B) = −?N−1

i=2(Zi−1XiZi+1+ BXi)

− X1Z2− BX1− ZN−1XN− BXN. (3)

The ground state of this Hamiltonian is non-degenerate,

and for B = 0 is given by the 1-D cluster state on a

line. Pachos and Plenio [14] have shown explicitly that

this model (with periodic boundary conditions) exhibits a

quantum phase transition at |B| = 1, and that the local-

izable entanglement length remains infinite for all values

|B| < 1. Their method makes use of the Jordan-Wigner

transformation to yield a linear fermionic system. We

provide a more direct transformation to a known model

– the transverse-field Ising model [15] – which provides a

natural generalization to higher-dimensional lattices.

Our duality transformation is as follows. On red (even)

sites, the Pauli operators transform as

X2j→¯ X2j,Z2j→??j

k=1¯ X2k−1

?¯Z2j.(4)

X

X

X

(a)

(b)

X

Z

Z

X

X

X

X

X

X

X

X

Z

Z

X

X

X

X

X

X

X

Z

Z

X

X

X

X

X

Z

X

X

Z

FIG. 1: (a) The duality transformation of Eqs. (4-5) on a 1-D

line. (b) A generalization of this duality transformation to a

2-D square lattice.

On blue (odd) sites, the Pauli operators transform as

X2j−1→¯ X2j−1,Z2j−1→¯Z2j−1

??N

k=j¯ X2k

?. (5)

This mapping is canonical, meaning the new Pauli matri-

ces¯ Xjand¯Zjsatisfy the correct commutation and anti-

commutation relations. An illustration of this transfor-

mation is presented in Fig. 1(a). We emphasize that this

duality transformation is non-local, and thus the proper-

ties of a system for MBQC are not preserved under this

mapping. However, as we now demonstrate, the phases

and order parameters of this dual model are well-studied

and will allow us to completely classify the phases as well

as calculate the fidelities of the MBQC quantum gates in

the original TFCM.

We consider only the case where N is even. In terms

of transformed Pauli operators, the Hamiltonian Hracts

only on red sites and has the form

Hr= −¯Z2−?N/2

i=2

?¯Z2(i−1)¯Z2i+ B¯ X2i

?, (6)

The Hamiltonian Hbis similar, acting only on blue sites,

with a¯Z boundary term at j = N. This mapping on the

TFCM, then, yields two identical transverse-field Ising

models, one on each of Lr and Lb. Each has a local

¯Z field term which breaks the symmetry in the ordered

(|B| < 1) phase and specifies a unique ground state. The

ground state of the total lattice is then non-degenerate

and is given by the product state of these two unique

ground states.

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The solution to this known model allows us, via the

duality transformation, to completely characterise the

TFCM. For example, the phases of the TFCM are spec-

ified by the well-studied phases of the transverse-field

Ising model; in particular, there is a unique quantum

phase transition at |B| = 1 [16]. Also, the well-known

order parameters for the transverse-field Ising model can

be mapped, using the duality transformation, to order

parameters for the TFCM. In the ordered phase of the

transverse-field Ising model, the correlation functions

?¯Z¯Z? (for both colors) are long ranged.

limk→∞?¯Zi¯Zi+k? = (1 − |B|2)1/4for |B| < 1 [16].) We

can use this result to make a corresponding statement

about correlation functions for the TFCM. By reversing

the duality transformation, we have

(Specifically,

?¯Z2i−1¯Z2j−1? → ?Z2i−1(?j−1

k=iX2k)Z2j−1? = ??j−1

k=iK2k?,

(7)

?¯Z2i¯Z2j? → ?Z2i(?j−1

k=iX2k+1)Z2j? = ??j−1

k=iK2k+1?.

(8)

That is, in the phase |B| < 1 wherein ?¯Z¯Z? is long-

ranged, the string-like operators corresponding to the

product of even (or odd) stabilizers Kiin the TFCM are

also long-ranged, with the limiting value (1 − |B|2)1/4.

These two correlation functions are all that is needed to

calculate the fidelity of the resource state for the identity

gate with the ideal maximally-entangled state, and it is

found to be > 1/4 for all |B| < 1. (The average fidelity

of a randomly chosen state yields 1/4.) The same calcu-

lation for other single-qubit Clifford gates [17] and for an

arbitrary Z-rotation Uz(θ) = exp(−iθZ) (a non-Clifford

gate), yields the same result [9].

Thus, this duality transformation has allowed us to

prove our desired results: First, that the TFCM does in-

deed possess a phase, given by |B| < 1, which we denote

the cluster phase. The order parameters of this phase,

given by products of even or odd stabilizer operators Ki,

demonstrate that quantum gates can be performed with

high fidelity (relative to a randomly-chosen state) using

any state within this phase. The ground states in this

phase are indeed “robust” against variations in the pre-

cise value of B. However, the one-dimensional cluster

state is not a universal resource for MBQC, and so we

direct our attention to a two-dimensional model.

Two dimensions.

We consider a square lattice; the

cluster state on this lattice is a universal resource for

MBQC. This lattice is bipartite, and thus we can define

the commuting Hamiltonians Hrand Hb as above. We

use a natural generalization of the 1-D duality transfor-

mation, as follows. On red sites, Pauli operators trans-

form as Xµ→¯ Xµand Zµ→??

blue sites, Xµ→¯ Xµ and Zµ→¯Zµ

µ′> µ (µ′< µ) denotes that µ′lies in the upper (lower)

cone relative to µ as in Fig. 1(b). Again, one can easily

verify that this transformation is canonical.

µ′>µ¯ Xµ′?¯Zµ, whereas on

??

µ′<µ¯ Xµ′?. Here,

XX

XX

X

X

XX

X

aout

ain

aout

ain

bin

bout

(a)(b)

Z

Z

Z

Z

Z

Z

1

4

3

2

FIG. 2: (a) A measurement pattern on the cluster state that

localizes entanglement between sites ain and aout, where X

(Z) denotes a measurement in the X-basis (Z-basis). The

two string-like stabilizers, centred on sites connected by the

shaded red and shaded blue diagonal lines, have long-ranged

expectation values in the |B| < 1 phase; these correlation

functions directly quantify the fidelities of single-qubit gates

between ain and aout in MBQC. (b) The measurement se-

quence corresponding to the CSIGN gate between a and b.

The expectation of four stabilizers characterises the CSIGN

gate: KainK3Kaout, KbinK4Kbout, K1K4 and K2K3. These

stabilizers can be appended with diagonal strings of red (blue)

stabilizers in the direction of the arrows (and terminated with

Z measurements as in (a)) to reach distant qubits. With X

measurements on qubits 1-4, the resulting state provides the

CSIGN transformation.

Under this mapping,

monochromatic operator consisting only of ¯Z terms.

Non-boundary stabilizers map to products of four¯Z op-

erators on the corners of a fundamental plaquette ?;

boundary conditions can be chosen such that boundary

stabilizers map to two-¯Z and one-¯Z terms. The Hamil-

tonian Hr(Hb) on Lr(Lb) maps to

each stabilizer maps to a

H = −

?

?

¯Z

¯Z

¯Z

¯Z− B

?

µ

¯ Xµ, (9)

plus boundary terms (not shown) which ensure a non-

degenerate ground state for all B. This model possesses

a phase transition at |B| = 1 [18, 19]. Thus, through

this duality map, we know that the 2-D TFCM has a

phase transition at |B| = 1, and we use the term clus-

ter phase to denote the |B| < 1 phase. In addition, this

model of Eq. (9) is dual to the anisotropic quantum or-

bital compass model (AQOCM) [20, 21, 22, 23], with a

mapping that also locally maps the boundary terms).

The key advantage of the AQOCM is that it contains

only two-body terms in the Hamiltonian, and is therefore

very amenable to numerical investigation. For example,

the projected entangled-pair state algorithm applied to

this model provides very strong evidence that the phase

transition is first order [24]. The model also possesses

correlation functions for an Ising order parameter that

simulations indicate are long-ranged for |B| < 1 [24].

Inverting this duality transformation and returning to

the TFCM, these Ising-type correlation functions map

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onto strings of monochromatic stabilizers along diagonal

lines in the square lattice (see Fig. 2(a)). Again using the

correlation functions for single-qubit gates given in [9], we

find that these strings of monochromatic stabilizers char-

acterize the fidelities of the identity gate and a generating

set of single-qubit gates between two distant points, and

serve as order parameters for the cluster phase.

In addition, in this 2-D model we can consider two-

qubit gates. We make use of the elementary measurement

pattern for a CSIGN gate on two qubits which are sub-

sequently swapped, as given in Refs. [2, 9] and shown in

Fig. 2(b). The desired long-ranged correlation functions

on the AQOCM are of the form of 4-body correlators

?˜Z(i,j0)˜Z(i,j∗)˜Z(i+1,j∗)˜Z(i+1,j1)?, where j∗is an intermedi-

ate column between j0and j1. Such 4-body correlation

functions should be possible to numerically evaluate in

the AQOCM using recent techniques. The CSIGN to-

gether with the above single-qubit gates yields a universal

gate set, and thus the cluster phase is indeed character-

ized by the fidelities of a universal gate set for MBQC.

Discussion. Using the TFCM as an example, we have

demonstrated the utility of correlation functions corre-

sponding to quantum gates as order parameters to iden-

tify a phase according to its usefulness for MBQC. The

perspective of quantum-computational universality of a

state as a new type of quantum order may assist in identi-

fying new quantum systems that can be used for MBQC.

The behaviour of the TFCM contrasts with the model

considered in [13], which is the cluster state Hamilto-

nian perturbed by a local Z-field.

gate correlation functions discussed in this paper become

short-ranged at any non-zero perturbation. However, by

pre-processing with certain local filtering operations it is

still possible to perform MBQC for sufficiently low field

and sufficiently low temperature [13]. Unlike the TFCM,

the model of [13] does not undergo a phase transition.

These behaviours are very reminiscent of the quantum

Ising model in one dimension; where there is a broken

symmetry that disappears at a phase transition for suf-

ficiently large transverse field but longitudinal fields de-

stroy the ground state order without any phase transi-

tion.

One could also ask whether these ordered phases per-

sist to finite temperature. As our model is gapped ex-

cept at the phase transition, it is possible for a finite-

sized thermal system to be cooled to have arbitrarily high

overlap with the ground state (although this becomes a

challenge close to the phase transition). In one dimen-

sion, the fact that the transverse-field Ising model does

not maintain an ordered phase at any finite temperature

demonstrates that the 1-D TFCM does not either. In two

In that model the

dimensions, it is less clear. For this reason it would be

worth investigating the TFCM on a three-dimensional

lattice such as in [11], for which the B = 0 model is

known to allow for fault-tolerant MBQC at finite tem-

perature [11, 13, 25].

Acknowledgments. We acknowledge helpful discussions

with Sean Barrett, Dan Browne and Terry Rudolph, and

the support of the Australian Research Council.

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