arXiv:0802.4314v3 [quant-ph] 16 Jul 2009
Identifying phases of quantum many-body systems that are universal for quantum
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-
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 . 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 .) 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 . 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 .
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
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
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 ; 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
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
− 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  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  – which provides a
natural generalization to higher-dimensional lattices.
Our duality transformation is as follows. On red (even)
sites, the Pauli operators transform as
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
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
?¯Z2(i−1)¯Z2i+ B¯ X2i
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
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 . 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 .) We
can use this result to make a corresponding statement
about correlation functions for the TFCM. By reversing
the duality transformation, we have
?¯Z2i−1¯Z2j−1? → ?Z2i−1(?j−1
k=iX2k)Z2j−1? = ??j−1
?¯Z2i¯Z2j? → ?Z2i(?j−1
k=iX2k+1)Z2j? = ??j−1
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  and for an
arbitrary Z-rotation Uz(θ) = exp(−iθZ) (a non-Clifford
gate), yields the same result .
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.
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,
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
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 = −
¯ 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 . The model also possesses
correlation functions for an Ising order parameter that
simulations indicate are long-ranged for |B| < 1 .
Inverting this duality transformation and returning to
the TFCM, these Ising-type correlation functions map
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 , 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 , 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 . Unlike the TFCM,
the model of  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-
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 , 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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