Identifying phases of quantum manybody systems that are universal for quantum computation.
ABSTRACT Quantum computation can proceed solely through singlequbit measurements on an appropriate quantum state, such as the ground state of an interacting manybody system. We investigate a simple spinlattice system based on the clusterstate model, and by using nonlocal correlation functions that quantify the fidelity of quantum gates performed between distant qubits, we demonstrate that it possesses a quantum (zerotemperature) phase transition between a disordered phase and an ordered "cluster phase" in which it is possible to perform a universal set of quantum gates.

 SourceAvailable from: Gregory M Crosswhite
Article: Adiabatic Quantum Transistors
[Show abstract] [Hide abstract]
ABSTRACT: We describe a manybody quantum system which can be made to quantum compute by the adiabatic application of a large applied field to the system. Prior to the application of the field quantum information is localized on one boundary of the device, and after the application of the field this information has propagated to the other side of the device with a quantum circuit applied to the information. The applied circuit depends on the manybody Hamiltonian of the material, and the computation takes place in a degenerate ground space with symmetryprotected topological order. Such adiabatic quantum transistors are universal adiabatic quantum computing devices which have the added benefit of being modular. Here we describe this model, provide arguments for why it is an efficient model of quantum computing, and examine these manybody systems in the presence of a noisy environment.Physical Review X. 07/2012; 3(2).  SourceAvailable from: iopscience.iop.org[Show abstract] [Hide abstract]
ABSTRACT: The twodimensional cluster state, a universal resource for measurementbased quantum computation, is also the gapped ground state of a shortranged Hamiltonian. Here, we examine the effect of perturbations to this Hamiltonian. We prove that, provided the perturbation is sufficiently small and respects a certain symmetry, the perturbed ground state remains a universal resource. We do this by characterising the operation of an adaptive measurement protocol throughout a suitable symmetryprotected quantum phase, relying on generic properties of the phase rather than any analytic control over the ground state.New Journal of Physics 07/2012; 14(11). · 4.06 Impact Factor
Page 1
arXiv:0802.4314v3 [quantph] 16 Jul 2009
Identifying phases of quantum manybody 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 singlequbit measurements on an appropriate
quantum state, such as the ground state of an interacting manybody system. We investigate a
simple spinlattice system based on the clusterstate 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 (zerotemperature) phase transition between a disordered phase and an
ordered “cluster phase” in which it is possible to perform a universal set of quantum gates.
Measurementbased 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 socalled 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 stronglycoupled quantum manybody 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 nonlocal 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 muchstudied
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 manybody 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 manybody interactions, it
can be realized as the effective lowenergy theory of a
Hamiltonian consisting only of twobody 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 transversefield cluster model (TFCM), and we will
demonstrate the existence of a single zerotemperature
phase transition in the ground state of such models
on both a 1D line and a 2D 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
wellunderstood 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 singlequbit
Page 2
2
gates as well. In addition, in two dimensions, we per
form a similar analysis of the twoqubit CSIGN gate,
exp(iπ1??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 twodimensional models in
detail. An immediate observation is that this model is
selfdual. 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 selfduality 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 1D lattice
with fixed boundary conditions – a line. A state of a 1D
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 nondegenerate,
and for B = 0 is given by the 1D 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 JordanWigner
transformation to yield a linear fermionic system. We
provide a more direct transformation to a known model
– the transversefield Ising model [15] – which provides a
natural generalization to higherdimensional 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. (45) on a 1D
line. (b) A generalization of this duality transformation to a
2D 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 nonlocal, 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 wellstudied
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 transversefield 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 nondegenerate
and is given by the product state of these two unique
ground states.
Page 3
3
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 wellstudied phases of the transversefield
Ising model; in particular, there is a unique quantum
phase transition at B = 1 [16]. Also, the wellknown
order parameters for the transversefield Ising model can
be mapped, using the duality transformation, to order
parameters for the TFCM. In the ordered phase of the
transversefield Ising model, the correlation functions
?¯Z¯Z? (for both colors) are long ranged.
limk→∞?¯Zi¯Zi+k? = (1 − B2)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 stringlike operators corresponding to the
product of even (or odd) stabilizers Kiin the TFCM are
also longranged, with the limiting value (1 − B2)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 maximallyentangled 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 singlequbit Clifford gates [17] and for an
arbitrary Zrotation Uz(θ) = exp(−iθZ) (a nonClifford
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 randomlychosen 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 onedimensional cluster
state is not a universal resource for MBQC, and so we
direct our attention to a twodimensional 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 1D 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 Xbasis (Zbasis). The
two stringlike stabilizers, centred on sites connected by the
shaded red and shaded blue diagonal lines, have longranged
expectation values in the B < 1 phase; these correlation
functions directly quantify the fidelities of singlequbit 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 14, the resulting state provides the
CSIGN transformation.
Under this mapping,
monochromatic operator consisting only of ¯Z terms.
Nonboundary 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 2D 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 twobody terms in the Hamiltonian, and is therefore
very amenable to numerical investigation. For example,
the projected entangledpair 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 longranged for B < 1 [24].
Inverting this duality transformation and returning to
the TFCM, these Isingtype correlation functions map
Page 4
4
onto strings of monochromatic stabilizers along diagonal
lines in the square lattice (see Fig. 2(a)). Again using the
correlation functions for singlequbit 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 singlequbit gates between two distant points, and
serve as order parameters for the cluster phase.
In addition, in this 2D 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 longranged correlation functions
on the AQOCM are of the form of 4body 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 4body correlation
functions should be possible to numerically evaluate in
the AQOCM using recent techniques. The CSIGN to
gether with the above singlequbit 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 quantumcomputational 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 Zfield.
gate correlation functions discussed in this paper become
shortranged at any nonzero perturbation. However, by
preprocessing 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 transversefield Ising model does
not maintain an ordered phase at any finite temperature
demonstrates that the 1D 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 threedimensional
lattice such as in [11], for which the B = 0 model is
known to allow for faulttolerant 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.
[1] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86,
5188 (2001).
[2] R. Raussendorf, D. E. Browne and H.J. Briegel, Phys.
Rev. A 68, 022312 (2003).
[3] D. Gross and J. Eisert, Phys. Rev. Lett. 98, 220503
(2007).
[4] D. Gross et al., Phys. Rev. A 76, 052315 (2007).
[5] M. Van den Nest et al., New J. Phys. 9, 204 (2007).
[6] G. K. Brennen and A. Miyake, Phys. Rev. Lett. 101,
010502 (2008).
[7] D. Gottesman and I. Chuang, Nature (London) 402, 390
(1999).
[8] A. M. Childs, D. W. Leung and M. A. Nielsen, Phys.
Rev. A 71, 032318 (2005).
[9] T. Chung, S. D. Bartlett and A. C. Doherty, Can. J.
Phys. 87, 219 (2009).
[10] M. Popp et al., Phys. Rev. A 71, 042306 (2005).
[11] R. Raussendorf, S. Bravyi, J. Harrington, Phys. Rev. A
71, 062313 (2005).
[12] S. D. Bartlett and T. Rudolph, Phys. Rev. A 74,
040302(R) (2006); T. Griffin and S. D. Bartlett, Phys.
Rev. A 78, 062306 (2008).
[13] S. D. Barrett et al., arXiv:0807.4797.
[14] J. K. Pachos and M. B. Plenio, Phys. Rev. Lett. 93,
056402 (2004).
[15] J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979).
[16] P. Pfeuty, Ann. Phys. 57, 79 (1970).
[17] M. A. Nielsen and I. L. Chuang. Quantum Computation
and Quantum Information (Cambridge University Press,
Cambridge, England, 2000).
[18] C. Xu and J. E. Moore, Phys. Rev. Lett. 93, 047003
(2004).
[19] C. Xu and J. E. Moore, Nucl. Phys. B 716, 487 (2005).
[20] Z. Nussinov and E. Fradkin, Phys. Rev. B 71, 195120
(2005).
[21] B. Doucot et al., Phys. Rev. B 71, 024505 (2005).
[22] J. Dorier, F. Becca, and F. Mila, Phys. Rev. B 72, 024448
(2005).
[23] D. Bacon, Phys. Rev. A 73, 012340 (2006).
[24] R. Orus, A. C. Doherty, and G. Vidal, Phys. Rev. Lett.
102, 077203 (2009).
[25] R. Raussendorf, J. Harrington, and K. Goyal, Ann. Phys.
321, 2242 (2006); R. Raussendorf, J. Harrington, and K.
Goyal, New J. Phys. 9, 199 (2007).