dissipate (or, more precisely, evacuate) the entropy of the system
Finally, optical pumping of the ancilla ion back to its initial state
j0i
constitutes the dissipative element in the sequence, which renders the dynamics of the
four system spins irreversible and enables to carry away entropy and thereby \cool" the
system qubits.
http://arxiv.org/pdf/1104.2507.pdf
http://www.univie.ac.at/qfp/publications3/pdffiles/Quantum%20computation%20and%20quantum-state%20engineering%20driven%20by%20dissipation.pdf
LETTERS
PUBLISHED ONLINE: 20 JULY 2009 |
DOI: 10.1038/NPHYS1342
Quantum computation and quantum-state
engineering driven by dissipation
Frank Verstraete
1*, Michael M.Wolf2 and J. Ignacio Cirac3*
The strongest adversary in quantum information science is
decoherence, which arises owing to the coupling of a system
with its environment
1. The induced dissipation tends to destroy
and wash out the interesting quantum effects that give rise
to the power of quantum computation
2, cryptography2 and
simulation
3. Whereas such a statement is true for many
forms of dissipation, we show here that dissipation can also
have exactly the opposite effect: it can be a fully fledged
resource for universal quantum computation without any
coherent dynamics needed to complement it. The coupling to
the environment drives the system to a steady state where
the outcome of the computation is encoded. In a similar
vein, we show that dissipation can be used to engineer a
large variety of strongly correlated states in steady state,
including all stabilizer codes, matrix product states
4, and their
generalization to higher dimensions
5.
The situation we have in mind is shown in Fig. 1. A quantum
system composed of
N particles (such as qubits) is organized in
space according to a particular geometry (in the figure, a one-
dimensional lattice). Neighbouring systems are coupled to some
local environments, which are dissipative in nature and tend to
drive the system to a steady state. Our idea is to engineer those
couplings, so that the environments drive the system to a desired
final state. The coupling to the environment will be static, so that the
desired state is obtained after some time without having to actively
control the system. Note that the role of the environments is to
dissipate (or, more precisely, evacuate) the entropy of the system,
and by choosing the couplings appropriately we can use this effect
to drive our system.
We will show first how to design the interactions with
the environment to implement universal quantum computation.
This new method, which we refer to as dissipative quantum
computation (DQC), defies some of the standard criteria for
quantum computation because it requires neither state preparation,
nor unitary dynamics
6. However, it is nevertheless as powerful as
standard quantum computation. Then we will show that dissipation
can be engineered
7 to prepare ground states of frustration-free
Hamiltonians. Those include matrix product states
4,8,9 (MPSs) and
projected entangled pair states
5,9 (PEPSs), such as graph states10
and Kitaev
11 and Levin Wen12 topological codes. Both DQC and
dissipative state engineering (DSE) are robust in the sense that,
given the dissipative nature of the process, the system is driven
towards its steady state independent of the initial state and hence
of eventual perturbations along the way.
Here, we will concentrate first on DQC, showing how given
any quantum circuit one can construct a locally acting master
equation for which the steady state is unique, encodes the outcome
of the circuit and is reached in polynomial time (with respect to
the one corresponding to the circuit). Then we will show how
1
Fakultät für Physik, UniversitätWien, 1090Wien, Austria, 2Niels Bohr Institute, 2100 Copenhagen, Denmark, 3Max-Planck-Institut für Quantenoptik,
85748 Garching, Germany. *e-mail: fverstraete@gmail.com; ignacio.cirac@mpq.mpg.de.
to construct dissipative processes that drive the system to the
ground state of any frustration-free Hamiltonian. In the Methods
section, we will prove that MPS (ref. 9) and certain kinds of
PEPS (ref. 9) can be efficiently prepared using this method, and
in Supplementary Information we will give details of the proofs.
In this letter we will not consider specific physical set-ups where
our ideas can be implemented. Nevertheless, the Methods section
will provide a universal way of engineering the master equations
required for DQC and DSE, which can be easily adapted to current
experiments
13 based on, for example, atoms in optical lattices14
or trapped ions
15. Thus, we expect that our predictions may be
experimentally tested in the near future.
Let us start with DQC by considering
N qubits in a line and a
quantum circuit specified by a sequence of nearest-neighbour qubit
operations
fUt gTt
D
1. We define j t i VDUtUt1 :::U1j0i1:::j0iN, so
that
j T i is the final state after the computation. Our goal is to find
a master equation
DL( ) with a Liouvillian in Lindblad form16
L
( )D
X
k
L
k Ly
k
1
2
L
y
k
Lk ;
C
(1)
where the
Lk acts locally and has a steady state, 0: (1) that is unique;
(2) that can be reached in a time poly(
T); (3) such that T can be
extracted from it in a time poly(
T). As in Feynman's construction
of a quantum simulator
3, we consider another auxiliary register
with states
fjt igTt
D
0, which will represent the time. We choose
the Lindblad operators
L
i Dj0iih1jj0it h0j
L
t DUt jt C1iht jCUy
t
jt iht C1j
where
i D 1;:::;N and t D 0;:::;T. It is clear that the L terms act
locally except for the interaction with the extra register, which can
be made local as well. Furthermore,
0 D
1
T
C1
X
t
j
t ih t jjt iht j
is a steady state, that is,
L( 0)D0. Given such a state, the result of the
actual quantum computation can be read out with probability 1
=T
by measuring the time register. In Supplementary Information, we
show that
0 is the unique steady state and that the Liouvillian has
a spectral gap
1D 2=(2T C3)2. This means indeed that the steady
state will be reached in polynomial time in
T. Note that this gap is
independent of
N as well as of the actual quantum computation that
is carried out (that is, independent of the
Ut ). It is also shown that
the same gap is retained if the clock register is encoded in the unary
NATURE PHYSICS
j VOL 5 j SEPTEMBER 2009 j www.nature.com/naturephysics 633
©
2009 Macmillan Publishers Limited. All rights reserved.
LETTERS
NATURE PHYSICS DOI: 10.1038/NPHYS1342
Figure 1
j Schematic representation of the set-up.We consider a collection of N quantum particles, locally coupled to a set of environments. The
couplings are engineered in such a way that the system reaches the desired state in the long-time limit.
way proposed by Kitaev and co-workers
17, making the Lindblad
operators strictly local. A sketch of the proof is as follows. First, we
do a similarity transformation on
Lthat replaces all gatesUi with the
identity gates, showing that its spectrum is independent of the actual
quantum computation. Second, another similarity transformation
is done that makes
L Hermitian and block-diagonal. Each block can
then be diagonalized exactly leading to the claimed gap.
In some sense, the present formalism can be seen as a robust
way of doing adiabatic quantum computation
18 (errors do not
accumulate and the path does not have to be engineered carefully)
and implementing quantum random walks
19, and it might therefore
be easier to tackle interesting open questions, such as the quantum
probabilistically-checkable-proofs theorem, in this setting
20. In
addition, it seems that the dissipative way of preparing ground states
is more natural than to use adiabatic time evolution, as nature itself
prepares them by cooling.
Let us now turn to DSE and consider again a quantum system
with
N particles on a lattice in any dimension. We are interested in
ground states
, of Hamiltonians
H
D
X
H
that are frustration-free, meaning that
minimizes the energy of
each
H individually, and local in the sense thatH acts non-trivially
only on a small set
f1; ::: ;Ng of sites (for example, nearest
neighbours). We can assume the terms
H to be projectors and
we will denote the orthogonal projectors by
P D1H . States
of the considered form are, for example, all PEPS (including MPS
and stabilizer states
21).
We will consider discrete time evolution generated by a trace-
preserving completely positive map instead of a master equation.
These two approaches are basically equivalent
22 as every local
completely positive map
T can be associated with a local Liouvillian
through
L( )DNTT ( ) U, which leads to the same fixed points
and spectrum.Wechoose completely positive maps of the form
T
( )D
X
p
"
P
P C
1
m
X
m
i
D1
U
;iH H Uy
;
i
#
(2)
where the
p terms are probabilities and U ;1; :::;U ;m is a set of
unitaries acting non-trivially only within region
. They effectively
rotate part of the high-energy space (with support of
H ) to the
zero-energy space, so that tr
TT ( ) U trT U increases. As for
Liouvillians (1), we could similarly take
L ;i D UiH , or the ones
associated with the completely positive map.
We show now that for every frustration-free Hamiltonian,
the completely positive map in equation (2) converges to the
ground-state space if we choose the unitaries
U ;i to be completely
depolarizing, that is,
T ( ) /
P
P P C 1 tr TH U=trT1 U.
For ease of notation, we will explain the proof for the case of a
one-dimensional ring with nearest-neighbour interactions labelled
by the first site
D1;:::;N. Assume is such that its expectation
value with respect to the projector
onto the ground-state space
of
H is non-increasing under applications of T , that is, in particular
tr
T UDtrTT N ( ) U. Expressing this in the Heisenberg picture in
which
T ( )D C
P
H tr ( )=(d2N), we get
tr
T U trT UC
1
(
d2N)N tr
"
X
N
D1
Y
N
D1
H
C tr C
(
)
#
trT UC
N
(
d2N)N trT HU
where the first inequality comes from discarding (positive) terms in
the sum and the second one is due to bounding all partial traces
of
H from below by the respective smallest eigenvalue . Note
that the latter is strictly positive unless
H has a product state as
the ground state (in which case the statement becomes trivial).
Hence, we must have tr
T HUD0; that is, is a ground state of H.
It is easily seen that the same argument applies for more general
interactions on arbitrary lattices.
Once we have shown that the steady state after the application
of the completely positive map lies within the desired subspace
(the ground-state space of the frustration-free Hamiltonian), the
next question to be addressed is how efficient the process is. This
depends on the spectral gap,
, of the completely positive map (or,
equivalently, of the corresponding Liouvillian), as the time to reach
the steady state,
D O(1= ). Thus, the above procedure will be
efficient as long as the gap vanishes only polynomially with the
number of systems,
N. Similarly to what occurs with many-body
Hamiltonians, the determination of such a gap is, in general, very
complicated. For a wide range of interesting models, however, it
can be proved that this gap scales favourably. This is the case for
all MPS as well as for a rich subfamily of PEPS that includes all
stabilizer states (such as Kitaev's toric code
11 and the Levin Wen
states
12). In the Methods section, we characterize such a subfamily
of states, and in Supplementary Information we give the technical
proofs of our statements. Here, we will qualitatively explain how
our method works efficiently for some families of states. For that
we note that the action of the completely positive map (2) can
be interpreted as randomly choosing a region
(according to p ,
which we may set equal to 1
=N), then measuring P and applying
a correction according to the unitaries if the outcome was negative.
We denote by
Rn the set of regions where ' satisfies the condition
H
j'i D 0. If we measure now in one of those regions, we will
obviously obtain a positive result, and thus
Rn will remain the
same. If we measure in another region, we may have a positive or
negative result, something that may change the set
Rn. By imposing
certain conditions on the operators
H and U ;i, we can make sure
that in each step
Rn cannot be reduced and that the probability of
634
NATURE PHYSICS j VOL 5 j SEPTEMBER 2009 j www.nature.com/naturephysics
©
2009 Macmillan Publishers Limited. All rights reserved.
NATURE PHYSICS
DOI: 10.1038/NPHYS1342 LETTERS
being enlarged is non-vanishing. This automatically ensures that
the
scales only polynomially with the number of systems. In
one dimension, however, one can get rid of all those restrictions
and show that any MPS can be prepared in a time that also scales
favourably with
N. The fact that all MPS states can be prepared
with our method, together with the results reported in refs 23, 24,
automatically implies the existence of phase transitions driven by
dissipation in the following sense. By changing the parameters of
the operators
H appearing in the completely positive map (2), we
change the steady state of that map. It is possible to choose models
for which that state changes abruptly at some particular value of
that parameter in such a way that the correlation length diverges
and an order parameter appears (an example can be found in the
Supplementary Information).
We have investigated the computational power of purely
dissipative processes, and proved that it is equivalent to that of
the quantum circuit model of quantum computation. We have
also shown that dissipative dynamics can be used to create ground
states (such as MPS or PEPS) of frustration-free Hamiltonians of
strongly correlated quantum spin systems. We believe that these
new methods can be experimentally tested using atoms or ions with
current set-ups (see the Methods section).
Let us stress that we have been concerned here with a proof-
of-principle demonstration that dissipation provides us with an
alternative way of carrying out quantum computations or state
engineering. We believe, however, that much more efficient and
practical schemes can be developed and adapted to specific
implementations. We also think that these results open up
some interesting questions that deserve further investigation: for
example, how the use of fault-tolerant computations can make
our scheme more robust, or how one can design translationally
invariant completely positive maps that prepare MPS more
efficiently, or the importance and generality of the set of commuting
Hamiltonians (see the Methods section), which is intimately
connected to the fixed points of the renormalization group
transformations on PEPS (as it happens with MPS; ref. 25).
Furthermore, the model ofDQCmight well lead to the construction
of new quantum algorithms, as, for example, quantum random
walks can more easily be formulated within this context. Finally,
other ideas related to this work can be easily addressed using the
methods introduced; for example, thermal states of commuting
Hamiltonians can be engineered using DSE because the Metropolis
way of sampling over classical spin configurations can be adopted
to the case of commuting operators. Similar techniques could be
applied to free fermionic and bosonic systems, and, more generally,
it should be possible to devise DSE schemes converging to the
ground or thermal states of frustrated Hamiltonians by combining
unitary and dissipative dynamics.
Note added.
Concurrently with the submission of this paper,
refs 26 and 27 appeared in which a similar quantum-reservoir
engineering was used to prepare many-body states and non-
equilibrium quantum phases.
Methods
Engineering dissipation.
Here we show how to engineer the local dissipation that
gives rise to the master equations (1) and completely positive maps (2). They are
composed of local terms, involving few particles (typically two), so that we just have
to show how to implement those. To simplify the exposition, we will treat those
particles as a single one and assume that one has full control over its dynamics (for
example, one can apply arbitrary gates).
Let us start with the completely positive maps. It is clear that by applying a
quantum gate to the particle and a `fresh' ancilla and then tracing the ancilla one
can generate any physical action (that is, completely positive map) on the system.
Furthermore, by repeating the same process with short time intervals one can
subject the system to an arbitrary time-independent master equation. This last
process may not be efficient. An alternative way works as follows. Let us assume
that the ancilla is a qubit interacting with a reservoir such that it fulfils a master
equation with Liouville operator
La D
p
, where Dj0ih1j. Now, we couple
the ancilla to the system with a Hamiltonian
H D( Ly C
y
L). In the limit
, one can adiabatically eliminate the level j1i of the ancilla28 by applying
second-order perturbation theory to the Liouvillian (albeit for non-Hermitian
operators). In this way we obtain an effective master equation for
describing
the system alone, with Liouville operator
=
p
L. By using several ancillas
with Hamiltonians
H D( Li C
y
Lyi
) and following the same procedure we
obtain the desired master equation. Although we have not specified here a physical
system, one could use atoms. In that case, the ancilla could be an atom itself with
j
0i and j1i an electronic ground and excited level, respectively, so that spontaneous
emission gives rise to the dissipation. The coupling to the system (other atoms)
could be achieved using standard ideas used in the implementation of quantum
computation using those systems
13.
Efficient state preparation.
We have shown that it is possible to engineer
dissipative processes that prepare ground states of frustration-free Hamiltonians in
steady state. In the proof, the time for this preparation scales as
NN , which may be
an issue for experiments with large number of particles. Here we give much more
efficient methods for certain classes of frustration-free Hamiltonians.
We consider first frustration-free Hamiltonians for which
TH ;H UD0 and
show that, under certain conditions, the corresponding ground states can be
prepared in a time that scales only polynomially with the number of particles. The
corresponding set of ground states contains important families, such as stabilizer
states (for example, cluster states and topological codes), or certain kinds of PEPS,
namely, those that have (commuting) parent Hamiltonians with the injectivity
condition (as defined in refs 8, 29). Note that there was no known way of efficient
preparation for the latter.
Loosely speaking, we will consider two classes of Hamiltonians.
(1) Hamiltonians for which all excitations can be locally annihilated. In this case the
time of convergence scales as
DO(logN). (2) Interactions where excitations have
to be moved along the lattice before they can annihilate and
DO(NlogN).
To see how the first case can occur notice that, when iterating
T , the
correction on
does not change the outcome of previous measurements on
neighbouring regions because
8
6D 0V TU ;i;H 0 UD0 (3)
In fact, this can always be achieved by regrouping the regions into larger ones
having an interior
I ( ) on which only H acts non-trivially and letting the
U
;i solely act on I ( ). Denote by q the largest probability for obtaining twice a
negative measurement outcome on the same region
. The energy trTHT M( )U
after
M applications of T decreases then as N(1(1q)=N)M such that it takes
O
((NlogN)=(1q)) steps to converge to a ground state. The relaxation time of the
corresponding Liouvillian is thus
DO(logN1=1q). Clearly, this is a reasonable
bound only if
q<1, a condition possibly incompatible with equation (3).
Note that for all stabilizer states we can achieve
qD0, because there exists
always a local unitary (acting on a single qubit) so that
H U H D0. A class of
stabilizer states where this is compatible with equation (3) are the so-called graph
states
10. In this case, labels (with some abuse of notation) a vertex of a graph and
H
D(1 ( )
x
Q
(
; )2E ( )
z
)=2, where ( ) is a Pauli operator acting on site and
E
is the set of edges of the graph. Obviously, U D ( )
z
does the job. In this special
case, we can get even faster convergence when using the Liouvillian
L
( )D
X
U
H H Uy
1
2
n
H
;
o
C
The corresponding relaxation time can be determined exactly by realizing
that the spectrum of
L equals that of (H 1C1H)=2 so that D1 (see
Supplementary Information).
For the second type of commuting Hamiltonians, equation (3) and
q<1 are
incompatible. However, we can still prove fast convergence by relaxing equation (3)
such that within each region
the U acts on a site closest to a predetermined
site (say the origin) on the lattice and thus commutes with all terms
H that are
further away (see Supplementary Information for details). In this way excitations
are moved over the lattice before they can annihilate. As this requires extra time
proportional to the system's size, we get
DO(NlogN).
We turn now to another family of ground states of frustration-free
Hamiltonians, namely MPS (ref. 9). For the sake of clearness, we will consider
here translationally invariant Hamiltonians, although the analysis can be
straightforwardly extended to systems without that symmetry. We will specify a
completely positive map to prepare states of the form
j
iD
X
d
i
D1
tr(
Ai1 :::AiN )ji1 :::iNi
where the
A terms are DD matrices. We assume the injectivity property29, which
implies that
is the unique ground state of a nearest-neighbour frustration-free
NATURE PHYSICS
j VOL 5 j SEPTEMBER 2009 j www.nature.com/naturephysics 635
©
2009 Macmillan Publishers Limited. All rights reserved.
LETTERS
NATURE PHYSICS DOI: 10.1038/NPHYS1342
`parent' Hamiltonian that has a gap. Denoting by
the reduced density operator
corresponding to particles
k and k C1, Hk and Pk D 1Hk will denote the
projectors onto its kernel and range, respectively. Note that tr(
Pk ) D D2. We
take
N D2n for simplicity, but this is clearly not necessary. We construct the
channel
T in several steps. We first define a channel acting on two neighbouring
particles
k;kC1, as follows
R
r;c (X) VD PkXPk C
P
k
D
2 tr(HkX)
Here,
k D2r1(2c 1), where r D1;:::;n and c D1;:::;2nr . The action of these
maps has a tree structure, where the index
r indicates the row in the tree, whereas c
does it for the column. Now we define recursively,
S
r;c VD
(1
r )
2
(
Sr1;2c CSr1;2cC1)C rRr;c
Here,
r D2;:::;n, c D1;:::;2nr , S1;c VDR1;c and rC1 D1=Mr , where M DCN2
and
C 1 (see Supplementary Information). Note that Sr;1 acts on the first 2r
particles,
Sr;2 on the next 2r and so on. We finally define
T
VD (1 nC1)Sn;1C nC1Rn;2 (4)
In the Supplementary Information, we show that this map achieves the fixed point
(up to an exponentially small error in
C) in a time O(Nlog2 (N)). The intuition
behind the completely positive map (4) is that the channels
S1;c , which are the ones
that most often applied, project the state of every second nearest neighbour onto
the right subspace. Then
S2;c do the same with half of the pairs that have not been
projected. Then
S3;c does the same on half of the rest, and so on.
Received 11 March 2008; accepted 18 June 2009; published online
20 July 2009
References
1. Aliferis, P., Gottesman, D. & Preskill, J. Quantum accuracy threshold for
concatenated distance-3 codes.
Quant. Inf. Comput. 6, 97 165 (2006).
2. Nielsen, M. A. & Chuang, I. L.
Quantum Computation and Quantum
Information
(Cambridge Univ. Press, 2000).
3. Feynman, R. P. Simulating physics with computers.
Int. J. Theor. Phys. 21,
467 488 (1982).
4. Fannes, M., Nachtergaele, B. & Werner, R. F. Finitely correlated states on
quantum spin chains.
Commun. Math. Phys. 144, 443 490 (1992).
5. Verstraete, F. & Cirac, J. I. Renormalization algorithms for
quantum-many body systems in two and higher dimensions. Preprint at
<
http://arxiv.org/abs/cond-mat/0407066> (2004).
6. DiVincenzo, D. P. The physical implementation of quantum computation.
Fortschr. Phys.
48, 771 783 (2000).
7. Poyatos, J. F., Cirac, J. I. & Zoller, P. Quantum Reservoir Engineering with
laser cooled trapped ions.
Phys. Rev. Lett. 77, 4728 4731 (1996).
8. Perez-Garcia, D., Verstraete, F., Wolf, M. M. & Cirac, J. I. Matrix product state
representations.
Quant. Inf. Comput. 7, 401 430 (2007).
9. Verstraete, F., Murg, V. & Cirac, J. I. Matrix product states, projected entangled
pair states, and variational renormalization group methods for quantum spin
systems.
Adv. Phys. 57, 143 224 (2008).
10. Briegel, H. J. & Raussendorf, R. Persistent entanglement in arrays of interacting
qubits.
Phys. Rev. Lett. 86, 910 913 (2001).
11. Kitaev, A. Y. Fault-tolerant quantum computation by anyons.
Ann. Phys. 303,
2 30 (2003).
12. Levin, M. A. & Wen, X. G. String-net condensation: A physical mechanism for
topological phases.
Phys. Rev. B 71, 045110 (2005).
13. Cirac, J. I. & Zoller, P. New frontiers in quantum information with atoms and
ions.
Phys. Today 57, 38 44 (2004).
14. Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases.
Rev. Mod. Phys.
80, 885 964 (2008).
15. Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. Quantum dynamics of
single trapped ions.
Rev. Mod. Phys. 75, 281 324 (2003).
16. Lindblad, G. On the generators of quantum dynamical semigroups.
Commun. Math. Phys.
48, 119 130 (1976).
17. Kempe, J., Kitaev, A. Y. & Regev, O. The complexity of the local Hamiltonian
problem.
SIAM J. Comput. 35, 1070 1097 (2004).
18. Aharonov, D.
et al. Adiabatic quantum computation is equivalent to standard
quantum computation.
SIAM J. Comput. 37, 166 194 (2007).
19. Kempe, J. Quantum random walks an introductory overview.
Contemp. Phys.
44,
307 327 (2003).
20. Arora, S. & Safra, S. Probabilistic checking of proofs: A new characterization of
NP.
J. ACM 45, 70 122 (1998).
21. Gottesman, D. A theory of fault-tolerant quantum computation.
Phys. Rev. A
57,
127 137 (1998).
22. Wolf, M. M. & Cirac, J. I. Dividing quantum channels.
Commun. Math. Phys.
279,
147 168 (2008).
23. Wolf, M. M., Ortiz, G., Verstraete, F. & Cirac, J. I. Quantum phase transitions
in matrix product systems.
Phys. Rev. Lett. 97, 110403 (2006).
24. Verstraete, F., Wolf, M. M., Perez-Garcia, D. & Cirac, J. I. Criticality, the area
law, and the computational power of PEPS.
Phys. Rev. Lett. 96, 220601 (2006).
25. Verstraete, F., Cirac, J. I., Latorre, J. I., Rico, E. & Wolf, M. M.
Renormalization-group transformations on quantum states.
Phys. Rev. Lett.
94,
140601 (2005).
26. Diehl, S.
et al. Quantum states and phases in driven open quantum systems
with cold atoms.
Nature Phys. 4, 878 883 (2008).
27. Kraus, B.
et al. Preparation of entangled states by quantum Markov processes.
Phys. Rev. A
78, 042307 (2008).
28. Cohen-Tannoudji, C., Dupont-Roc, J. & Grynberg, G.
Atom-Photon
Interactions
(Wiley, 1992).
29. Perez-Garcia, D., Verstraete, F., Cirac, J. I. & Wolf, M. M. PEPS as unique
ground states of local Hamiltonians.
Quant. Inf. Comput. 8, 0650 0663 (2008).
Acknowledgements
We thank D. Perez-Garcia for discussions and acknowledge financial support by
the EU projects QUEVADIS, SCALA, the FWF, QUANTOP, FNU, SFB FoQuS, the
DFG, Forschungsgruppe 635, the Munich Center for Advanced Photonics (MAP)
and Caixa Manresa.
Author contributions
All authors have contributed equally to this paper.
Additional information
Supplementary information accompanies this paper on www.nature.com/naturephysics.
Reprints and permissions information is available online at http://npg.nature.com/
reprintsandpermissions. Correspondence and requests for materials should be
addressed to F.V. or J.I.C.
636
NATURE PHYSICS j VOL 5 j SEPTEMBER 2009 j www.nature.com/naturephysics
©
2009 Macmillan Publishers Limited. All rights reserved.
No comments:
Post a Comment