We introduce the multi-scale entanglement renormalization ansatz (MERA), an
efficient representation of certain quantum many-body states on a D-dimensional
lattice. Equivalent to a quantum circuit with logarithmic depth and distinctive
causal structure, the MERA allows for an exact evaluation of local expectation
values. It is also the structure underlying entanglement renormalization, a
coarse-graining scheme for quantum systems on a lattice that is focused on
preserving entanglement.Comment: 4 pages, 5 figure

G. Vidal

P. Calabrese

English

arXiv

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Class of Quantum Many-Body States That Can Be Efficiently Simulated

We introduce the multiscale entanglement renormalization ansatz, a class of quantum many-body states on a D-dimensional lattice that can be efficiently simulated with a classical computer, in that the expectation value of local observables can be computed exactly and efficiently. The multiscale entanglement renormalization ansatz is equivalent to a quantum circuit of logarithmic depth that has a very characteristic causal structure. It is also the ansatz underlying entanglement renormalization, a novel coarse-graining scheme for many-body quantum systems on a lattice

Vidal, G.

University of Queensland eSpace

Class of Quantum Many-Body States That Can Be Efficiently Simulated
G. Vidal
School of Physical Sciences, The University of Queensland, QLD 4072, Australia
(Received 3 December 2006; published 12 September 2008)
We introduce the multiscale entanglement renormalization ansatz, a class of quantum many-body states
on a D-dimensional lattice that can be efficiently simulated with a classical computer, in that the
expectation value of local observables can be computed exactly and efficiently. The multiscale entangle-
ment renormalization ansatz is equivalent to a quantum circuit of logarithmic depth that has a very
characteristic causal structure. It is also the ansatz underlying entanglement renormalization, a novel
coarse-graining scheme for many-body quantum systems on a lattice.
DOI: 10.1103/PhysRevLett.101.110501 PACS numbers: 03.67.Lx, 03.65.Ud, 03.67.Hk
A better understanding of quantum entanglement has
enabled significant progress in the numerical simulation
of quantummany-body systems over the past few years [1–
4]. Building on the density matrix renormalization group
[5]—a well-established technique for systems on a 1D
lattice—new insight from quantum information science
has led, e.g., to efficient algorithms to simulate time evo-
lution [1] and address 2D systems [2].
A key ingredient of such algorithms is the use of a
network of tensors to efficiently represent quantum
many-body states. Examples of tensor networks are matrix
product states (MPSs) [6] for 1D systems, tree tensor net-
works (TTNs) [3] for systems with a tree shape, and
projected entangled-pair states (PEPSs) [2] for 2D systems
and beyond. The three structures differ in the graph that
defines how the tensors are interconnected into a network:
The graphs for MPSs, TTNs, and 2D PEPSs are, respec-
tively, a chain, a tree, and a 2D lattice. Importantly, from
these tensor networks the expectation value of local ob-
servables can be computed efficiently. But whereas from a
MPS and a TTN such calculations are exact, from a PEPS
local expectation values can be obtained efficiently only
after a number of approximations.
In this Letter, we present the multiscale entanglement
renormalization ansatz (MERA), a structure that efficiently
encodes quantum many-body states of D-dimensional lat-
tice systems and from which local expectation values can
be computed exactly. A MERA consists of a network of
isometric tensors in Dþ 1 dimensions, where the extra
dimension can be interpreted in two alternative ways:
either as the time of a peculiar class of quantum computa-
tions or as parametrizing different length scales in the
system, according to successive applications of a lattice
coarse-graining procedure known as entanglement renor-
malization [7]. The MERA is a promising candidate to
describe emergent quantum phenomena, including quan-
tum phase transitions, quasiparticle excitations, and topo-
logical order. Here we establish its connection to
entanglement renormalization and explore some of its
basic properties: the efficient contractibility of the net-
work, leading to an efficient evaluation of local expectation
values; the inherent support of power-law correlations; the
versatility to adapt to both the local and the global structure
of the system’s underlying lattice; and the ability to as-
similate symmetries such as invariance under translations
or rescaling—which result in substantial additional gains
in computational efficiency.
Let us consider a lattice L in D spatial dimensions
consisting of N sites, where each site s 2 L is described
by a complex vector space V of finite dimension m. We
study states ji 2 VN of the lattice L that are generated
by a special type of quantum circuit C (Fig. 1). Each site in
L is identified with one outgoing wire of C, labeled with
the same index s. In order to further characterize the class
of circuits C, we need to introduce the notion of a causal
cone.
Definition.—The causal cone C½s of an outgoing wire s
of circuit C (correspondingly, a site s ofL) is the subcircuit
causally connected with s, that is, the subset of wires and
gates that influence the state of s. The width of the causal
cone C½s is the maximum number of wires involved in a
time slice C½s , where  is a discrete time label and wires
00 00 000 0
0000
00
0 0
θ
0
1
2
N
C
s
Ψ
O
(lo
g N
)
FIG. 1 (color online). Example of quantum circuit C that trans-
forms the product state j0iN into the state ji of lattice L, case
D ¼ 1. This circuit contains 2N  1 gates organized in OðlogNÞ
layers that are labeled by a discrete time . Notice the periodic
boundary conditions.
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that are in a product state j0i are not counted in the width of
C½s.
The defining feature of circuit C is that the width of the
causal cone of any of its N outgoing wires is bounded by a
constant that is independent of N. For instance, in the
example of Fig. 1, corresponding to a lattice in D ¼ 1
dimensions, the causal cone of any outgoing wire involves
at most four wires at any time , as highlighted in Fig. 2.
Circuits with this property can be built for any lattice L in
any spatial dimension D. Also, given a lattice L, there are
still many circuits with bounded causal cone width. For a
cubic lattice in D dimensions, Fig. 3 shows a possible
construction where causal cones have at most 3D14 wires
at any given time . The above constraint in the causal cone
is easily seen to imply that the depth of C is at most OðNÞ.
If, in addition, we assume that the circuit is organized in a
pattern that resembles a tree as in Fig. 1, then its depth is
OðlogNÞ.
The MERA is a network of tensorsM made from any
such circuit C with a few cosmetic changes. For concrete-
ness, we consider the circuit of Fig. 1. Each unitary gate u
of C gives rise to a tensor of the MERAM. But incoming
wires in the fixed state j0i are absorbed, producing three
kinds of tensors: (i) The top tensor t ¼ uj0ij0i ofM has
two indices and is normalized to 1:
ðtÞ  ðuÞj;¼0;
X

ðtÞðtÞ ¼ 1; (1)
(ii) tensors in every second row are isometries w ¼ uj0i:
ðwÞ  ðuÞj¼0;
X

ðwÞðwÞ0 ¼ ~~0 ; (2)
(iii) the rest of the tensors inM are unitary gates u:
X

ðuÞðuÞ00 ¼
X

ðuÞðuÞ00 ¼ ~~00 ; (3)
that we call disentanglers. Notice that in this example
storing M requires computational space that grows as
Oðm4NÞ, that is, linearly in N, given that there are 2N 
3 tensors and each tensor depends on at most m4
parameters.
Similarly, in the case of a generic lattice L in D spatial
dimensions, the MERA is an efficient representation for
certain pure states ji of L that consists of a tensor net-
work M in Dþ 1 dimensions with two properties:
(i) Tensors are constrained to be unitary or isometric as
in Eqs. (1)–(3); (ii) each open wire s, associated to one site
of the underlying lattice L, has a causal cone C½s with
bounded width. As a consequence of its peculiar causal
structure, the reduced density matrix of a small number of
lattice sites can be computed exactly with remarkably
small cost. In what follows, p1 and p2 are finite integers
that depend on the choice of C, and we assume that the
underlying circuit has depth OðlogNÞ.
Lemma 1.—The density matrix ½s  trsðjihjÞ for
one site s 2 L can be obtained from the MERA with
computational effort that scales as Oðmp1 logNÞ.
Proof.—Let  denote the reduced density matrix for
the time slice C½s of the causal cone C
½s. As illustrated in
Figs. 4 and 5 for the particular MERA of Fig. 2, þ1 can
be computed from , with a cost polynomial in m and
independent of N. Iterating, we obtain a sequence
f1; 2; . . . ; g of ¼ OðlogNÞ density matrices, where
½s ¼ . This completes the proof.
Lemma 2.—The two-site density matrix ½s1s2 can be
computed with time Oðmp2 logNÞ.
Proof.—Again, the causal cone C½s1s2 has logarithmic
depth and a width independent of N (see Fig. 7), and the
reduced density matrix þ1 for time slice C
½s1s2
þ1 can be
obtained from  for C
½s1s2
 with a cost Oðmp2Þ.
More generally, the reduced density matrix ½s1sq for q
sites can be computed with cost Oðmp logNÞ, where p
grows exponentially in q. The above lemmas readily imply
that the expectation value of local observables, such as
two-site correlators
θ
τ
0
1
2
0
1
2
t
w
u
M
s
C[s][s]
FIG. 2 (color online). The MERAM is made of three types of
tensors: t, w, and u; see Eqs. (1) and (2). Its causal structure is
inherited from that of the circuit C: The causal cone C½s for site s
has bounded width. When reversing the arrow of time , M
corresponds to OðlogNÞ entanglement renormalization transfor-
mations labeled by 	.
A
B C
CBAθ
x y
τ
FIG. 3 (color online). Detail of a causal cone of a simple
MERA construction for a square lattice, D ¼ 2. 3 3 wires
(bottom left) are mapped into 3 3 wires (top right) by layers of
tensors that act along the y and x directions [some stages involve
3 4 wires]. The generalization to D> 2 is straightforward.
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C2ðs1; s2Þ  hja½s1b½s2ji ¼ trð½s1s2a½s1b½s2Þ; (4)
can be computed efficiently and exactly from the MERA.
We now show how the MERA relates to a real-space
renormalization group transformation. We reverse the ar-
row of time in the quantum circuit C, and, starting from
j0i  ji, we consider the sequence of states
fj0i; j1i; j2i; . . .g, where j	i is the [nonfactorizable
part of the] state of C at time  ¼  2	 (Fig. 6). Notice
that j	þ1i is obtained from j	i by applying one layer of
disentanglers u and one layer of isometries w. In other
words, M implements entanglement renormalization
transformations [7]. More generally, we can use the
MERA to transform the sites of lattice L0  L, as well
as linear operators defined onL such as a HamiltonianH0,
so as to obtain a sequence of increasingly coarse-grained
lattices fL0;L1;L2; . . .g and corresponding effective
Hamiltonians fH0; H1; H2; . . .g. An operator defined on
site s 2 L is mapped into an operator contained in the
causal cone C½s, so that local operators in L0 remain local
in L	.
Numerical evidence for 1D lattices [7] shows that a
MERA encodes, in a markedly more efficient way than a
MPS [8], accurate approximations to the ground state near
to or at quantum criticality. At a critical point, correlators
C2ðs1; s2Þ decay as a power law with the distance r between
sites s1 and s2. Next we argue that the MERA supports
algebraic decay of correlations. This is the case in any
dimension D.
It follows from the geometry of the 
-shaped causal
cone C½s1s2 that we can compute ½s1s2 from the density
matrix  	 corresponding to a block of contiguous sites of
L 	, where 	  log2r, by means of a sequence of density
matrices f 	; . . . ; 2; 1; ½s1s2g; see Fig. 7. That is, ½s1s2
is obtained from  	 after OðlogrÞ transformations. If, as it
can be argued in a (scale-invariant) critical ground state,
each of these transformations reduces correlations by a
constant factor z < 1, we readily obtain a power-law decay
for C2ðs1; s2Þ:
C2ðs1; s2Þ  zlogr ¼ rq; q  log1z > 0: (5)
1σ
s s
2σ
3σ
σΘ
FIG. 4 (color online). Left: Tensor network for ½s ¼
trLnsjihj in Lemma 1. Right: The gates outside the causal
cone of s can be removed by using the unitarity constraints of
Eqs. (2) and (3). The resulting tensor network can now be
contracted by sequentially computing OðlogNÞ density matrices
f1; 2; . . .g.
FIG. 7 (color online). Causal cone C½s1s2 for D ¼ 1. A small
square at the end of an open wire indicates a trace over the
corresponding degrees of freedom. The states ½s1s2 are obtained
from  	 through a sequence of OðlogrÞ transformations. Similar
constructions hold for D> 1.
Ψ
Ψ ⊗
Ψ ⊗
FIG. 6 (color online). Renormalization group transformation
associated to the MERA. The tensors in M define a sequence
fL0;L1; . . .g of increasingly coarser lattices in states
fj0i; j1i; . . .g.
A B C
D GE H
I
J K L
M
(ii)(i)
(iii)
(iv)
2 1kσ −
A B C
A B C
FE G H
FE G H
2kσ
J K L
J K L
2 1kσ +
= A B C
D FE G H I
2 1kσ −
=
FGE H
J K L
M
2kσ
2kθ =
2 1kθ = +
2 1kθ = −
F
FIG. 5 (color online). Efficient computation of ½s, Lemma 1.
(i) Part of the causal cone C½s of ½s. (ii) Sequence of density
matrices 2k1 ! 2k ! 2kþ1 represented as tensors with 6 or
8 indices. (iii) 2k expressed in terms of 2k1 and isometries w.
(iv) 2kþ1 expressed in terms of 2k and disentanglers u. By
contracting all of the indices shared by two tensors in (iii) and
(iv), 2k and 2kþ1 are obtained with cost Oðmp1 Þ.
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This property, valid in any spatial dimension D, is enabled
by the fact that two sites separated a distance r in L are
connected through a path of length OðlogrÞ inM, and it
indicates that a MERA is particularly well suited to de-
scribe states with quasi-long-range order, such as critical
ground states. As demonstrated in Refs. [7,9], the MERA
can also reproduce the logL scaling of entanglement for a
block of L contiguous sites that characterizes critical
ground states in D ¼ 1 dimensions [10] and the boundary
law LðD1Þ for a hypercube of LD sites in D> 1
dimensions.
For the sake of concreteness, we have focused on a
specific MERA for aD ¼ 1 lattice and indicated a possible
extension to a square (cubic) lattice for D> 1. A generic
lattice can be addressed by properly modifying circuit C
while keeping its characteristic causal structure. For in-
stance, in D ¼ 2 dimensions, a specific MERA can be
engineered to represent states of a triangular lattice, or of
a lattice with random vacancies or linear or bulk defects, or
to account for a variety of boundary conditions (e.g., plane,
cylinder, sphere, or torus). In addition, the number of levels
m can vary throughout M. When used as the basis for
numerical simulations, adjusting the MERA to the specif-
ics of a problem leads to computational gains.
In particular, the symmetries of ji can be assimilated
into the MERA. An internal symmetry, such as SUð2Þ
invariance, results in a series of constraints for the tensors
inM [11], which then depend on fewer parameters. More
substantial gains are obtained when ji is invariant under
translations (that is, cyclic shifts by one lattice site in a
system with periodic boundary conditions), since all of the
tensors in a layer ofM can be chosen to be the same and
the MERA depends on Oðm4log2NÞ parameters instead of
Oðm4NÞ. The most dramatic simplification occurs for
states that are invariant under entanglement renormaliza-
tion transformations, even in an infinite lattice [7]. Now all
disentanglers and isometries in M are identical and the
MERA depends on just Oðm4Þ parameters. Some 1D criti-
cal ground states are numerically seen to belong to this
class [7].
We conclude with a few pointers to future work. On the
one hand, most techniques to simulate quantum systems
with a MPS can be generalized to a MERA. This includes
algorithms to compute the ground and thermal states and to
simulate time evolution. Importantly, in order to updateM
after a unitary gate v½s1s2 acts on sites s1 and s2, only the
tensors in the causal cone C½s1s2 need to be modified, with a
cost logarithmic in N.
On the other hand, the potential of the MERA goes
beyond representing individual states. Notice that, by feed-
ing the incoming wires, labeled by r, of quantum circuit C
with an arbitrary product state Nr¼1j½rr i instead of
Nr¼1j0½ri, we can generate a different entangled statejfrgi for the lattice, this one represented by a MERA
that differs from the original one only in the isometric
tensors w and such that, given the unitarity of C, fulfills
hfrgjf0rgi ¼
Y
r
h½rr j0½rr i: (6)
This can be used to encode, in just one (generalized)
MERA, not only the ground state of a Hamiltonian H but
also its quasiparticle excitations. Moreover, in systems
with topological order, this property allows one to repre-
sent sets of topologically inequivalent ground states in a
single MERA, by storing the topological information on
the top tensor t [12].
The author acknowledges support from the Australian
Research Council, No. FF0668731 and No. DP0878830.
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[8] In a MPS, correlations between the left and right halves of
a chain are accounted for through mMPS terms in the
Schmidt decomposition. A MERA can account for about
2mlog2N terms in the Schmidt decomposition. Thus, it
takes a MPS with mMPS  2mlog2N to represent a state
stored with a MERA of m-level wires.
[9] G. Evenbly and G. Vidal, arXiv:0710.0692v2;
arXiv:0801.2449v1.
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Class of quantum many-body states that can be efficiently simulated

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A class of quantum many-body states that can be efficiently simulated

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