low entanglement wavefunctions

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Focus Article

Low entanglement wavefunctionsGarnet Kin-Lic Chan

We review a class of efficient wavefunction approximations that are based around

the limit of low entanglement. These wavefunctions, which go by such namesas matrix product states and tensor network states, occupy a different regionof Hilbert space from wavefunctions built around the HartreeFock limit. Thebest known class of low entanglement wavefunctions, the matrix product states,forms the variational space of the density matrix renormalization group algo-rithm. Because of their different structure to many other quantum chemistrywavefunctions, low entanglement approximations hold promise for problemsconventionally considered hard in quantum chemistry, and in particular prob-lems which have a multireference or strong correlation nature. In this review,we describe low entanglement wavefunctions at an introductory level, focus-ing on the main theoretical ideas. Topics covered include the theory of effi-cient wavefunction approximations, entanglement, matrix product states, andtensor network states including the tree tensor network, projected entangled pairstates, and the multiscale entanglement renormalization ansatz. C 2012 John Wiley &Sons, Ltd.

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WIREs Comput Mol Sci 2012, 2: 907920 doi: 10.1002/wcms.1095

INTRODUCTION

I n wavefunction-based quantum chemistry, wewish to determine the stationary states of the elec-tronic Schrodinger equation. This is a difficult task

because the dimension of the Hilbert space growsexponentially with the number of electrons. Thus,any practical calculation must use some simplifyingapproximations, or ansatz, for the state we wish tostudy.

Figure 1 illustrates schematically the full Hilbertspace of wavefunctions. Complete expansions, for ex-ample, via full configuration interaction or full reso-nance valence bond theory, span the entire Hilbertspace at exponential cost. Practical wavefunction ap-proximations are of polynomial cost, and are nec-essarily biased toward a given portion of the Hilbert

space. In opposite corners of the rectangle in Figure 1,we show two families of approximations salient toour discussion. One is biased toward the HartreeFock mean-field or independent particle limit. Suchwavefunctions are well developed in quantum chem-istry and can be systematically made more flexible

Correspondence to: [email protected]

Department of Chemistry and Chemical Biology, Cornell Univer-sity, Ithaca, NY 14853, USA

DOI: 10.1002/wcms.1095

by increasing the excitation level in the wavefunc-tion. The other family of wavefunctions, which arethe main subject of this review, are biased toward theindependent local subsystem, or zero entanglement,

limit. These wavefunctions are systematically mademore flexible by increasing the amount of encodedentanglement.

Low entanglement wavefunctions first becamewidely applied with the advent of the density ma-trix renormalization group (DMRG).18 Within afew years, it was understood that the DMRG algo-rithm is an energy minimization algorithm within aclass of low entanglement wavefunctions known asmatrix product states (MPS).57,915 Decimation inthe renormalization group (RG) algorithm is equiv-alent to encoding limited entanglement in the state,linking the RG picture with quantum information

perspectives.6,13,1517 In the last decade, MPSs alsohave been used in quantum chemistry via the DMRGalgorithm.8,1750 Quantum chemical DMRG has sig-nificantly widened the scope of ab initio calcula-tions with respect to strongly correlated electrons.In particular, it has allowed multireference activespace calculations with more than 30, and in somecases up to 100, active orbitals (see Refs 8,22,23,2530,42,43,4749 and Ref 7 for a summary of recentcalculations).

Vo l u m e 2 , N o v e m b e r / De c e m b e r 2 0 1 2 907c 2 0 1 2 Jo h n Wi l e y & S o n s , L td .


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Focus Article wires.wiley.com/wcms

F I G U R E 1 | The Hilbert space of many-electron wavefunctions.

The independent particle and independent local subsystem limits lie in

different regions of the Hilbert space. By increasing the excitation level

or entanglement, respectively, in approximations built around theselimits, they can be made to span the full Hilbert space.

F I G U R E 2 | Area laws express locality in physical systems. In a

one-dimensional system, the boundary area is independent of theoverall system size, thus entanglement entropy in a physical ground

state is expected to be independent of system size, unless at a

quantum critical point. In two-dimensional system, the boundary area

can scale as the system width, and the entanglement entropy scales

similarly, except for critical and topological corrections.

More recently, generalizations of MPSs, whichremove some fundamental limitations, for example,with respect to system dimensionality, have beenintroduced.15,5164 Such tensor network wavefunc-tions are still in their infancy, and efficient compu-tational techniques to work with them remain under

development. Indeed, they have yet to be applied toquantum chemistry (also, see Refs 50, 60, 65 for earlywork in this direction). Nonetheless, they hold consid-erable promise due to their increased flexibility overMPSs. We describe them also in this review.

The structure of the discussion is as follows. Insection Wavefunctions and the Orthogonality Catas-trophe, we first consider why, despite the exponen-tial size of the Hilbert space, it is possible to faith-fully approximate a quantum state. In sections Wave-

functions around the HartreeFock Limit and LowEntanglement Wavefunctions, we describe the twofamilies of wavefunctions presented above. We be-gin with a brief discussion of wavefunctions builtaround the independent particle limit. Then we turnto our main focuslow entanglement wavefunctions.We introduce the mathematical definition of entan-glement, and proceed to describe in detail the theoryof MPS. We next describe more general tensor net-work wavefunctions, including tree tensor networks(TTNs), projected entangled pair states, and the mul-tiscale entanglement renormalization ansatz (MERA).We finish with our conclusions in section Conclu-sions.

WAVEFUNCTIONS AND THEORTHOGONALITY CATASTROPHE

How do we construct a many-electron wavefunctionto represent a quantum state? At first glance, thereappears to be a fundamental problem, as the orthog-onality catastrophe shows that in a large system, anyapproximate wavefunction is orthogonal to the stateof interest. This might even suggest that a wavefunc-tion is not a valid way to describe large quantum sys-tems (see, e.g, the discussion in Ref 66). To illustratethe catastrophe, consider a set of N noninteractingelectrons. For simplicity, we neglect antisymmetry.Then the ground-state wavefunction is a product ofNorbitals

= 1(r1)2(r2) . . . N(rN). (1)We imagine representing each orbital by an approx-imate , with a small error,

| = 1 . (2)The overlap of the approximate and exact wavefunc-tions is a product of the overlaps of the orbitals

| = 1|12|2 . . . N|N= (1 )N

exp(N). (3)

We see that the overlap of the true and approxi-mate determinants decreases exponentially with thenumber of particles. This is not essentially changedif include antisymmetry, or add in the interactionsbetween the electrons. This suggests that any approx-imate wavefunction is a very poor representation ofthe quantum state in a large systems.

The practical success of quantum chemistry,however, indicates that approximate many-electronwavefunctions do contain useful information. Indeed,

908 Vo l u m e 2 , N o v e m b e r / De c e m b e r 2 0 1 2c 2 0 1 2 J o h n W i l e y & S o n s , L t d .


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WIREs Computational Molecular Science Low entanglement wavefunctions

even with a poor overlap, expectation values of op-erators can be very good. In the above case, for asingle-particle operator Q =i qi , with expectationvalue Q = | Q|,

| Q| =i

i |qi |i = Q+ NO(). (4)

We see that the error in the expectation value perparticle is O(), independent of the number of par-ticles. Thus, wavefunctions with poor overlap canyield meaningful expectation values even in the limitof large particle number, that is, the thermodynamiclimit.

Further examination of the original argumentalso shows that the catastrophe is not actually as se-vere as it first appears. We assumed that we computeeach orbital to within an error of, regardless of howlarge the system is. However, we could try to repre-sent each orbital more accurately as the number of

electrons increases. Indeed, to defeat the exponentialloss of overlap we need the error in each orbital todecrease such as

| 1 N. (5)

In this case, the approximate wavefunction can retainunit overlap with the true state. For the scheme tobe practical, we must be able to determine an orbitalto a given error of in a time polynomial in 1.Current methods to solve the single-particle molecu-lar Schrodinger equation (e.g., by Gaussian basis setexpansion) satisfy this criterion.6770 Thus, even using

the total wavefunction overlap as a measure of accu-racy, it is possible in some cases to faithfully representthe state of a large system with a wavefunction.

Now that we have established that approximatewavefunctions can in principle provide meaningfuldescriptions of N electron systems, we still have theproblem of determining appropriate approximationsfor physical problems. We first describe wavefunc-tions that are built around the HartreeFock limit,then move to our main focus of low entanglementwavefunctions.

WAVEFUNCTIONS AROUND THEHARTREEFOCK LIMIT

Wavefunctions built around the HartreeFock mean-field limit are widely used in quantum chemistry. TheHartreeFock wavefunction is the determinant whichminimizes the energy E = |H|/|71,72. Start-ing from a determinant allows a separation of orbitalsinto an occupied space (labeled by i, j, . . .) and a vir-tual space (labeled by a, b, . . .), thus defining excita-

tions relative to the determinant. In the standard ex-citation hierarchy, one expands a general wavefunc-tion as a linear combination of excitations around theHartreeFock reference,

| = |0 +ia

cai |ai

+

i>j,a>b

cabi j |abi j + . . . . (6)

By mixing in successively higher excitations, one cov-ers the full quantum Hilbert space. Compact wave-function approximations are constructed by restrict-ing the generality of the excitation expansion. Intruncated configuration interaction and perturbationtheories,71,72 we restrict the maximum excitationlevel in the expansion (6), whereas truncated coupledcluster theory rewrites the expansion first in clusterform7174

| = exp( T)|0, (7)and truncates the excitation level in the cluster oper-ator T,

T=

ia

tai aa ai +

i>ja>b

tabi j aa a

bai aj + . . . . (8)

There are other ways to construct wavefunc-tions around the HartreeFock limit aside from anexcitation expansion. Another class of compact wave-functions is obtained by applying diagonal operators,known as Jastrow factors (sometimes known as cor-relation factors), to the determinant.7577 Jastrow fac-

tors take the general form

J =

i

ji ni +i>j

ji j ni nj + . . . , (9)

where ni is a number operator ai ai in some basis,

not necessarily the molecular orbital basis of the de-terminant. The Jastrow determinant wavefunction isthen

| = J|. (10)As successively more terms are included in the Jas-trow factor, the Jastrow determinant wavefunction

can also cover the full quantum Hilbert space.In both the excitation and Jastrow expansions,

the choice of starting determinant is not unique.For example, the HartreeFock energy minimizationhas many local minima, each of which may forma suitable reference. Some solutions of the energyminimization do not preserve the symmetry of theoriginal Hamiltonian. Symmetry-broken solutionscan often be associated with different kinds of elec-tronic phases and can thus be chosen to reflect the



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nature of the final state we wish to capture. Break-ing spin symmetry yields unrestricted and generalHartreeFock determinants, while breaking numbersymmetry leads to the BardeenCooperSchrieffer(BCS) wavefunction.78 (We refer to the BCS wave-function also as a determinant for simplicity, althoughit is a determinant of quasiparticles, not electrons.) Inmolecular systems, the BCS wavefunction is typicallyprojected to restore the particle number symmetry,and the projected states are known as the antisym-metrized geminal power79,80 or (if they break spinsymmetry also) Pfaffian wavefunctions.81

Despite the flexibility in the choice of start-ing reference, there are some situations when nogood starting determinant can be found. Determinantwavefunctions are eigenfunctions of one-particle ef-fective Hamiltonians

h =i j

ti j ai aj + i j (ai aj + aj ai ), (11)

where the first term is a number conserving one par-ticle term, and the second is the pairing field associ-ated with breaking particle number symmetry in theBCS wavefunction. The quality of the starting refer-ence is governed by the magnitude of the perturbationV= H h, where H is the full electronic Hamilto-nian of the problem. Strongly correlated electronicstructure, commonly found in transition metal chem-istry and excited electronic states, is characterized bya large magnitude of V relative to the energy spacingof h.

For small numbers of strongly correlated elec-trons, one can augment the starting reference deter-minant by a linear combination of starting determi-nants. This gives rise to multireference wavefunctionapproximations. Perturbative, configuration interac-tion, and coupled cluster theories from a multirefer-ence starting point can be formulated,72,8286 as canmultireference Jastrow wavefunctions.87 However,multireference approximations are compact only solong as the number of starting references remainspolynomial in the system size. This is not the casefor large numbers of strongly correlated electrons.We therefore turn our attention to a different class of

wavefunction approximations that seems well suitedto strong correlation problems: the low entanglementwavefunctions.

LOW ENTANGLEMENTWAVEFUNCTIONS

Valence Bond TheoryThe motivations for constructing wavefunctions byencoding entanglement between local objects are il-

lustrated by the stretched two-electron bond in the hy-drogen molecule. Using an excitation expansion suchas Eq. (6), the wavefunction at large separation maybe written as

| = 21/2(A[g(1)g(2)]

A[u(1)u(2)]), (12)where g and u are the bonding and antibondingmolecular orbitals. The stretched bond wavefunctionrequires a multireference approximation as there is nodominant determinant. At large separation, however,it seems more natural to build the wavefunction fromthe atomic states. Denoting the atomic orbitals as 1saand 1sb, we write

| = 12

[1sa (1)1sb(2) + 1sa (2)1sb(1)]

[a (1)b(2) b(1)a (2)]. (13)

The coupling between the atomic spins reflects strongcorrelations. For example, if we measure spin onatom A, the corresponding spin measurement on atomB will yield spin . This type of strong correlation isknown as entanglement, and is associated with thechemical bond. (Note that as discussed above, strongcorrelation is commonly defined with respect to aperturbative partitioning of the Hamiltonian; strongcorrelation and entanglement are thus not synonyms.One of the attractions of entanglement as a descriptor,rather than strong correlation, is the ability to defineprecise mathematical measures of entanglement, as il-

lustrated in the next section). However, in a molecule,not all atoms are bonded to all other atoms, that is,there are only spin couplings of the above form alongspecific chemical bonds. This suggests that compactwavefunction approximations may be constructed byencoding only the entanglement between local statesto reflect the bonding network of the molecule.

Equation (13) is also the starting point for va-lence bond (VB) theory. In VB theory, the wavefunc-tion is expanded in a valence bond basis, correspond-ing to the classical resonance structures of chemistry.For example, for three hydrogen atoms, there arethree neutral doublet resonance structures. However,

transforming to the valence bond basis does not byitself yield a compact wavefunction, as the basis is ex-ponentially large. For a compact parametrization, wehave to limit the number of resonance structures in theexpansion. If only a single-resonance structure is used,we obtain a method known as generalized valencebond theory.88,89 Moving beyond a single-resonancestructure theory in a scalable, fully general, and com-pact way, however, proves quite difficult.9092 Lowentanglement wavefunctions change the focus from



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the bonds to the local states and provide a more prac-tical framework for compact approximations. Wenow turn to a mathematical discussion of entangle-ment.

EntanglementBipartite entanglement,93 or the entanglement be-tween two systems, is defined as follows: let systems1, 2 be spanned by sets of (orthonormal) states {|n1},{|n2}, respectively. Any state in the combined system1, 2 is expanded in the product space {|n1n2} as

| =n1n2

n1n2 |n1n2. (14)

(Here we work in a second quantized formulation,where fermion antisymmetry is expressed in the com-mutation relations between operators, rather than byantisymmetrizing products of states). The bipartite

entanglement entropy is defined from a singular valuedecomposition of the coefficients n1n2 . Writing thisas a matrix , then

= UVT, (15)where is the diagonal matrix of singular values. Theentanglement entropy is

S =

i

2i log2 2i . (16)

The entanglement entropy quantifies the corre-lations between the systems. If the wavefunction is aproduct wavefunction, there is only a single nonzerosingular value, and the entanglement entropy is zero.In the maximally entangled state, all singular valuesare the same. The maximally entangled state is thuswritten as

| = 1M

Mi

|ri li , (17)

where {|ri} and {|li} denote the basis in which theen-tangling is performed, and the entanglement entropyis log2M. The hydrogen molecule at infinite separa-tion is in a maximally entangled state. Consideringthe spin part of the wavefunction

|spin =1

2[(1)(2) (1)(2)], (18)

we find that the entanglement entropy is log22 = 1,that is, there is one pair of entangled spins.

Any wavefunction can be written in a diagonalform similar to the maximally entangled state. FromEq. (15), the matrices U and V define a transforma-tion of the original bases. We define {|li} and {|ri}

through

|ri =

n1

Un1i |n1, (19)

|li =n2

Vn2i |n2. (20)

In terms of the transformed basis, the wavefunctionthen assumes a diagonal form

| =

i

i |ri li. (21)

Matrix Product StatesWhen constructing a wavefunction from the statesof two local systems, the wavefunction can always bewritten in the diagonal form | =ii|rili. Relatingthis to the original basis |n1n2, this gives

| =

n1n2,i

An1i An2i |n1n2, (22)

where we have for notational purposes absorbed thesingular values in Eq. (15) into the definitions of theA matrices. This form is very suggestive, as we canthink of the index i (which we call an auxiliary index)as entangling the original states {|n1}, {|n2}, in thetwo systems.

We can now try to generalize the above to morethan two local systems. Consider three systems 1, 2,and 3. We imagine entangling system 1 with system 2,

and then system 2 with system 3. The natural gener-alization of Eq. (22) is to use two entangling auxiliaryindices i1, i2, and to write the wavefunction as

| =

n1n2n3

i1i2

An1i1 An2i1i2

An3i2 |n1n2n3

=

n1n2n3

An1 An2 An3 |n1n2n3, (23)

where in the last line we have switched to a matrixnotation, and An1 and An3 are thin matrices (a rowvector and a column vector, respectively). The mul-tiplication of the matrices An1 , An2 , An3 , yields the

wavefunction coefficient n1n2n3 associated with theconfiguration |n1n2n3. Since the wavefunction coef-ficient is obtained as a matrix product, this wavefunc-tion approximation is known as a matrix productstate (MPS).57,915 The accuracy and complexity ofthe MPS is controlled by the dimension of the auxil-iary indices i1, i2 of the matrices. If we fix this at somedimension M, we say that the MPS is of dimensionM.



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We can generalize the MPS to k systems, wherewe entangle them in a sequential manner,

| =

n

An1 An2 . . .Ank|n, (24)

where n denotes the configuration |n1n2. . .nk. Notethat the MPS construction is general and does notdepend on the definition of the local system or thenature of the constituent states |n. For example, wemay take each system to be an atom (in which casethe local states span the Fock space of each atom,where by Fock space we mean the full Hilbert spaceacross all particle number sectors). Or, we could takeeach system to be as small as a single orbital, spannedby the four states, | , |, |, |. To emphasizethis generality, we now introduce the term site as asynonym for local system. This is the terminologyprimarily used in the MPS and DMRG literature. Thewavefunction approximation in Eq. (24) is thus an

MPS wavefunction for k sites.One way to analyze the accuracy of the MPS ap-

proximation is to consider how much (bipartite) en-tanglement entropy can be described by such a state,if we imagine dividing our k sites into two sets. If thedimension of the auxiliary indices is M, then the max-imum entanglement expressed by the MPS for any cutbetween the sites is log2M, irrespective of the numberof sites in the problem. We might, however, more nat-urally expect the true entropy to capture in a physicalproblem to scale with the size of the problem.

The link between entanglement entropy, sys-

tem size, and dimensionality is given rigorously byso-called area laws.9496 Whereas thermodynamic en-tropy is an extensive quantity (i.e., it is simply propor-tional to volume), entanglement entropy is believedto scale only with the size (i.e., area) of the bound-ary between the two systems used to define the en-tropy. At an intuitive level, this expresses locality.Given local interactions in the Hamiltonian, then inthe ground state, only quantum degrees of freedomalong the boundary can be entangled with each other,and thus the entropy scales with boundary size. Theboundary size depends on the overall system size in adimensionally dependent way. In one dimension, any

division of the system yields a boundary size indepen-dent of system size, thus entanglement entropy in aone-dimensional ground state is independent of sys-tem size. However, in a two- or higher-dimensionalsystem, the boundary of a division of the system canscale like the width L of the system, and thus the en-tanglement entropy scales as the system width (seeFigure 2).

As we argued above, for an MPS of dimen-sion M, the maximum entanglement entropy log2M

is independent of system size, and we see that thisis clearly appropriate to a one-dimensional systemground state. If we apply an MPS to describe a twoor higher-dimensional system, however, to capturethe growing entanglement entropy, the matrix di-mensions must grow exponentially with the boundaryarea, that is, M

eL.

This may suggest that using an MPS to simulatetwo- and three-dimensional systems is a bad idea. Itis indeed possible to devise states with more generalentanglement structures than the MPS, which natu-rally capture the area law in higher dimensions andsuch states are discussed in section General TensorNetwork States. However, crucially, the MPS cap-tures the correct structure of entanglement for oneof the spatial dimensions, and thus the exponentialcost is associated only with the remaining dimensions,that is, an area. This is a clear improvement overfull configuration interaction, where the cost scales

exponentially with the system volume. In practice,therefore, MPS offers significant computational ad-vantages when simulating systems even with two- orthree-dimensional connectivities.97

We now discuss a few miscellaneous topics re-lated to MPS: algorithms, their interpretation as pro-jected wavefunctions, and finally their graphical rep-resentation.

MPS AlgorithmsMPSs are compatible with many efficient computa-tional algorithms. For example, overlaps and expec-

tation values can be obtained efficiently.5,6,15

We il-lustrate this with the computation of the norm. Thenorm of a matrix product state is

| =

n

(An1 An2 . . .Ank) (An1 An2 . . .Ank)

= E1E2 . . .Ek, (25)where the E matrices are of dimension M2 M2,

Ei =

ni

Ani Ani . (26)

The evaluation of the norm is itself a matrix prod-

uct. From Eq. (25), it would appear to take O(M4k)time; however, using the fact that the E matrices havean outer product structure, it requires only O(M3k)time. By a similar construction, one finds that expec-tation values of local operators can also be evaluatedin O(M3k) time.6

The expectation value structure in Eq. (25) isreminiscent of a transfer matrix expression in statis-tical mechanics, where the E matrices are the trans-fer operators. This structure means that correlation



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F I G U R E 4 | (Bottom) A norm can be represented graphically by

fusing the vertical indices of the matrix product state (MPS) bra

(pointing downward) with the vertical indices of the MPS ket (pointing

upward), denoting summation over all the state indices, n1, n2. . .nk.

(Top) Performing a state summation at each site leads to an

intermedate matrix E of dimension M2, and the norm evaluation is a

product of the E matrices. (Note that the leftmost and rightmost

matrices are, in practice, row and column vectors.)

and a vertical line (the site index n). Lines that areconnected represent indices that are summed over.

The graphical representation is useful not onlyfor representing the states, but also operators and

expectation values. The computation of an overlapis shown in Figure 4. The graphical representationclearly shows the nature of the index contractions.This becomes even more essential when consideringstates with a more general entanglement structurethan the MPSs, to which we now turn.

General Tensor Network StatesThe limitation of MPS is that they can only efficientlyencode a sequential structure of entanglement. Forarbitrary systems, more general encodings can be ex-plored. This is a rapidly expanding area of investi-

gation, and many questions remain to be answered.Here we discuss three promising kinds of tensor net-works: TTN,57,58,60 which extend MPSs and maintaina similar computational efficiency, but which still re-strict the structure of the entanglement; projected en-tangled pair states (PEPS),,15,52,54,55 which are a verygeneral way to encode a network of entanglement;and the MERA,6164 which combines some of thepractical strengths of the TTN while achieving someof the formal advantages of PEPS.

F I G U R E 5 | (Left) For every local state |np, a tree tensor network(TTN) associates a tensor with zauxiliary indices. Here shown is a TTN

where each tensor has three auxiliary indices. (Right) The

wavefunction coefficient n1n2...nk

is obtained by contracting theauxiliary indices according to the tree connectivity

Tree Tensor NetworksIn an MPS, a local system is represented by a matrixAn with two auxiliary indices, which allows entan-gling to the left and right neighbors. In the general-ization to a tree structure, a local system is representedby a tensor with z auxiliary indices, which allows cou-plings to z neighbors. The TTN state can be writtenas

|

=n z i Anizi

|n

, (33)

where zi denotes the set of z indices on each tensor,and

z denotes the contraction with the indices ofneighboring tensors according to the connectivity ofthe tree, shown in Figure 5. The accuracy of the TTNis controlled by the dimensions of the auxiliary indicesz, which we can denote by M.

One advantage of the TTN over the MPS is thatthe maximum distance between two sites in a tree islogarithmic in the total number of sites, as opposedto linear in the case of the MPS. This allows the TTNto describe states with correlation functions, which

decay algebraically with respect to site separation,as opposed to exponentially in the case of MPS, asdescribed in section MPS Algorithms.

A second advantage of the TTN is that, just asin the MPS, expectation values can be efficiently ob-tained. By contracting the tensors from out to in, wesee that the cost for calculating a norm is O(M)z+1k.While the scaling of a z > 2 TTN computation ishigher than that for a corresponding MPS (z = 2),the ability of TTN to better represent long-range



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F I G U R E 6 | Projected entangled pair states (PEPS) are associated

with arbitrary networks of entanglement. Shown here is a

representation of the PEPS for a square lattice of sites. Each local state

|np is then associated with a tensor with four auxiliary indices l, r, u, dthat point along the bonds of the square lattice. The PEPS

approximation for the wavefunction coefficient n1n2...nk is obtained

by contracting all the auxiliary indices.

correlations means that we expect a TTN will reachthe accuracy of a given MPS calculation with smallerauxiliary dimension of the tensors. Preliminary inves-tigations appear to confirm this.60

Projected Entangled Pair StatesThe TTN and the MPS possess a special connectivity,with no cycles between the tensors in the wavefunc-tions. This is why these states are compatible withvery simple and efficient algorithms for calculatingnorms and expectation values. As one contracts the

expectation value network, shown in Figure 4 for theMPS, one proceeds through intermediates where thenumber of dangling bonds is independent of systemsize.

States with arbitrary networks of entanglementon the other hand, including cycles, are known asPEPS. To illustrate the PEPS construction, we con-sider an appropriate state to describe a square latticeof sites. In the PEPS, each site is associated with atensor Anz , where z denotes the four auxiliary indicesl, r, u, d that point toward the neighbors in the di-rections left, right, up, and down. In this notation,the PEPS wavefunction takes exactly the same form

as the TTN (33), except the auxiliary indices are con-tracted according to the bonds of the square lattice(see Figure 6).

The name PEPS arises from the projected wave-function interpretation of the state, described in sec-tion Projected Wavefunctions for MPSs. Every site issurrounded by four bonds on which we place maxi-mally entangled states of auxiliary particles,

m |,

and each site is associated with a projector operatorwhich projects from the auxiliary particles onto the

F I G U R E 7 | (Bottom) Projected entangled pair states (PEPS) norm

evaluation can be represented by fusing a PEPS bra lattice (pointing

downward) with a PEPS ket lattice (pointing upward). The result is a

square lattice which must be contracted over the auxiliary indices.

(Top) Each contraction of the local states |np results in an E tensorwith four auxiliary indices, each of dimension M2.

physical particles,

A =

nlrud

Anlrud|nlrud|. (34)

The whole PEPS wavefunction is the product of suchcommuting projection operators (associated with thesites) on a reference, which is a product of entangledpairs on the bonds.

PEPS wavefunctions can be constructed to sat-isfy the correct entanglement area laws in two andthree dimensions. Consequently, they are expected toprovide compact descriptions of the ground states ofalmost arbitrary physical systems. However, the abil-ity to represent the wavefunction compactly does notguarantee that computations of observables are effi-cient. For the square lattice PEPS above, the evalu-ation of an expectation value requires contracting asquare lattice network of E-type tensors, as shownin Figure 7, where Ezz =

n A

nz A

nz . As we contract

column by column, we form intermediates where thenumber of dangling bonds is proportional to the sys-

tem width. The cost of forming these intermediates,and thus the cost of the contraction, grows expo-nentially with system width. However, for physicalsystems, it appears to be possible to approximatethese intermediates by simpler forms. For example,a two column contracted intermediate can be accu-rately approximated by a single-column intermediatewith slightly larger auxiliary dimension (Figure 8).This means that an iterative approximation algo-rithm can be constructed where the evaluation of an



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F I G U R E 8 | Naive contraction of the square lattice ofE tensorsthat arises from the projected entangled pair states (PEPS) norm

evaluation is of exponential cost. Instead, we consider first a

contraction of two E columns, starting from the left edge. We then

view the dangling bonds (pointing toward the right, here r1, r2, r3, r4)

as the local state indices of a fictitious wavefunction Fr1r2r3r4 . We can

imagine approximating such a wavefunction by a matrix product state

(MPS) of a small auxiliary dimension, Ar1Ar2Ar3Ar4 . This effectively

approximates the two column structure by a single-column structure.

Repeating this for the remaining columns of the E-square lattice, we

can contract the full PEPS from left to right efficiently. The final

top-to-bottom contraction involves only a single column and is of the

same complexity as an MPS norm evaluation.

observable reduces finally to a single-column con-traction, as in the case of an MPS observableevaluation.55,56

Multiscale Entanglement RenormalizationAnsatzThe MERA is a flexible entanglement structure that isclosely related to the original real space renormaliza-tion group used in statistical problems. For simplicity,we describe the MERA in one dimension, althoughanalogous constructions can be considered in two and

three dimensions.MERAs consist of a set of operators known as

isometries, which perform coarse graining, and disen-tanglers, which are unitary rotations. We first discussthe isometries. We divide the k sites under considera-tion into blocks. For concreteness, we take each blockto contain three sites. The first block spans the space{|n1n2n3}. We now coarse grain from the Hilbertspace of the three site block {|n1n2n3} to a singleeffective site with a reduced Hilbert space {|n1} ofspecified dimension M. The operator which performsthis projection is an isometry,

W=

n1,n1n2n3

|n1Wn1

n1n2n3n1n2n3|, (35)

as illustrated in Figure 9. In a similar way, we coarsegrain all the three site blocks onto effective sites, thusgiving a lattice with k/3 effective sites, each with an M-dimensional Hilbert space. Once all the blocks havebeen coarse grained, the coarse graining can be iter-ated. For example, we can use another isometry tomap a block of three coarse-grained sites to a single,

F I G U R E 9 | The multiscale entanglement renormalization ansatz is

constructed from two types of tensors: isometries W, which map the

states of a block of sites to a single effective sites; and disentanglers U,

which perform a unitary rotation of the basis of two or more sites.

even more coarse-grained site {|n1n2n3 }{ |n1}.After log3k levels of coarse graining, we obtain a finaleffective site with an M-dimensional Hilbert space,in which the ground state of the entire problem is

expressed. The iterated isometries are connected ina tree structure (of different nature to the TTN de-scribed above) and the coefficient of the isometries Ware the variational parameters of the state.

The problem with using only isometries in a treestructure is that the coarse graining does not treat thesites uniformly, and thus neglects some entanglementthat may be present in the ground state. For exam-ple, in the above, the effective coarse-grained statesin the first two blocks {|n1}, {n2} build in entangle-ment within the first and second blocks of three sitesin the ground state, but do not between sites 3 and4, which are on the boundaries of the blocks. Theidea behind the MERA is that one can incorporatethis missing entanglement between the block bound-aries, by applying a unitary operator to these sites.61

For example, in the above example, we would use aunitary operator which acts on the product space ofsites 3 and 4. The MERA consists of a layer of disen-tanglers, which first handle the entanglement betweenthe boundaries of the blocks, alternating with layersof isometries, which handle the entanglement withinthe blocks,

| = ( W1 U1 W2 U2 . . . Wl)n

|n, (36)

where l is log3k, the number of layers in the MERA.The MERA structure is illustrated in Figure 10.

The MERA possesses several unique strengths.First, the presence of disentanglers allows the MERAto correctly describe the entanglement area laws inany dimension, even including the corrections toarea laws that are present in gapless one-dimensionalsystems. Second, the close relationship between theMERA and a tree structure means that unlike the



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F I G U R E 1 0 | (Left) The original real space RG is associated with a

tree structure for the wavefunction, where a layer of sites is subjected

to successive coarse grainings (isometries, W). However, the coarse

grainings within the blocks neglect to take into account entanglement

at the boundaries of the blocks. (Right) This is rectified within the

MERA by first performing a disentangling operation Ubetween the

block boundaries before coarse graining.

PEPS, expectation values and observables can be ex-actly computed in polynomial time.61 Nonetheless,the computational effort of using the MERA remainshigh, and applications remain at a very early stage ofdevelopment.

CONCLUSIONS

In this review, we have described the basic theorybehind low entanglement wavefunction approxima-tions. Because low entanglement wavefunctions oc-cupy a region of Hilbert space far from the Hartree

Fock mean-field limit, they offer the intriguing pos-sibility to efficiently model chemical problems thatchallenge standard quantum chemistry methods. Inrecent years, practical evidence of this has been pro-vided by the DMRG algorithm (based on low en-tanglement wavefunctions known as MPSs). This hasbeen used to describe large multireference, strong cor-relation problems involving transition metals and ex-cited states. In our discussion, in addition to MPSs, wehave also described more general low entanglementapproximations, such as TTNs, PEPS, and MERA.These states are considerably more complex thanMPSs and their properties are not yet completely un-derstood. Nonetheless, it is clear at the mathematicallevel that they overcome many of the formal limita-tions of MPSs, particularly in relation to the encodingof entanglement as a function of dimensionality. Akey remaining challenge is the development of practi-cal numerical algorithms to work with these flexible

states. This is an area where we can expect the wealthof knowledge in quantum chemistry in computationalalgorithms for complex wavefunctions to play a ma-jor part. Numerical computation with general lowentanglement approximations is thus positioned tobe an exciting possibility for quantum chemistry toexplore in this coming decade.

ACKNOWLEDGMENTS

The author acknowledges support from the NSF programs CHE-0645380 and CHE-1004603.

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