# Ground states of unfrustrated spin Hamiltonians satisfy an area law

**ABSTRACT** We show that ground states of unfrustrated quantum spin-1/2 systems on general lattices satisfy an entanglement area law, provided that the Hamiltonian can be decomposed into nearest-neighbor interaction terms which have entangled excited states. The ground state manifold can be efficiently described as the image of a low-dimensional subspace of low Schmidt measure, under an efficiently contractible tree-tensor network. This structure gives rise to the possibility of efficiently simulating the complete ground space (which is in general degenerate). We briefly discuss "non-generic" cases, including highly degenerate interactions with product eigenbases, using a relationship to percolation theory. We finally assess the possibility of using such tree tensor networks to simulate almost frustration-free spin models. Comment: 14 pages, 4 figures

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Page 1

Ground states of unfrustrated spin Hamiltonians satisfy an area law

Niel de Beaudrap,1Tobias J. Osborne,2and Jens Eisert1,2

1Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany

2Institute for Advanced Study Berlin, 14193 Berlin, Germany

(Dated: November 11, 2010)

We show that ground states of unfrustrated quantum spin-1/2 systems on general lattices satisfy an entan-

glement area law, provided that the Hamiltonian can be decomposed into nearest-neighbor interaction terms

which have entangled excited states. The ground state manifold can be efficiently described as the image of a

low-dimensional subspace of low Schmidt measure, under an efficiently contractible tree-tensor network. This

structure gives rise to the possibility of efficiently simulating the complete ground space (which is in general

degenerate). We briefly discuss “non-generic” cases, including highly degenerate interactions with product

eigenbases, using a relationship to percolation theory. We finally assess the possibility of using such tree tensor

networks to simulate almost frustration-free spin models.

I. INTRODUCTION

An important insight in the study of quantum many-body

systems is related to the observation that common states that

naturally occur do not quite exhaust the entire Hilbert space

available to them, but instead a much smaller subspace. This

insight is at the heart of powerful numerical methods that have

been devised in recent years. Ideas such as the density-matrix

renormalization group approach, and new ideas that allow for

the simulation of higher-dimensional quantum lattice models

[1–3], work exactly because they model well quantum states

that in a certain sense have little entanglement. More pre-

cisely, the states which are tractible by these approaches sat-

isfy what is called an area law [1, 4–17], so the entropy of

a subregion scales at most as the boundary area of that re-

gion (for a review, see Ref. [1]). For practical purposes, and

in particular for 1D systems, these methods in particular give

accurate accounts of ground state properties.

Now, not all ground states of local quantum lattice mod-

els can be efficiently approximated. This holds even true for

1D chains: indeed, one can construct models for which ap-

proximating the ground state energy is provably NP-hard [18]

— albeit using a fairly sophisticated construction involinvg

large local dimensions [20]. An important feature of these

constructions is that the difficulty of their solution appears to

be strongly related to whether the system is frustrated or not.

Thissuggeststhatwhetherornotthesystemisfrustratedisan-

other criterion for whether a quantum lattice model should be

considered “easy” or “hard”, in addition to its ground states

having “a lot” or “little” entanglement. This intuition that

frustrated systems should be hard to simulate is indeed true

for classical systems, where the frustrated or glassy models

are the hard ones to describe. For quantum systems, there is

evidence that the situation should be more complex [21].

In this work, we explore a class of models where the in-

tuition of frustration-free models being easy to solve holds

true. Building upon work in Ref. [22] and in Ref. [23], for

a natural class of two-local Hamiltonians acting on spin-1/2

particles (simply “spins” henceforth), we show that ground

states can be reduced to a completely characterized and low-

dimensional subspace, and then re-constructed by identifying

the ground state-space of each interaction of the Hamiltonian

term-by-term. Specifically, the ground space is the image of

a symmetric subspace under an explicitly constructible, and

efficiently contractible, tensor network. It follows that the

ground states satisfy an area law, and hence contain little en-

tanglement in the above sense. This generalizes recent results

regarding the existence of states which have little entangle-

ment in the ground-state manifolds of such Hamiltonians [24].

We discuss how to efficiently simulate the ground state mani-

fold, and suggest how this could be used to simulate “almost”

frustration-free quantum lattice models.

II.PRELIMINARIES

A.Frustration-free Hamiltonians and area laws

We consider spin-1/2 Hamiltonians on a lattice. The lattice

is described by some graph, the vertex set of which we de-

note by V . Naturally, the Hamiltonian will be local, or more

specifically include only nearest-neighbor interaction terms.

We represent the Hamiltonian as

?

for some terms ha,bacting on pairs of spins in the lattice de-

scribed by V . By rescaling, we may without loss of generality

require that the ground state energy of each interaction term

ha,bis zero. We wish to describe properties of the ground

state manifold M of such Hamiltonians, given the list of the

individual two-spin terms ha,bas input. An important class of

Hamiltonians are those which are frustration-free (or unfrus-

trated), for which each ground state vector |Φ? ∈ M is also a

ground state of the individual coupling terms: i.e. for which

H =

{a,b}⊆V

ha,b

(1)

ha,b|Φ?= 0

(2)

holds for all ha,band all |Φ? ∈ M. The actual ground state ρ

is the maximally mixed state over M, and so is mixed unless

the ground state manifold M is non-degenerate.

arXiv:1009.3051v2 [quant-ph] 10 Nov 2010

Page 2

2

Our main results pertain to frustration-free spin Hamiltoni-

ans as above, with the further constraint that each term ha,b

has at least one entangled excited state. In Section VI, we

show that the ground states ρ of such Hamiltonians satisfy an

area law [1, 4–17]: that is, for a contiguous region of spins

A ⊆ V , the entanglement of formation. If the ground state

is non-degenerate and hence pure, the entanglement of for-

mation is nothing but the usual entanglement entropy. of the

ground state satisfies

EF(ρ) ? C|∂A|

(3)

for C > 0 constant, where |∂A| is the “boundary area” (i.e.

the number of edges in the interaction graph of H, which are

incident to both A and V ? A): see Fig. 1. The entanglement

of formation is the largest asymptotically continuous entan-

glement monotone, so this also implies an entanglement area

law for e.g. the distillable entanglement. Thus, ground states

of frustration-free Hamiltonians contain little entanglement.

FIG. 1. The entanglement scales at most as the boundary area |∂A|

of a distinguished region A in some lattice, here a cubic lattice, with

tighter bounds for special cases.

This follows as a consequence of the fact that the ground-

state manifold is the image of the symmetric subspace on n

spins(forsomenboundedabovebythenumberofspinsofthe

Hamiltonian) under an efficiently simulatable, and explicitly

constructible, tree-tensor network. We show this in Section V,

in which we demonstrate how this allows us to efficiently sim-

ulate the ground space of the spin models we consider. For the

cases where we have a tree tensor network with a single top

root, the problem considered here may be viewed as exactly

the converse problem to the one discussed in Ref. [25].

Our result of an analytical area law complements results on

area laws in harmonic bosonic systems [7, 10, 11], fermionic

[13–15] on cubic lattice and general gapped models in one-

dimensional quantum chains [16]. For a comprehensive re-

view on area laws — and on implications on the simulatability

of quantum many-body systems — see Ref. [1].

B.Quantum 2-SAT problem

The arguments behind our analysis builds upon and extends

the ideas of Ref. [22], which defined the problem of quantum

satisfiability, and presented Bravyi’s algorithm for QUANTUM

2-SAT. We describe here the connection between this problem

and frustration in spin Hamiltonians.

QUANTUM 2-SAT is the quantum analogue of the “clas-

sical” 2-SAT problem on boolean formulae.

asks when there exists an assignment of boolean variables

x1,...,xn which simultaneously “satisfy” a collection of

constraints on pairs of those variables. This problem is ef-

ficiently solvable [26]; in contrast, the similar problem 3-

SAT (in which constraints apply to triples of variables) is NP-

complete [27]. In QUANTUM 2-SAT, individual clauses on

boolean variables are replaced by projectors pa,bwith

The latter

p2

a,b= pa,b

(4)

on pairs of spins: an instance of QUANTUM 2-SAT is satisfi-

able if there is a vector which is a zero eigenvector of each

projector simultaneously.

For an instance of QUANTUM 2-SAT, determining whether

there exist such simultaneous zero eigenvectors is equivalent

todeterminingwhethertheHamiltonianobtainedbysumming

the projectors is unfrustrated. Conversely, the problem of de-

termining when a 2-local spin Hamiltonian H is frustration-

free may be reduced to QUANTUM 2-SAT, by rescaling the

terms of the Hamiltonian H so that each term ha,bhas a min-

imal eigenvalue of 0, and replacing each rescaled term ha,b

with the projector pa,bonto img(ha,b). By construction, such

a substituation does not affect the ground space of the terms.

Thus, solving QUANTUM 2-SAT is equivalent to determining

whether a 2-local spin Hamiltonian is frustration-free.

Recently, random instances of QUANTUM k-SAT with rank-

1 projectors [28] have been studied for k ? 2, delineating

the “boundary” of frustration in k-local spin Hamiltonians in

terms of the density of interactions [29–32]. We instead ex-

tendthefindingsofRef.[22]fork = 2, remarkingonimplica-

tions for simulating the ground space manifold in frustration-

free Hamiltonians. In Section III, we review Bravyi’s algo-

rithm for QUANTUM 2-SAT, in order to demonstrate impor-

tant features of the reductions involved when they are applied

to unfrustrated Hamiltonians satisfying natural constraints.

III.REDUCTION TOOLS FOR FRUSTRATION-FREE

HAMILTONIANS

Bravyi’s algorithm for QUANTUM 2-SAT [22] efficiently

demonstrates the satisfiability of an instance of QUANTUM 2-

SAT by a sequence of reductions of Hamiltonians, yielding a

homogeneous instance (in which all projectors have rank 1),

and then verifying the satisfiability of these instances. We

may similarly use Bravyi’s algorithm to detect frustration in

2-local spin Hamiltonians, and consider the features of these

Hamiltonian reductions when applied to particular classes of

frustration-free Hamiltonians.

Throughout the following, we admit representations of

Hamiltonians H?with non-zero single spin terms ha,

?

and again describe H?as unfrustrated if there exists a joint

ground state with eigenvalue zero of all terms (including the

single-spin terms ha).

H?=

{a,b}⊆V

ha,b +

?

a∈V

ha,

(5)

Page 3

3

A. Reductions by isometries

Condensing the analysis of Ref. [22], we consider a re-

duction for 2-local Hamiltonians H to Hamiltonians H?on

fewer spins, provided that H contains only positive semidef-

inite terms which have non-trivial kernels. Throughout, we

denote C2by H2.

1.Two-spin isometric contractions

Consider a Hamiltonian term hu,v of rank 2 or 3. If H

is frustration-free, hu,v fixes a subspace of H⊗{u,v}

mension at most 2, over which the reduced state ρu,v of a

state vector |Φ? ∈ ker(H) must be a mixture. We describe

this reduced state by an encoding of one spin into two. Let

{|ψ0?,|ψ1?,|ψ2?,|ψ3?} be an orthonormal basis for H2⊗H2

such that

2

of di-

ker(hu,v) ⊆ span{|ψ0?,|ψ1?}.

Define an isometry Uu:u,v: H⊗{u}

?

This is an isometric reduction, similar to those in a tree tensor

network or a MERA ansatz [33]. By construction, the support

of the reduced state ρu,vlies in img(Uu:uv). We may then

define a Hamiltonian

(6)

2

−→H⊗{u,v}

?

2

such that

x∈{0,1}

αx|x?u

?−→

x∈{0,1}

αx|ψx?u,v.

(7)

H?= U†

u:u,vHUu:u,v

(8)

on the subsystem V?= V ? {v}, where the spin v is essen-

tially deleted; any state vector |Φ?∈ ker(H) then has the form

|Φ?= Uu:uv|Φ??,

We may express H?as a sum of terms

|Φ??∈ ker(H?).

(9)

h?

a,b= U†

u:u,vha,bUu:u,v,h?

a= U†

u:u,vh?

aUu:u,v.

(10)

(In the case that h is of rank 3, these will include a non-

zero single spin operator h?

contains non-zero terms ha,uand ha,v, we obtain two terms

h?

Hamiltonian H, which both act on the spins u and a. We sum

these to obtain a combined term

u,vwhich acts on u alone.) If H

a,u= U†

u:u,vha,uUu:u,vand h?

a,v= U†

u:u,vha,vUu:u,vin the

¯h?

a,u= h?

a,u+ h?

a,v

(11)

in the reduced Hamiltonian, which may be of higher rank than

either h?

contributions h?

terms huand hvin H.) Fig. 2 illustrates the effect of mul-

tiple reductions on the interaction graph of the Hamiltonian.

For example: Consider an anti-ferromagnetic four-spin

Hamiltonian, with interactions

a,uor h?

a,v. (We similarly accumulate any single-spin

uand h?

vwhich may arise from single-spin

hj,k =

1

2σ(j)

xσ(k)

x

+

1

2σ(j)

yσ(k)

y

(12)

FIG. 2. Illustration of transformations of the interaction graph of a

Hamiltonian H by two-spin isometries. Darker, thicker edges repre-

sent interaction terms ha,bof rank 2 or 3, which may be eliminated

by contraction (corresponding to a reduction H ?→ U†

In the case of an interaction term with rank(hr,u) = 3 as illustrated

above, a single spin operator (represented above by a loop) is pro-

duced on the contracted vertex.

a:abHUa:ab).

on a four-spin cycle with interacting pairs (1,2), (2,3), (3,4),

and (4,1). Here we denote

σx =

?0 1

?0 −i

1 0

?

= |1??0| + |0??1|,

(13a)

σy =

i0

?

= i|1??0| − i|0??1|.

(13b)

Rescaling the interactions to have ground energy zero (and

taking these for the hj,kinstead) gives us

hj,k = |0,0??0,0|j,k+ |1,1??1,1|j,k.

(14)

The kernel of this operator is clearly spanned by |0,1? and

|1,0?. We may consider the effect of contracting spins 3 and

4 into a renormalized spin, using the isometry

R3:3,4=?|0?3⊗ |1?4

By construction, we have R†

is disjoint from {3,4}, we have h?

h1,2. We compute the renormalized terms h?

??0|3+?|1?3⊗ |0?4

3:3,4h3,4R3:3,4= 0; and as {1,2}

1,2= R†

??1|3.

(15)

3:3,4h1,2R3:3,4 =

2,3and h?

4,1as

h?

2,3= R†

3:3,4

?

|0,0??0,0|2,3⊗ 1 14+ |1,1??1,1|2,3⊗ 1 14

= |0,0??0,0|2,3+ |1,1??1,1|2,3,

?

= |1,0??1,0|3,1+ |0,1??0,1|3,1;

?

R3:3,4

(16a)

?

(16b)

h?

4,1= R†

3:3,4

1 13⊗ |0,0??0,0|4,1+ 1 13⊗ |1,1??1,1|4,1

R3:3,4

Page 4

4

uptorescaling, theresultingrenormalizedHamiltonianisthen

??σ(1)

+

?σ(3)

with a ferromagnetic coupling between the site 1 and the

renormalized site 3. Arbitrary rank-2 or rank-3 interactions

may be contracted similarly.

H?= R†

3:3,4HR3:3,4∼1

2

xσ(2)

?σ(2)

x

+ σ(1)

yσ(2)

y

?

xσ(3)

x

+ σ(2)

yσ(3)

y

?

−

xσ(1)

x

+ σ(3)

yσ(1)

y

??

, (17)

2. Single-spin deletions

For a 2-local Hamiltonian H containing non-zero single-

spin operators hv(e.g., such as may arise from the preceding

reduction), any state vector |Φ? ∈ ker(H) must be factoriz-

able into a single-spin pure state vector |ψ? ∈ ker(hv) acting

on v, and some state of the remaining spins. If the operator hv

has full rank, it follows |ψ? = 0, so that H has trivial kernel.

Otherwise, |ψ?may be taken to be a unit vector spanning the

kernel of hv, and we may form a Hamiltonian

H?=??ψ|v⊗ 1?H?|ψ?v⊗ 1?

on the subsystem V?= V ?{v}, consisting of a sum of terms

h?

a,b=??ψ|v⊗ 1?ha,b

acting on pairs of spins {a,b} ⊆ V . In the latter case, if

v ∈ {a,b}, then h?

spin; otherwise, we have h?

of single-spin terms h?

term hapresent in H, we may accumulate these into a term

¯h?

We may again describe H?using an isometric reduction in

this case: if we define Pv= |ψv?⊗1V?, then Pvis an isometry

whoseimage containsanystatevector|Φ?∈ ker(H). Wemay

then rewrite Eq. (18) as

(18)

a= haacting on individual spins a ?= v, and terms

h?

?|ψ?v⊗ 1?,

(19)

a,bwill be an operator acting on a single

a,b= ha,b. Again, in the case

a,bacting on a spin a, if there was a

a= ha+ h?

a,b.

H?= P†

vHPv.

(20)

expressing it in a form more explicitly similar to Eq. (8).

3.Remarks on these reductions

The reductions above correspond to the reductions pre-

sented in Ref. [22] for instances of QUANTUM 2-SAT which

contain two-spin operators of rank greater than 1. This allows

us to reduce QUANTUM 2-SAT to the special case of “homo-

geneous” instances (in which all terms have rank 1).

The key feature of both reductions above is that the kernels

of H and H?are related by isometries, and so have the same

dimension. If the Hamiltonian H?has any terms of full rank

(acting on either one or two spins), it follows that the Hamil-

tonian H?has trivial kernel; then the same holds for H as

well. If we do not encounter any full-rank terms, each reduc-

tion produces a Hamiltonian acting on one fewer spins, even-

tually yielding a “homogeneous” Hamiltonian (extending the

terminologyofRef.[22]toHamiltoniansingeneral, including

those with single-spin terms of rank 1).

The choice of the reduction at each stage does not matter, in

the following sense. As long as we have a Hamiltonian con-

taining two-spin terms of rank at least 2, and which does not

contain full-rank terms, we may extend any sequence of re-

ductions to one which terminates with a Hamiltonian˜H which

is either homogeneous or contains a full-rank term. In the

latter case, the original Hamiltonian has trivial kernel, and is

therefore frustrated; otherwise, we obtain a “homogeneous”

Hamiltonian whose kernel may be mapped to that of the orig-

inal Hamiltonain H by a sequence of known isometries. If

we can solve the homogeneous case, we may then choose the

reductions according to whichever criteria are convenient.

Note that this reduction process, from an input Hamiltonian

H to a homogeneous Hamiltonian, amounts to a tree tensor

network of isometries (albeit applied to a vector subspace):

fromatemporaltoplayerdefinedbyaHamiltoniancontaining

only terms of rank 1, one constructs the ground space of the

full Hamiltonian H by sequential applications of isometries

with a simple topology. We develop this observation further,

and remark on implications for simulating the ground space of

H, in Section V. Note also that for a one-dimensional quan-

tum chain and a sequential contraction, this construction gives

rise to a sequential preparation of a quantum state and hence

to a matrix-product state of small bond dimension.

B.The homogeneous case

Given a homogeneous Hamiltonian H?(containing only

terms of rank 1) acting on some system V?, consider a col-

lection of vectors |βa,b?∈ H⊗{a,b}

ha,b= |βa,b??βa,b| .

We interpret each two-spin Hamiltonian term h?

straint on the corresponding two-spin marginal state ρa,bof

a state |Φ? ∈ ker(H), and attempt to obtain additional con-

straints on pairs of spins a,b ∈ V?by combinations of the

constraints which are already known. Ref. [22] shows that if

|Φ? lies in the kernel of ?βa,b| and ?βb,c| acting on the corre-

sponding spins, it also lies in the kernel of the functional

?β?

acting onthe spinsa andc, where|Ψ−?∝ |0?|1?−|1?|0?is the

two-spin antisymmetric state vector. We call such a constraint

?β?

gives rise to?β?

we may add a term

˜hu,v=??β?

2

such that

(21)

a,bas a con-

a,c

??=??βa,b| ⊗?βb,c|??1 ⊗??Ψ−?⊗ 1?

??an “induced” constraint, and use the term induction of

a,c

For each induced constraint?β?

(22)

a,c

constraints to refer to the operation on?βa,b| and?βb,c| which

??in Eq. (22), up to a scalar factor.

u,v

??on a pair of spins u,v,

??β?

u,v u,v

??

(23)

Page 5

5

to the Hamiltonian H?, obtaining a Hamiltonian˜H which (by

construction) has the same kernel as H?. If H?already con-

tains a term hu,vwhich is not colinear to the induced term

˜hu,v, these may be accumulated into a term¯hu,vwhose rank

is at least 2, and one may apply a two-spin contraction as

described in Section IIIA. Otherwise, we may induce fur-

ther constraints from the terms of˜H, until we obtain a com-

plete homogeneous Hamiltonian Hc: a Hamiltonian in which

the two-spin constraints ?βu,v| are closed under constraint-

induction.

By inducing constraints on pairs of spins, possibly perform-

ing two-spin contractions as in Section IIIA when we obtain

terms of rank 2 or more, we may efficiently obtain a com-

plete homogeneous Hamiltonian Hcfrom a frustration-free,

2-local Hamiltonian H. Furthermore, Ref. [22] shows that a

complete homogeneous Hamiltonian Hcacting on at least one

spin (and which lacks single-spin operators [34]) has a ground

space which contains product states. Thus, for homogeneous

Hamiltonians H, we may either efficiently determine that it is

frustrated, or efficiently obtain a Hamiltonian which is closed

under constraint-induction. In the latter case, we may con-

struct product states in the kernel of H by selecting states for

each spin consistent with the two-spin constraints [35].

IV. UNFRUSTRATED NATURAL HAMILTONIANS

We now present results concerning the ground state man-

ifold of a “physical” class of 2-local spin Hamiltonians. We

will say that a Hamiltonian H is natural if it is 2-local, con-

tains no isolated subsystems, and each term ha,b(acting on

H2⊗ H2) has at least one entangled excited state (i.e., there

exists an entangled state orthogonal to the ground state man-

ifold of ha,b). Without loss of generality, we may further

require that the ground energy of each term in H is zero.

This is a natural assumption that typical physical interactions

will satisfy: for instance, ferromagnetic or anti-ferromagnetic

Ising interactions (which have excited eigenstates |0?|1? −

|1?|0?and |0?|0?+|1?|1?respectively), ferromagnetic or anti-

ferromagnetic XXX models (which also have those respective

eigenstates), or indeed any interaction which is inequivalent to

either diag(0,0,Ea,Eb) or diag(0,Ea,0,Eb) up to rescaling

and a choice of basis for each spin.

Using the reductions of Section IIIA, we show strong

bounds on the dimension of the ground space of an unfrus-

trated natural Hamiltonian on spins. This will allow us in

Section V to describe a scheme for efficiently simulating the

ground space of frustration-free natural Hamiltonians, and

in Section VI to demonstrate that the ground states of such

Hamiltonians satisfy an entanglement area law.

A.Ground-spaces of unfrustrated, natural, homogeneous

Hamiltonians

We now present an extension of the analysis of Ref. [22]

for homogeneous and complete Hamiltonians Hc(acting on

a set Vcof spins), to examine the ground-state manifold of

Hcin the case that Hcis also natural. We show, using tech-

niques similar to those used in Ref. [30, Section III A], that

the ground space of such a Hamiltonian is equivalent to the

symmetric subspace Symm(H⊗Vc

ficiently constructible choice of invertible operations on each

spin. As we may reduce more general Hamiltonians, i.e. hav-

ing terms of rank 2 or 3 (extending beyond those Hamiltoni-

ans considered in Ref. [30]) to homogeneous natural Hamil-

tonians via the reductions of Section IIIA, these results yield

important consequences for natural frustration-free Hamilto-

nians in general.

Consider a Hamiltonian Hcacting on Vc, where Hchas

no single-spin terms. Because the two-spin constraints de-

scribed by the terms of Hcare closed under the induction of

constraints (as described by Eq. (22)), and as there are no iso-

lated subsystems, it is easy to show that every pair of spins

is acted on by a non-zero term in Hc. For such a Hamilto-

nian, the excited states |βa,b?for the terms ha,bin Hcare en-

tangled states. We may then construct a family of operators

{Lv}v∈Vc⊆ GL(2) such that

?βu,v| ∝

for each pair of spins u,v, where |Ψ−? is again the two-spin

antisymmetric state vector. For instance, one may fix La= 1

for an arbitrarily chosen spin a ∈ Vc, and determine linear

operators Lvsatisfying Eq. (24) for each v ∈ Vcand opera-

tor ?βa,v|. Any such choice of operators {Lv}v∈Vcsatisfies

Eq. (24) for all u,v, which follows from the closure of the

constraints?βu,v| under induction:

?βu,v| ∝

∝

∝

We define scalars λu,v

?=

λu,v?Ψ−|?Lu⊗ Lv

??βu,v| ⊗ 1?T =?Ψ−??

where we define Tu,vby

2

) ⊆ H⊗Vc

2

, up to some ef-

?Ψ−??

u,v

?Lu⊗ Lv

?

(24)

??βua| ⊗?βa,v|??1 ⊗??Ψ−?⊗ 1?

?Ψ−???Lu⊗ Lv

?

??Ψ−??⊗?Ψ−????Lu⊗??Ψ−?⊗ Lv

0 such that ?βu,v|

for each u,v ∈ Vc, and let T =

v∈VcLv)−1: we then have

?

?.

(25)

=

(?

u,v⊗ Tu,v,

(26a)

Tu,v

= λu,v

?

w?=u,v

L−1

w.

(26b)

As the operators Tu,vhave full rank, the operator??βu,v| ⊗

eigenspace of the SWAP operator acting on u and v. It follows

that the kernel of the Hamiltonian

?

=

1?T then has the same kernel as ?Ψ−|u,v, which is the +1-

T†HcT =

u,v∈Vc

?

T†?|βu,v??βu,v| ⊗ 1?T

??Ψ−??Ψ−??

2

u,v∈Vc

u,v⊗ T†

u,vTu,v

(27)

is the symmetric subspace Symm(H⊗Vc

the result of [30, Section III A].

); this corresponds to

Page 6

6

We remark on some important properties of Symm(H⊗nc

where nc= |Vc|. This subspace is spanned by uniform super-

positions |Wk? of the standard basis states having Hamming

weight 0 ? k ? nc,

2

),

|Wk?∝

?

x∈{0,1}nc

?x?1=k

|x? ;

(28)

thus dimSymm(H⊗nc

spanned by product state vectors |α0?⊗nc,...,|αnc?⊗ncfor

anysetofnc+1pairwiseindependentstatevectors|αj?∈ H2.

Thus, any natural Hamiltonian Hcwhich is also complete and

homogenous has a ground space of dimension nc+1, and can

be spanned by a family of classically efficiently simulatable

state vectors

?

for some choice of pair-wise independent single-spin state

vectors |α0?,...,|αnc? ∈ H2; we use this fact in Section V.

Note that if even this efficient method should be too compu-

tationally costly for very large systems, one can also Monte-

Carlo sample from the ground state manifold in this way.

2

) = nc+1. This subspace may also be

|Φj? =

v∈Vc

?Lv|αj??,

(29)

B.Preservation of natural Hamiltonians under reductions

A key feature of natural Hamiltonians (defined on page 5)

is that the class of frustration-free natural Hamiltonians on

spins is preserved by the two-spin contractions described in

Eq. (8). This implies that the reductions of Section IIIA map

the ground-state manifold of an unfrustrated natural Hamilto-

nian H provided as input to that of a complete, homogeneous,

natural Hamiltonian; we may then apply the results of the pre-

ceding section to describe the ground-state manifold of H.

Consider an isometry Uu:u,v : H⊗{u}

rived from a two-spin Hamiltonian term hu,vas described in

Section IIIA1. We may show that for any term ha,uin H, the

corresponding term h?

Hamiltonian H?= U†

state if the same holds for ha,u. We require the following two

lemmas, whose proofs we defer to Appendix A:

2

−→ H⊗{u,v}

2

de-

a,u= U†

u:u,vHUu:u,vhas an entangled excited

u:u,vha,uUu:u,vin the reduced

Lemma 1 (Product states). For two-spin state vectors |ψ?and

|φ?, we have??ψ|⊗1??1⊗|φ??= 0 only if both |ψ?and |φ?

Lemma 2 (Product operators). Let U : H2−→ H2⊗ H2be

an isometry which is not a product operator. Let η ? 0 be an

operator on two spins, and η?=?U†⊗12)(12⊗η)(U ⊗12).

only if η is a product operator.

are product states.

If η?is not of full rank, then η?is a product operator if and

We show that frustration-free natural Hamiltonians are pre-

served by the reductions of Section IIIA1 as follows. Let H

be a natural 2-local Hamiltonian, and hu,vbe a two-spin term

in H. Define

Uu:u,v = |ψ0??0| + |ψ1??1|

(30)

for orthonormal two-spin state vectors |ψ0?,|ψ1?whose span

contains ker(hu,v); we require that |ψ1? be entangled, which

ensures that Uu:u,vis not a product operator. Consider the

terms h?

nian H?= U†

acting on v and some other spin a, the fact that hv,ahas an en-

tangled excited state implies in particular that it is not a prod-

uct operator. Thus, h?

rank. If H is frustration-free, h?

h?

entangled excited states. As h?

it follows that H?is a natural Hamiltonian; and as H?has a

kernel of the same dimension as H, it is frustration-free as

well.

We may strengthen this result, to show that if H is natu-

ral and frustration-free, and also contains no two-spin terms

of rank 1, then the same is true of H?= U†

well. For any two-spin term hv,aacting on v with rank at least

2, consider states |ϕ0?,|ϕ1? ∈ img(hv,a) such that |ϕ0? =

|α?|β?is a product state and |ϕ1?is entangled; Any subspace

of H2⊗ H2of dimension at least 2, such as img(hu,v), con-

tains a product state vector |ϕ0?; the existence of |ϕ1?is guar-

anteed by the definition of a natural Hamiltonian (compare

also Ref. [37]). and choose real parameters λ0,λ1> 0 such

that

a,b= U†

u:u,vha,bUu:u,vwhich occur in the Hamilto-

u:u,vHUu:u,v. For any two-spin operator hv,a

v,ais a product operator only if it has full

v,acannot have full rank; then

v,ais not a product operator, and in particular it will have

a,b= ha,bwhen a,b / ∈ {u,v},

u:u,vHUu:u,vas

hv,a− λ0|ϕ0??ϕ0| − λ1|ϕ1??ϕ1| ? 0.

Let ηk= |ϕk??ϕk| for k ∈ {0,1}, and consider the images η?

under contraction by Uu:u,v:

(31)

k

η?

k= U†

u:u,vηkUu:u,v

?

?

=

j,?

?|j??ψj| ⊗ 1??1 ⊗ ηk

|j???| ⊗ Mj,kM†

??|ψ????| ⊗ 1?

=

j,?

?,k,

(32)

where we define Mj,k=??ψj|⊗1??1⊗|ϕk??. By Lemma 1,

operators; this implies that the operators M1,kin particular

are non-zero, so that η?

we have Mj,k = 0 only if both |ψj? and |ϕk? are product

k?= 0 for any k. Note that

h?

v,a= U†

? λ0U†

= λ0η?

u:u,vhv,aUu:u,v

u:u,vη0Uu:u,v + λ1U†

0+ λ1η?

u:u,vη1Uu:u,v

1;

(33)

because H?has a non-trivial kernel, h?

in which case neither operator η?

η?

operators are linearly independent. Then, λ0η?

rank at least 2; by Eq. (33), the same is true of h?

the terms in H have rank 2 or higher, the same then holds for

H?as well.

Thus, ifweapplythereductionsofSectionIIIAtoaninitial

Hamiltonian which is both natural and frustration-free, the re-

sulting Hamiltonians will also be natural and frustration free.

v,ahas rank at most 3,

khas full rank. By Lemma 2,

0is a product operator, these

0+ λ1η?

1is not a product operator; as η?

1has

v,a. If all of

Page 7

7

Furthermore, if H contains no terms of rank 1, then neither

will the reduced Hamiltonians. Because the process of induc-

ing constraints described in Section IIIB also preserves the

property of each term ha,bhaving entangled excited states,

these invariants ensure that initial Hamiltonians with these

properties (natural and frustation-free, and possibly contain-

ing no terms of rank 1) may be reduced to homogeneous and

complete Hamiltonians which have these same properties. We

may then apply the results of Section IVA to these reduced

Hamiltonians.

FIG. 3. A simple tree tensor network.

V.SIMULATING GROUND SPACES OF

FRUSTRATION-FREE NATURAL HAMILTONIANS

Building on the results of Section IV, we now show how

the reductions of Section IIIA may be used to obtain a proce-

dure for simulating states from the ground-state manifold of

frustration-free natural Hamiltonians H.

A. Tree tensor networks and matrix-product states

As we noted on page 4, implicit in the reductions of Sec-

tionIIIAisthattheisometricreductionfromgeneralHamilto-

nians to homogeneous instances has the form of a tree-tensor

network. Thus, simulating the ground state manifold of any

unfrustrated 2-local Hamiltonian on spins may be reduced

to that of a complete homogeneous Hamiltonian acting on a

smaller system. In this section, we sketch this reduction.

For any unfrustrated 2-local Hamiltonian H on n spins, we

may apply two-spin reductions as described in Section IIIA

until we obtain a homogeneous instance without single-spin

terms. We then attempt to induce additional constraints via

Eq. (22), and apply further two-spin reductions if we obtain

terms of rank 2 or 3. If H is frustration-free, this process will

ultimately terminate in a complete homogeneous Hamiltonian

Hcon a subset Vc⊆ V .

Consider the tensor network T which performs the com-

plete reduction as above; we describe T in reverse order, as

introducing new spins to represent a unitary embedding of

ker(Hc) into ker(H). The various spin contraction isometries

Uu:u,vas in Eq. (7) each have a single input-index and two

output indices; the spin deletion isometries Pvhave no input

indices at all. These are applied sequentially, giving rise to an

acyclic directed network. As the in-degree of each tensor is

at most 1, it follows that the network contains no cycles at all

(neither directed nor undirected): the output indices of each

tensor represent spins whose state depends on only a single

spin at the input. Put another way: any spin v which is intro-

duced by an isometry Uu:u,vmay be considered a “daughter

spin” of a unique parent U, which imposes a tree-like hierar-

chy on the tensor network T, as illustrated in Fig. 3. Strictly

speaking, the quantum circuit or tree tensor network will have

the structure of a forest graph, which is a graph which may

have more than one connected component, each of which are

trees.

The roots of each tree are spins u which are either prepared

by an isometry Puderived from the removal of single-spin

terms, or which correspond to free indices at the input of the

tensor network T.

In the case that H is non-degenerate, the resulting tensor

network will (by that fact) simply be a tree-tensor network

with no free input indices. Conversely, if the input Hamilto-

nian H is degenerate, there will necessarily be free input in-

dices, representing a domain consisting of a state space of di-

mension at least 2. In the latter case, the tensor network T will

yield ground states of the original unfrustrated Hamiltonian H

if and only if it operates on a state |ϕ?∈ ker(Hc) at the input,

where Hcis the complete homogeneous instance obtained by

the Hamiltonian reductions. Thus, if one may efficiently sim-

ulate states from the ground space of such a Hamiltonian, we

may apply the network T to simulate the ground space of the

original Hamiltonian H.

B.Efficiently simulating ground spaces of unfrustrated

natural Hamiltonians

Tensor networks T with free input indices, and with a tree-

like structure such as described above, can be efficiently sim-

ulated over inputs with low Schmidt measure [38], as follows.

For any observable Ω acting on m spins, one may eval-

uate ?Ω?Hfor the maximally mixed state over the ground-

state manifold of H by computing the expectation?T†ΩT?

Hamiltonian Hcobtained as described in Section III. As the

tensor T has tree-stucture, the observable T†ΩT also acts on

at most m spins. If we can obtain an orthonormal basis for

ker(Hc) which may be succinctly described in terms of prod-

uct states, we may evaluate expectation values of T†ΩT with

respect to m-fold products of single spin states.

As we noted in Section IVA, ker(Hc) can be spanned by

a collection of nc+ 1 product vectors (where ncis the num-

ber of spins on which Hcacts). Let |Φ?

ker(Hc) be a collection of independent product vectors,

??Φ?

We may efficiently compute a projection of Ωconto ker(Hc)

by performing a suitable transformation of the matrix

Hc

over the ground-state manifold of the complete homogeneous

0?,|Φ?

1?,...,??Φ?

nc

?∈

(34)

j

?

= |ϕj,1?⊗ |ϕj,2?⊗ ··· ⊗ |ϕj,nc? .

W(Ωc) =

nc

?

j,k=0

|j??Φ?

j

??Ωc|Φ?

k??k| ,

(35)

Page 8

8

as follows. The operator W(1 1) in particular is positive defi-

nite; wethushaveW(1 1) = U∆U†forsomeunitaryU unitary

and positive diagonal matrix ∆. It is not difficult to show that

∆−1/2U†

nc

?

j=0

|j??Φ?

j

??=

nc

?

j=0

|j??Φj| ,

(36)

for some orthonormal basis |Φ0?,...,|Φnc? of ker(Hc), by

takingtheproductoftheaboveoperatorwithitsadjoint. Thus,

the restriction of Ωcto ker(Hc) with respect to the basis of

states |Φj?may be computed as

¯Ω = ∆−1/2U†W(Ωc)U∆−1/2.

(37)

By considering operators Ωc= T†ΩT, this allows us to com-

pute the restriction of operators to the ground-space of HU:

see Fig. 4.

Tree tensor network

Tree tensor network

FIG. 4. Schematic diagram showing the isometric decomposition for

efficient simulating the ground state manifold. A local Hamiltonian

term supported on two sites is marked yellow.

We may thus efficiently estimate such observables with re-

spect to ground-states of HU: for constant m, the required

inner products may be calculated as (sums of) scalar products

of at most ncinner products over vector spaces of bounded

dimension. To evaltuate the value of Ω with respect to ground

states of the input Hamiltonian H, it suffices to analyse the

polynomially-sized operator¯Ω representing the action of Ω

on the ground-state manifold.

VI.

OF FRUSTRATION-FREE NATURAL HAMILTONIANS

ENTANGLEMENT BOUNDS FOR GROUND STATES

The fact that the reductions of Section IIIA preserve the

class of natural Hamiltonians (as defined on page 5) allow us

also to make more global statements about ground states for

frustration-free Hamiltonians, again by reduction to the com-

plete homogeneous case described in Section IVA. In this

section, considering frustration-free natural Hamiltonians H,

we demonstrate an area law for the entanglement possible in a

ground state of H between any subsystem A ⊆ V and its en-

vironment V ? A. We also consider some very general cases

in which still stronger upper bounds on the entanglement may

be obtained.

A.Area law for frustration-free natural Hamiltonians

We consider first the case where A is a contiguous subsys-

tem (i.e. for which there is a path between any pair of spins

in A, following edges in the interaction graph of the Hamilto-

nian H), and subsequently generalize the observation in this

case to arbitrary subsystems A. We first decompose

H = HA+ HB+ HA,B,

(38)

for B = V ? A, and where HAand HBcontain all terms

internal to A and B, respectively. We then apply the reduc-

tions of Section IIIA to the subsystem A. That is, we per-

form two-spin contractions as described in Section IIIA1 for

any two-spin terms in HAof rank 2 or 3, and perform spin-

deletions as described in Section IIIA2 for any single-spin

terms in HA. Performing the constraint-induction process of

Eq. (22) — again on the terms acting on A alone — and then

reducing further reductions as necessary, we eventually obtain

a Hamiltonian˜H of the form

˜H = ˜H˜

A+ HB+˜H˜

A,B,

(39)

where˜A ⊆ A is the set of spins remaining after the reduc-

tion process, and where˜H˜

Hamiltonian. The Hamiltonians˜H˜

terms derived from HAand HA,Brespectively by the reduc-

tion process. In other words, we perform a complete tree ten-

sor reduction on the subsystem A, until we obtain a Hamil-

tonian whose restriction to A is homogeneous and complete.

As dimker(˜H) = dimker(H) > 0, we have ker(˜H˜

and ker(HB) > 0 as well; in particular, ker(˜H) ⊆ ker(˜H˜

so that˜H˜

well, we then have

Ais a homogeneous and complete

Aand˜H˜

A,Bcontain the

A) > 0

A),

Ais frustation-free. Because ker(˜H) ⊆ ker(HB) as

ker(˜H) ⊆ ker(˜H˜

A) ⊗ ker(HB)

(40)

taking the restrictions of˜H˜

systems˜A and B. Let ˜ n = |˜A|: as˜H˜

neous and complete, it has kernel of dimension ˜ n + 1 by Sec-

tion IVA.

In the case where A contains multiple components

A1,A2,...,Ak with respect to the interaction graph of H

(where each Ajis disconnected from the others but connected

internally), we may perform the Hamiltonian reductions of

Section IIIA to each component independently. We may fur-

ther decompose the Hamiltonian˜H˜

Aand HBto their respective sub-

Ais also homoge-

Aobtained in Eq. (39) as

˜H˜

A= ˜H˜

A1+ ··· +˜H˜

Ak,

where˜Aj=˜A ∩ Aj.

(41)

As˜H˜

unfrustrated as well, in which case we may write

Ais unfrustrated, each of the sub-Hamiltonians˜H˜

Ajis

ker(˜H˜

A) ⊆ ker(˜H˜

A1) ⊗ ··· ⊗ ker(˜H˜

Ak),

(42)

similarly to Eq. (40). Then, the dimension of ker(H˜

bounded by the product of dimker(˜H˜

subsystem, where ˜ nj= |˜Aj|. Let αjbe the number of spins

A) is

Aj) = ˜ nj+ 1 for each

Page 9

9

in Ajwhich are adjacent in the spin lattice to spins in B, and

let ˜ αj ? αjbe the number of such spins in˜Aj. For any H

with nearest neighbor interactions on a lattice in finitely many

dimensions (in the graph theoretical sense), there exist scalars

c,K > 0 such that

αj? Knc

j

(43)

for each subsystem Aj. We may then bound on the dimension

of ker(˜H˜

A) in terms of these “boundary spins” as

log(dimker(˜H˜

A)) ?

k

?

j=1

log(dim ker(˜H˜

Aj))

?

k

?

j=1

log(˜ nj+ 1) ?

k

?

j=1

˜ nc

j?

α

K,

(44)

where α = α1+···+αkis the number of spins in A adjacent

to elements of B.

From this bound on dimker(˜H˜

state |Φ?in the ground space of˜H has Schmidt measure [38]

at most α/K across the partition˜A + B, and so can support

at most this many e-bits of entanglement between˜A and B.

Because any state vector |Ψ? ∈ ker(H) can be obtained from

some vector |Φ? ∈ ker(˜H) by a network of isometries act-

ing only on spins in A, it follows that any state vector |Ψ?

in the ground-state manifold of H contains at most α/K e-

bits of entanglement between A and B. Note the similarity to

ground states of low Schmidt rank close to factorizing ground

states in Heisenberg models [36]. In case the ground state ρ is

degenerate, each pure state in the spectral decomposition of ρ

will have that property. As the entanglement of formation is

convex (this usually being taken as a necessary property of an

entanglement monotone), one obtains the bound

A), it follows that any pure

EF(ρ) ?α

K

(45)

for the entanglement of formation between A and B. Finally,

let ∂A be the set of edges between A and B. By definition,

for each edge in ∂A, there is a spin in A which is adjacent to

some spin in B; then we have α ? |∂A|, so that

EF(ρ) ?|∂A|

K

.

(46)

Thus, the amount of entanglement which can be supported by

a ground state of H between A and B is governed by an area

law. We summarize:

Proposition 1 (Area law). Let H be an unfrustrated natural

Hamiltonian on a lattice V , and denote with ρ its (possibly

degenerate) ground state. Then for any subsystem A ⊆ V , the

entanglement of formation of ρ with respect to A and V ? A

satisfies an area law, i.e. there exists a constant C > 0 of the

lattice model such that

EF(ρ) ? C|∂A|.

(47)

B. Stronger entanglement bounds for contiguous subsystems

The above analysis imposes no additional constraints, be-

yond the requirement that H be natural and frustration-free.

We may obtain still stronger bounds — by the logarithm of

the system size, or even by a constant — on the entanglement

between A and its environment B = V ?A, under fairly gen-

eral conditions on the subsystem A when it is a contiguous

subsystem.

1.Contiguous subsystems in general

Implicit in the analysis of the previous subsection is a

stronger entanglement bound for contiguous subsystems in

general: we observe that if A consists of a single component,

we have

dimker(˜H˜

A) = ˜ n + 1

(48)

for ˜ n = |˜A| by the analysis of Section IVA (where H˜

reduced Hamiltonian acting on the subsystem A described in

Eq. (39)). By a similar analysis, if α is the number of spins in

A adjacent to at least one spin in B, we may use Eq. (43) to

obtain

Ais the

log(dimker(˜H˜

A)) = log(˜ n + 1) ?log(α/K + 1)

c

.

(49)

As α ? |∂A|, we may then obtain:

Proposition 2 (Logarithm law for contiguous systems). Let

H be an unfrustrated natural Hamiltonian on a lattice V , and

denote with ρ its (possibly degenerate) ground state. There

then exists a constant C > 0 of the lattice model such that,

for any contiguous subsystem A ⊆ V , the entanglement of

formation of ρ with respect to A and V ? A satisfies

EF(ρ) ? C log|∂A|.

(50)

2.Subsystems acted on by many high-rank Hamiltonian terms

In the above result, we have neglected the difference in the

sizes of the subsystem A, and the reduced subsystem˜A ⊆ A.

The difference in their sizes will be precisely the number of

isometric reductions performed to obtain˜A from A. Each iso-

metric reduction corresponds to either an edge contraction in

theinteractiongraphGoftheHamiltonianH, oravertexdele-

tioninG, yieldinganinteractiongraphG?fortheHamiltonian

H?. Two-spin isometries Uu:uvrepresent the reduced state-

space of the two-spin subsystem {u,v} as the image of a sin-

gle spin under an isometry: the corresponding reduction may

thus be represented as a “contraction” of two spins into one,

as illustrated in Fig. 2. Single-spin terms humay be repre-

sented by loops on vertices: isometric reductions arising from

terms hu,vof rank 3 also yield a loop on the contracted ver-

tex. Spin-removal reductions Pucorrespond to the deletion of

a vertex u with a loop, which removes all edges au incident

to u, possibly replacing them by loops on the neighbors a.

Page 10

10

This representation of Hamiltonian reductions in terms of

graphs is underdetermined, in that it is not always possible

to determine the ranks of the reduced Hamiltonian H?from

those of the Hamiltonian H prior to contraction. However,

the correspondence to graph reductionsmotivates a simple ob-

servation. Consider a subsystem A, and consider the Hamil-

tonian HAtogether with its interaction graph GA. We may

“colour” or “rank” the edges of GAaccording to whether the

term corresponding to each edge is rank-1 (which we call

“light” edges) or has rank 2 or 3 (which we call “heavy”

edges). The two-spin isometric reductions of Section IIIA1

required to obtain˜H˜

edges in GA. As such contractions preserve connectivity, this

implies that the interaction graph˜G˜

has as many vertices as there are connected components in

the “heavy subgraph” of GA. In particular, if the number of

“heavy” connected components (components connected only

by heavy edges) is bounded above by some parameter β, we

then obtain ˜ n = |˜A| ? β, so that

log(dimker(˜H˜

Acorresponds to contractions of all heavy

Acorresponding to˜H˜

A

A)) ? log(β + 1).

(51)

A consequence of this is that if H is a frustration-free nat-

ural Hamiltonian which contains only terms of rank 2 or 3,

all edges in GAwill be heavy, so that it consists of a single

heavy component; we then have log(dimker(˜H˜

this case, there is at most one e-bit of entanglement between

A and any other, disjoint subsystem in the lattice.

We may further refine this observation by considering the

impact of Hamiltonian terms of rank 3. More generally, we

may consider rank-1 single-spin terms in the reduced Hamil-

tonians, arising either from rank-1 terms in preceding Hamil-

tonians, or from performing an isometric reduction on terms

of rank 3. Such single-spin terms correspond to loops on ver-

tices in the interaction graphs G?

nians H?

nian containing such terms, we may show that the Hamilto-

nian H is non-degenerate with a ground state consisting es-

sentially of a product of single-spin states (together with a

single two-qubit entangled state if the original Hamiltonian

H contains a term of rank 3). Consider the effect of preferen-

tially performing single-spin deletions in the process of reduc-

ing H by isometries: for a natural Hamiltonian, we may eas-

ily verify that removal of such a vertex u (i.e. performing the

single-spin removal reduction of Section IIIA2) will induce

loops corresponding to single-spin terms on all neighbors of

u. These spins may then be removed in turn, inducing still

further loops; by the requirement that the original interaction

graph G be connected, this ultimately results in the removal

of every spin on which H acts, each by independent single-

spinisometrieswhichdescribeafixedsingle-qubitstatevector

|ϕu?. As a result, the entire lattice contains no entanglement,

or at most one e-bit if H contained a single rank-3 term giv-

ing rise to a “seed” loop; any subsystem A which is not acted

on by the rank-3 term therefore contains no entanglement, nor

has any entanglement with its environment.

The above exhibits the fragility of the condition of

frustration-freeness: it follows, for instance, that any natu-

ral Hamiltonian H which contains as many as two terms of

A)) = 1. In

A? of the reduced Hamilto-

A?. In the case of a frustration-free natural Hamilto-

rank 3 is necessarily frustrated (i.e. does not have a ground

space characterized by those of its interaction terms). Because

the same unique ground state must be produced by any re-

duction, e.g. in which we first perform two-qubit isometries,

it follows that each two-qubit isometry in such a reduction

must also map the single-spin states (describing the unique

ground state of the reduced Hamiltonians) to product states,

which is of course highly unlikely if instead one considers

arbitrarily chosen two-qubit isometries and single-product in-

put states. These observations may be used together with the

random satisfiability results of Ref. [30] to suggest that “ex-

act” frustration-freeness is likely to be rare in physical sys-

tems; small perturbations are likely to cause frustration. This

is nothing but a manifestation of a fragility against sponta-

neous symmetry breaking. However, in Section VIII, we sug-

gest ways in which systems which differ only slightly from

frustation-free systems may be examined using the techniques

of Sections V and VI.

VII.DIFFERENT MODELS OF FRUSTATION-FREE

HAMILTONIANS

In this section, we consider frustration-free Hamiltonians

H, but suspend our earlier restriction to natural Hamiltonians

(as described on page 5) in order to consider different mod-

els of Hamiltonians that are of interest. In doing so, we will

compare the resulting analysis to the case of frustration-free

natural Hamiltonians in Section VI.

A.Rank-two terms lacking entangled excited states

Any Hamiltonian term ha,bin H which has rank 2 and has

only product states orthogonal to its ground space is of the

form ha,b = |ϕ??ϕ|a⊗ ηb(or the reverse tensor product),

where η is a single-spin operator of full rank and |ϕ? ∈ H2.

This operator has the same kernel as the single-spin operator

|ϕ??ϕ|a⊗ 1b; therefore, if H is frustration-free, we may per-

form this substitution without any change to the ground state

manifold or its properties. As any rank-3 operator has entan-

gled states orthogonal to its (unique) ground state, we may

therefore restrict to the case where “non-natural” terms ha,b

occuring in H have rank 1, so long as we permit input Hamil-

tonians with single-spin terms.

B. Unfrustrated translationally invariant Hamiltonians

Consider a frustration-free Hamiltonian H in which the

interaction terms ha,bof each spin a is the same for all of

its neighbors b. If H is not natural, it follows that ha,b =

|α??α|a⊗ |β??β|bfor some states |α?,|β? ∈ H2; and by a

suitable choice of basis on each site, we may without loss of

generality let |α?= |β?= |1?.

Consider a ground state vector |Φ?of the Hamiltonian. For

each site a, if the state vector of |α? is not given by |1?, it

follows that all of the neighbors of a are in |1?; and conversely,

Page 11

11

if all of the neighbors of some site a are in |1?, the site a may

be in an arbitrary single-spin state without contributing to the

energy of the global state. It follows that the ground-state

manifold of H consists of all superpositions of product states

in which all sites are in |1?, except for some set of mutually

non-adjacent sites A ⊆ S, whose spins may have arbitrary

states (including states which are entangled with other sites

in S). In particular, for bipartite lattices, this includes states

in which the entire “even” sublattice of sites an even distance

from the origin are in |1?, and the opposite “odd” sublattice

may have an arbitrary entangled state.

Thus, if H is isotropic and frustration-free, then without

loss of generality it is either natural, or contains subspaces in

which large subsystems of the lattice are essentially uncon-

strained, and may occupy states with arbitrarily large entan-

glement content. Consequently, one may expect that unfrus-

trated translationally-invariant Hamiltonians should have in-

teraction terms with entangled excited states, i.e. be given by

natural Hamiltonians.

C. Unfrustated lattices with randomly located product terms

and percolation

Finally, we wish to consider a class of random Hamil-

tonians which includes non-natural Hamiltonians, and com-

pare the behaviour of their ground-state manifolds to natural

Hamiltonians. If one distributes random Hamiltonian terms

ha,bover nearest-neighbor pairs in an arbitrary lattice, then

they will almost certainly have an entangled excited state, as

the highest-energy eigenstate of each term will be a product

state with probability zero. This remains true even if one con-

strains each interaction term ha,bin the lattice to have ranks

described by integers ra,b ∈ {1,2,3} selected according to

any distribution, including the case where every term has rank

1. In order to obtain a random model of non-natural Hamilto-

nians, we must explicitly designate certain interactions ha,bto

be rank-1 product operators (non-natural terms of higher rank

being subject to the remarks of Section VIIA above), and con-

sider the scaling of the resulting lattice model.

Consider a d-dimensional rectangular lattice, in which each

term ha,bhas rank-1, and for each term we randomly deter-

mine whether ha,bis a product term (i.e. satisfies ha,b =

|α??α|a⊗ |β??β|bfor some |α?,|β? ∈ H2) or an entan-

gled term (satisfies ha,b = |γ??γ| for some entangled |γ? ∈

H2⊗ H2). The probability that ha,bis entangled is given

by some fixed p > 0, independently for each edge. Hav-

ing determined whether ha,bis entangled or not, we select a

random rank-1 projector for ha,bsubject to that constraint on

ha,b. Considering only frustration-free Hamiltonians H con-

structed under such a model [41] and subsystems A of the lat-

tice on which H acts, we wish to determine the dimension of

the ground-state manifold M = ker(HA); by a similar anal-

ysis as in Section VI, this will indicate how close the ground-

states of H come to obeying an entanglement area law.

AswenotedinSectionIVA, theprocessofinducingrank-1

constraints as in Eq. (22) will yield entangled (“natural”) con-

straints from two other entangled constraints. Consider the

subgraph of the lattice consisting of entangled constraints: it

follows that any subsystem A ⊆ V of the lattice which is con-

nected only by entangled constraints forms a subsystem for

which HAis a natural Hamiltonian, with a kernel of dimen-

sion at most |A|+1. Conversely, we may easily show that for

any product term ha,u= |α??α|a⊗ |β??β|u, the constraints

˜ha,vinduced by ha,btogether with any other constraint hu,v

will also be a product term (regardless of whether hu,vis a

product term). Thus, such product terms ha,uin the Hamilto-

nian represent obstacles to the induction of constraints which

would yield bounds on entanglement: as we noted in Sec-

tion VIIB above, the prevalence of product terms in a Hamil-

tonian H allow for the effective decoupling of large subsys-

tems in the ground-state manifold of H, yielding extremely

high degeneracy.

These observations suggest an approach to bounding

dim(M) using percolation theory [39] to bound the number

and size of components connected by entangled edges in a

large convex subset (for a review on applications of percola-

tion theory in quantum information, see Ref. [40]). We may

consider the worst case scenario in which no additional con-

straints may be induced between any two subregions A1,A2

which are internally connected by entangling terms, but sepa-

rated by a barrier of product terms which effectively decouple

the subsystems A1and A2. If the probability p is above the

percolation threshhold pcof the lattice, we may apply the fol-

lowing results:

Proposition 3 ([39, Theorem 4.2]). Let A be a hypercube

consisting of n vertices, in a d-dimensional rectangular lat-

tice with edge-percolation probability p. Then there exists a

positive real κp∈ R such that the number of connected com-

ponents in A grows as κpn, as n → ∞.

Proposition 4 ([39, Theorem 8.65]). Let C be a finite-size

connected component containing an arbitrary vertex (e.g.

the origin) in a d-dimensional rectangular lattice with edge-

percolation probability p. For p > pc, there exists a positive

ηp∈ R such that

?|C| = s?

Both κpand exp(−ηp) in the propositions above are an-

alytic for p > pc, and thus must converge to 0 in the limit

p → 1. We may thus describe an upper bound on the dimen-

sion of M as follows, for A a large cube containing n ? 1

spins. For p > pc, there is almost surely a unique maximum-

size component A0of the lattice which is connected by entan-

gled edges: because the percolation probability θpis strictly

positive (by definition) for p > pc, we will have |A0| = θpn

on average. Each subsystem A0,A1,...,Ak ⊆ A which is

connected by entangled edges induces a natural Hamiltonian

HAjwhich has a kernel of dimension at most |Aj| + 1: we

may bound dim(M) by noting that

Pr

p

? exp

?

− ηps(d−1)/d?

.

(52)

M ⊆ ker(HA0) ⊗ ker(HA1) ⊗ ··· ⊗ ker(HAk) ,

(53)

Page 12

12

as in Eq. (42). This allows us to obtain the bound

logdim(M) ?

?

?

j?0

logdim(ker(HAj))

?

j?0

log?|Aj| + 1?.

(54)

By Proposition 3, the expected number of components k

grows like κpn for some κp> 0 as n → ∞; using the prob-

ability bound on the typical finite component size of Proposi-

tion 4 as the probability of an indistinguished connected com-

ponent having size s, we obtain the upper bound

?

? log?|A0| + 1) +

κpn

?

? log(θpn + 1) + κpCpn ,

Exp

p

logdim(M)

?

?

j?1

log?|Aj| + 1?

?∞

s=1

? log(θpn + 1) +

j=1

?

log(s + 1)

exp?ηps(d−1)/d?

?

(55)

where Cpis the sum in square brackets (which is small for

exp(−ηp) small).

As logdim(M) is also the logarithm of the maximum

Schmidt rank of any state with respect to the bipartition into

A and V ? A for the lattice V , the amount of entanglement

scales with the logarithm of the size of the cube A, with a

small and tunable linear correction, for p ≈ 1. In this sense,

frustration-free Hamiltonians in such a “percolated” product-

model on rectangular lattices resemble frustration-free natural

Hamiltonians in the expected case as p → 1.

VIII. ALMOST FRUSTRATION-FREE HAMILTONIANS

The method of efficiently simulating ground state mani-

foldsoffrustration-freeHamiltonianscanbeextendedtoserve

as a method to simulate almost-frustration-free Hamiltonians,

albeit in a non-certified way. Consider a Hamiltonian

H = H0+ λH1

(56)

for λ ∈ R playing the role of a small perturbation, where

H0 =

?

{a,b}⊆V

ha,b

(57)

is exactly frustration-free (i.e. in the sense defined in Sec-

tion IIA), and H1is a small local perturbation. Then, one can

still efficiently compute

inf

|Φ?∈M?Φ|H |Φ?,

(58)

where M denotes the (in general, degenerate) ground state

manifold of H0. Again, we may characterize M as the im-

age of the low-dimensional subspace Symm(H⊗nc

tree-tensor network, as described in Section V; H1being a

local Hamiltonian, each term of the infumum above can be

efficiently computed using a suitable basis of Symm(H⊗nc

This is a variational approach that will always provide an up-

per bound to the true ground state energy.

In this way, one approximates the ground state manifold

of an almost frustration-free Hamiltonian with the ground

state manifold of an exactly frustration-free one. The inter-

esting aspect here is that one can consider the image of an

entire large subspace under a tensor network. In practice, one

would think of a Hamiltonian HUnear to a realistic one H,

where one may show that HUis frustration-free (which may

be efficiently verified using the algorithm of Ref. [22], as out-

lined in Section III), and then approximate the ground state

of the full Hamiltonian. This approach appears to be partic-

ularly suitable for slightly frustrated Hamiltonians reminding

of Shastry-Sutherland type [42] models, with — in a cubic lat-

tice and a frustration-free Hamiltonian — an additional bond

along the main diagonal renders the model frustrated.

2

) under a

2

).

IX. SUMMARY

In this work, we have investigated in great detail a class

of models whose ground-state manifolds can be completely

identified: those of physically realistic frustration-free mod-

els of spin-1/2 particles on a general lattice. We have seen

that the entire ground state manifold can be parametrized by

means of tensor networks applied to symmetric subspaces, by

essentially undoing a sequence of isometric reductions. We

also found that any ground state of such a system satisfies an

area law, and hence contains little entanglement. This is a

physically meaningful class of physical models — beyond the

case of free models — for which such an area law behaviour

can be rigorously proven. It is the hope that the idea of consid-

ering entire subspaces under tensor networks, and eventually

looking at the performance when being viewed as a numerical

method, will give rise to new insights into almost frustration-

free models.

X.ACKNOWLEDGEMENTS

This work has been supported by the EU (QESSENCE, MI-

NOS, COMPAS) and the EURYI award scheme. We would

like to thank D. Gross and S. Michalakis for discussions. Part

of this work was done while JE was visiting the KITP in Santa

Barbara as participant of the Quantum Information Program.

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|Φ? of H, the spins on which such terms act are disentangled

from the rest of the system.

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terms. Extending the analysis of Ref. [22], this may be effi-

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Appendix A: Technical lemmas

We now supply the proofs of technical lemmata required in

the preceding sections.

Proposition (Lemma 1). For two-spin state vectors |ψ? and

|φ?, we have??ψ|⊗1??1⊗|φ??= 0 only if both |ψ?and |φ?

Proof. Consider Schmidt decompositions

?

where without loss of generality we may require ?fr|gr? ?= 0

by an appropriate choice of labels. Then we have

?

which is only zero if µ0ν0= µ1ν1= 0, which implies µr= 0

and ν1−r= 0 for some value of r ∈ {0,1}.

Proposition (Lemma 2). Let U : H2 −→ H2⊗ H2be an

isometry which is not a product operator. Let η ? 0 be an

operator on two spins, and η?=?U†⊗12)(12⊗η)(U ⊗12).

only if η is a product operator.

are product states.

|ψ?=

r

µr|er?|fr?,|φ?=

?

s

νs|gs?|hs?,

(A1)

??ψ| ⊗ 1??1 ⊗ |φ??=

r,s

?µrνs?fr|gs??|er??hs| ,

(A2)

?

If η?is not of full rank, then η?is a product operator if and

Proof. Suppose that η?is not full rank, and is a product opera-

tor. As it is positive semidefinite, η?must either have the form

|α??α| ⊗ η??or η??⊗ |α??α| for some |α? ∈ H2. In particular,

there must exist a state vector |γ?∈ H2such that one of

??γ| ⊗ 1?η??|γ?⊗ 1?

= 0

or (A3a)

(A3b)

?1 ⊗?γ|?η??1 ⊗ |γ??

= 0

Page 14

14

holds. Decompose η in its spectral decomposition,

η =

?

k

λk|φk??φk|

(A4)

for λk> 0. Suppose that??γ|⊗1?η??|γ?⊗1?= 0: if we let

|Γ?= U |γ?, we have

0 =??γ|U†⊗ 1?(1 ⊗ η)?U |γ?⊗ 1?

=??Γ| ⊗ 1???

=λkTkT†

k

λk1 ⊗ |φk??φk|

??|Γ?⊗ 1?

?

k

k,

(A5)

where we define Tk=??Γ| ⊗ 1??1 ⊗ |φk??. By Lemma 1,

product vectors; we may then decompose |Γ? = |σ?|τ? and

|φk? = |τ??|φ?

each operator Tkis zero only if both |Γ? and |φk? are both

k?, where we require ?τ |τ?? = 0 for all k. We

then have

η = |τ???τ?| ⊗

??

k

λk|φ?

k??φ?

k|

?

.

(A6)

On the other hand, if?1 ⊗?γ|?η?1 ⊗ |γ??= 0, we obtain

0 =?U†⊗?γ|?η?U ⊗ |γ??

=

?

k

λkU†?1 ⊗ |φ?

k? =?1 ⊗?γ|?|φk?. We then require

posed as U?⊗ |u?for any state |u?∈ H2, this implies that the

vectors |φ?

some states |αk?∈ H2, and where ?γ|β? = 0. We then have

??

In either case, η?is a product operator only if η is a product

operator; the converse holds trivially.

k???1 ⊗?φ?

k|?U ,

(A7)

for single-spin states |φ?

U†?1 ⊗ |φ?

k? themselves are zero. Thus, |φk? = |αk?|β? for

k??= 0 for each k; because U cannot be decom-

η =

k

λk|α?

k??α?

k|

?

⊗ |β??β|.

(A8)

?

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