Ground states of unfrustrated spin Hamiltonians satisfy an area law

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.

3 the ground state energy is probably NP-hard 4 -albeit using a fairly sophisticated construction involving large local dimensions [19]. 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. This suggests that whether or not the system is frustrated is another criterion for whether a quantum lattice model should be considered 'easy' or 'hard', in addition to its ground states having 'a lot of' or 'little' entanglement. This intuition that a frustrated system 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 [20].
In this work, we explore a class of models where this intuition of frustration-free models being easy to solve holds true. Building upon the work in [21] and further extending the findings of [22], 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 entanglement in the above sense. We discuss how to efficiently simulate the ground state manifold, and suggest how this could be used to simulate 'almost' frustration-free quantum lattice models.

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 denote 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 h a,b acting on pairs of spins in the lattice described by V . By rescaling, we may without loss of generality require that the ground state energy of each interaction term h a,b be zero. We wish to describe the properties of the ground state manifold M of such Hamiltonians, given the list of individual two-spin terms h a,b as the input. An important class of Hamiltonians is those which are frustration-free (or unfrustrated), for which each ground state vector | ∈ M is also a ground state of the individual coupling terms, that is, for which holds for all h a,b and all | ∈ 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. Our main results pertain to frustration-free spin Hamiltonians, as above, with the further constraint that each term h a,b has at least one entangled excited state. In section 6, we show that 4 Figure 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.
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 5 of the ground state satisfies 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 figure 1). The entanglement of formation is the largest asymptotically continuous entanglement 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. This follows as a consequence of the fact that the ground state manifold is the image of the symmetric subspace on n spins (for some n bounded above by the number of spins of the Hamiltonian) under an efficiently simulatable, and explicitly constructible, tree tensor network. We show this in section 5, in which we demonstrate how this allows us to efficiently simulate 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 [23].
Our result of an analytical area law complements results on area laws in harmonic bosonic systems [7,10,11], fermionic [13]- [15] on cubic lattices and general gapped models in onedimensional (1D) quantum chains [16]. For a comprehensive review on area laws-and their implications on the simulatability of quantum many-body systems-see [1].

Quantum 2-sat problem
The arguments behind our analysis build upon and extend the ideas of [21], 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 'classical' 2-sat problem on boolean formulae. The latter asks when there exists an assignment of boolean variables x 1 , . . . , x n that 5 simultaneously 'satisfy' a collection of constraints on the pairs of those variables. This problem is efficiently solvable [24]; in contrast, the similar problem 3-sat (in which constraints apply to triples of variables) is NP-complete [25]. In quantum 2-sat, individual clauses on boolean variables are replaced by projectors p a,b with p 2 a,b = p a,b (4) on pairs of spins: an instance of quantum 2-sat is satisfiable if there is a vector that is a zero eigenvector of each projector simultaneously.
Given an instance of quantum 2-sat, the problem of determining if there exists such a simultaneous zero eigenvector is equivalent to determining whether the Hamiltonian obtained by summing the projectors is unfrustrated. Conversely, the problem of determining 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 h a,b has a minimal eigenvalue of 0, and replacing each rescaled term h a,b with the projector p a,b onto img(h a,b ). By construction, such a substitution 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 6 have been studied for k 2, delineating the 'boundary' of frustration in k-local spin Hamiltonians in terms of the density of interactions [26]- [29]. We instead extend the findings of [21] for k = 2, and remark on implications for simulating the ground space manifold in frustration-free Hamiltonians. In section 3, we review Bravyi's algorithm for quantum 2-sat, in order to demonstrate important features of the reductions involved when they are applied to unfrustrated Hamiltonians satisfying natural constraints.

Reduction tools for frustration-free Hamiltonians
Bravyi's algorithm for quantum 2-sat [21] 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 remainder of the paper, we admit representations of Hamiltonians H that include non-trivial single-spin terms h a , and again describe H as unfrustrated if there exists a joint ground state with eigenvalue zero of all terms (including the single-spin terms h a ).

Reductions by isometries
Condensing the analysis of [ This is an isometric reduction, similar to those in a tree tensor network or a MERA ansatz [30]. By construction, the support of the reduced state ρ u,v lies in img(U u:uv ). We may then define a Hamiltonian on the subsystem V = V {v}, where the spin v is essentially deleted; any state vector | ∈ ker(H ) then has the form We may express H as a sum of terms (In the case that h is of rank 3, these will include a non-zero single-spin operator h u,v that acts on u alone.) If H contains non-zero terms h a,u and h a,v , we obtain two terms h a,u = U † u:u,v h a,u U u:u,v and h a,v = U † u:u,v h a,v U u:u,v in the Hamiltonian H , which both act on the spins u and a. We sum these to obtain a combined term h a,u = h a,u + h a,v (11) in the reduced Hamiltonian, which may be of higher rank than either h a,u or h a,v . (We similarly accumulate any single-spin contributions h u and h v to H , arising from single-spin contributions h u and h v in H .) Figure 2 illustrates the effect of multiple transformations on the interaction graph of the Hamiltonian. For example, consider an anti-ferromagnetic four-spin Hamiltonian, with interactions on a four-spin cycle with interacting pairs (1,2), (2,3), (3,4) and (4,1). Here we denote Rescaling the interactions to have ground energy zero (and taking these for the h j,k instead) gives us h j,k = |0, 0 0, 0| j,k + |1, 1 1, 1| j,k .

Figure 2. Illustration of transformations of the interaction graph of a Hamiltonian
H by two-spin isometries. Darker, thicker edges represent interaction terms h a,b of rank 2 or 3, which may be eliminated by contraction (corresponding to a reduction H → U † a:ab HU a:ab ). In the case of an interaction term with rank(h r,u ) = 3, as illustrated above, a single-spin operator (represented above by a loop) is produced on the contracted vertex.
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 with a ferromagnetic coupling between site 1 and the renormalized site 3. Arbitrary rank-2 or rank-3 interactions may be contracted similarly.
on the subsystem V = V {v}, consisting of a sum of terms h a = h a acting on individual spins a = v, and terms acting on pairs of spins {a, b} ⊆ V . In the latter case, if v ∈ {a, b}, then h a,b will be an operator acting on a single spin; otherwise, we have h a,b = h a,b . Again, in the case of single-spin terms h a,b acting on a spin a, if there was a term h a present in H , we may accumulate these into a termh a = h a + h a,b . We may again describe H using an isometric reduction in this case: if we define P v = |ψ v ⊗ 1 V , then P v is an isometry whose image contains any state vector | ∈ ker(H ). We may then rewrite equation (18) as expressing it in a form more explicitly similar to equation (8).

3.1.3.
Remarks on these reductions. The reductions above correspond to the reductions presented in [21] for instances of quantum 2-sat that contain two-spin operators of rank greater than 1. This allows us to reduce quantum 2-sat to the special case of 'homogeneous' 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 Hamiltonian H has trivial kernel; then the same holds for H as well. If we do not encounter any full-rank terms, each reduction produces a Hamiltonian acting on one fewer spins, eventually yielding a 'homogeneous' Hamiltonian (extending the terminology of [21] to Hamiltonians in general, 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 that contains two-spin terms of rank at least 2 and which does not contain full-rank terms, we may extend any sequence of reductions to one that terminates with a HamiltonianH that 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 original Hamiltonian 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): from a temporal top layer defined by a Hamiltonian containing 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 5. Note also that for a 1D quantum 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.

The homogeneous case
Given a homogeneous Hamiltonian H (containing only terms of rank 1) acting on some system V , consider a collection of vectors |β a,b ∈ H ⊗{a,b} 2 such that We interpret each two-spin Hamiltonian term h a,b as a constraint on the corresponding two-spin marginal state ρ a,b of a state | ∈ ker(H ), and attempt to obtain additional constraints on pairs of spins a, b ∈ V by combinations of the constraints that are already known. Bravyi [21] has shown that if | lies in the kernel of β a,b | and β b,c | acting on the corresponding spins, it also lies in the kernel of the functional acting on spins a and c, where | − ∝ |0 |1 − |1 |0 is the two-spin antisymmetric state vector. We call such a constraint β a,c | an 'induced' constraint, and use the term induction of constraints to refer to the operation on β a,b | and β b,c | that gives rise to β a,c | in equation (22), up to a scalar factor.
For each induced constraint β u,v | on a pair of spins u, v, we may add a term to the Hamiltonian H , obtaining a HamiltonianH that (by construction) has the same kernel as H . If H already contains a term h u,v that is not colinear to the induced termh u,v , these may be accumulated into a termh u,v whose rank is at least 2, and one may apply a two-spin contraction, as described in section 3.1. Otherwise, we may induce further constraints from the terms ofH , until we obtain a complete homogeneous Hamiltonian H c : a Hamiltonian in which the two-spin constraints β u,v | are closed under constraint induction. By inducing constraints on pairs of spins, possibly performing two-spin contractions, as in section 3.1, when we obtain terms of rank 2 or more, we may efficiently obtain a complete homogeneous Hamiltonian H c from a frustration-free, 2-local Hamiltonian H . Furthermore, Bravyi [21] has shown that a complete homogeneous Hamiltonian H c that is acting on at least one spin (and which lacks single-spin operators 7 ) has a ground space that contains product states. Thus, for homogeneous Hamiltonians H , we may either efficiently determine that it is frustrated or efficiently obtain a Hamiltonian that is closed under constraint induction. In the latter case, we may construct product states in the kernel of H by selecting states for each spin consistent with the two-spin constraints. 8

Unfrustrated natural Hamiltonians
We now present results concerning the ground state manifold of a 'physical' class of 2-local spin Hamiltonians. We will say that a Hamiltonian H is natural if it is 2-local, contains no isolated subsystems, and each term h a,b (acting on H 2 ⊗ H 2 ) has at least one entangled excited state (i.e. there exists an entangled state orthogonal to the ground state manifold of h a,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 that is inequivalent to either diag(0, 0, E a , E b ) or diag(0, E a , 0, E b ) up to rescaling and a choice of basis for each spin.
Using the reductions of section 3.1, we show strong bounds on the dimension of the ground space of an unfrustrated natural Hamiltonian on spins. This will allow us, in section 5, to describe a scheme for efficiently simulating the ground space of frustration-free natural Hamiltonians and, in section 4, to demonstrate that the ground states of such Hamiltonians satisfy an entanglement area law.

Ground-spaces of unfrustrated, natural, homogeneous Hamiltonians
We now present an extension of the analysis of Bravyi [21] for homogeneous and complete Hamiltonians H c (acting on a set V c of spins) to examine the ground state manifold of H c in the case that H c is also natural. We show, using techniques similar to those used in [27, section III A], that the ground space of such a Hamiltonian is equivalent to the symmetric subspace Symm(H ⊗V c 2 ) ⊆ H ⊗V c 2 , up to some efficiently constructible choice of invertible operations on each spin. As we may reduce more general Hamiltonians, i.e. having terms of rank 2 or 3 (extending beyond those Hamiltonians considered in [27]) to homogeneous natural Hamiltonians via the reductions of section 3.1, these results yield important consequences for natural frustration-free Hamiltonians in general.
Consider a Hamiltonian H c acting on V c , where H c has no single-spin terms. Because the two-spin constraints described by the terms of H c are closed under the induction of constraints (as described by equation (22)), and as there are no isolated subsystems, it is easy to show that every pair of spins is acted on by a non-zero term in H c . For such a Hamiltonian, the excited states |β a,b for the terms h a,b in H c are entangled states. We may then construct a family of for each pair of spins u, v, where | − is again the two-spin antisymmetric state vector. For instance, one may fix L a = 1 for an arbitrarily chosen spin a ∈ V c , and determine linear operators L v satisfying equation (24) for each v ∈ V c and operator β a,v |. Any such choice of operators {L v } v∈V c satisfies equation (24) for all u, v, which follows from the closure of the constraints β u,v | under induction, We define scalars where we define T u,v by As the operators T u,v have full rank, the operator ( β u,v | ⊗ 1)T then has the same kernel as − | u,v , which is the +1-eigenspace of the swap operator acting on u and v. It follows that the kernel of the Hamiltonian is the symmetric subspace Symm(H ⊗V c 2 ); this corresponds to the result of [27, section III A]. We remark on some important properties of Symm(H ⊗n c 2 ), where n c = |V c |. This subspace is spanned by uniform superpositions |W k of the standard basis states having Hamming weight 0 k n c , thus dim Symm(H ⊗n c 2 ) = n c + 1. This subspace may also be spanned by product state vectors |α 0 ⊗n c , . . . , |α n c ⊗n c for any set of n c + 1 pairwise independent state vectors |α j ∈ H 2 . Thus, any natural Hamiltonian H c that is also complete and homogeneous has a ground space of dimension n c + 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 |α 0 , . . . , |α n c ∈ H 2 ; we use this fact in section 5. Note that if even this efficient method should be too computationally costly for very large systems, one can also Monte-Carlo sample from the ground state manifold in this way.

Preservation of natural Hamiltonians under reductions
A key feature of natural Hamiltonians (defined earlier in section 4) is that the class of frustrationfree natural Hamiltonians on spins is preserved by the two-spin contractions described in equation (8). This implies that the reductions of section 3. has an entangled excited state if the same holds for h a,u . We require the following two lemmas, whose proofs we defer to the appendix: 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 are product states. Lemma 2 (Product operators). Let U : H 2 → H 2 ⊗ H 2 be an isometry that is not a product operator. Let η 0 be an operator on two spins, and η = (U † ⊗ 1 2 )(1 2 ⊗ η)(U ⊗ 1 2 ). If η is not of full rank, then η is a product operator if and only if η is a product operator.
We show that frustration-free natural Hamiltonians are preserved by the reductions of section 3.1.1, as follows. Let H be a natural 2-local Hamiltonian, and h u,v be a two-spin term in H . Define U u:u,v = |ψ 0 0| + |ψ 1 1| (30) for orthonormal two-spin state vectors |ψ 0 and |ψ 1 whose span contains ker(h u,v ); we require that |ψ 1 be entangled, which ensures that U u:u,v is not a product operator. Consider the terms h a,b = U † u:u,v h a,b U u:u,v that occur in the Hamiltonian H = U † u:u,v HU u:u,v . For any two-spin operator h v,a acting on v and some other spin a, the fact that h v,a has an entangled excited state implies, in particular, that it is not a product operator. Thus, h v,a is a product operator only if it has full rank. If H is frustration-free, h v,a cannot have full rank; then h v,a is not a product operator, and in particular it will have entangled excited states. As h a,b = h a,b when a, b / ∈ {u, v}, 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 natural and frustration-free, and also contains no two-spin terms of rank 1, then the same is true of H = U † u:u,v HU u:u,v as well. For any two-spin term h v,a acting on v with rank at least 2, consider states |ϕ 0 , |ϕ 1 ∈ img(h v,a ) such that |ϕ 0 = |α |β is a product state and |ϕ 1 is entangled; 9 and choose real parameters λ 0 , λ 1 > 0 such that Let η k = |ϕ k ϕ k | for k ∈ {0, 1}, and consider the images η k under contraction by U u:u,v : where we define M j,k = ( ψ j | ⊗ 1)(1 ⊗ |ϕ k ). By lemma 1, we have M j,k = 0 only if both |ψ j and |ϕ k are product operators; this implies that the operators M 1,k , in particular, are non-zero, so that η k = 0 for any k. Note that because H has a non-trivial kernel, h v,a has rank at most 3, in which case neither operator η k has full rank. By lemma 2, η 1 is not a product operator; as η 0 is a product operator, these operators are linearly independent. Then, λ 0 η 0 + λ 1 η 1 has rank at least 2; by equation (33), the same is true of h v,a . If all of the terms in H have rank 2 or higher, the same then holds for H as well. Thus, if we apply the reductions of section 3.1 to an initial Hamiltonian that is both natural and frustration-free, the resulting Hamiltonians will also be natural and frustration free. Furthermore, if H contains no terms of rank 1, then neither will the reduced Hamiltonians. Because the process of inducing constraints, described in section 3.2, also preserves the property of each term h a,b having entangled excited states, these invariants ensure that initial Hamiltonians with these properties (natural and frustration-free, and possibly containing no terms of rank 1) may be reduced to homogeneous and complete Hamiltonians that have these same properties. We may then apply the results of section 4.1 to these reduced Hamiltonians.

Simulating ground spaces of frustration-free natural Hamiltonians
Building on the results of section 4, we now show how the reductions of section 3.1 may be used to obtain a procedure for simulating states from the ground state manifold of frustrationfree natural Hamiltonians H .

Tree tensor networks and matrix-product states
As we noted in section 3.1.3, implicit in the reductions of section 3.1 is that the isometric reduction from general Hamiltonians 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 3.1, until we obtain a homogeneous instance without single-spin terms. We then attempt to induce additional constraints via equation (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 H c on a subset V c ⊆ V .
Consider the tensor network T that performs the complete reduction as above. We describe T in reverse order, as introducing new spins to represent a unitary embedding of ker(H c ) into ker(H ). The various spin contraction isometries U u:u,v , as in equation (7), each have a single input index and two output indices; the spin deletion isometries P v have 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 that is introduced by an isometry U u:u,v may be considered a 'daughter spin' of a unique parent U , which imposes a tree-like hierarchy 10 on the tensor network T , as illustrated in figure 3. The roots of each tree are spins u which are prepared by an isometry P u derived 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 Hamiltonian H is degenerate, there will necessarily be free input indices, representing a domain consisting of a state space of dimension 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(H c ) at the input, where H c is the complete homogeneous instance obtained by the Hamiltonian reductions. Thus, if one may efficiently simulate 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 .

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 simulated over inputs with low Schmidt measure [33], as follows.
For any observable acting on m spins, one may evaluate H for the maximally mixed state over the ground state manifold of H by computing the expectation T † T H c over the ground state manifold of the complete homogeneous Hamiltonian H c obtained as described in section 3. As the tensor T has tree structure, the observable T † T also acts on at most m spins. If we can obtain an orthonormal basis for ker(H c ) which may be succinctly described in terms of product states, we may evaluate expectation values of T † T with respect to m-fold products of single-spin states.
As we noted in section 4.1, ker(H c ) can be spanned by a collection of n c + 1 product vectors (where n c is the number of spins on which H c acts). Let | 0 , | 1 , . . . , | n c ∈ ker(H c ) be a collection of independent product vectors, We may efficiently compute a projection of c onto ker(H c ) by performing a suitable transformation of the matrix Tree tensor network Tree tensor network for some orthonormal basis | 0 , . . . , | n c of ker(H c ), by taking the product of the above operator with its adjoint. Thus, the restriction of c to ker(H c ) with respect to the basis of states | j may be computed as By considering operators c = T † T , this allows us to compute the restriction of operators to the ground space of H U (see figure 4). We may thus efficiently estimate such observables with respect to ground states of H U : for constant m, the required inner products may be calculated as (sums of) scalar products of at most n c inner products over vector spaces of bounded dimension. To evaluate the value of with respect to ground states of the input Hamiltonian H , it suffices to analyze the polynomially sized operator¯ representing the action of on the ground state manifold.

Entanglement bounds for ground states of frustration-free natural Hamiltonians
The fact that the reductions of section 3.1 preserve the class of natural Hamiltonians (as defined in section 4) allows us also to make more global statements about ground states for frustration-free Hamiltonians, again by reduction to the complete homogeneous case described in section 4.1. 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 environment V A. We also consider some very general cases in which still stronger upper bounds on the entanglement may be obtained.

Area law for frustration-free natural Hamiltonians
We consider first the case where A is a contiguous subsystem (i.e. for which there is a path between any pair of spins in A, following edges in the interaction graph of the Hamiltonian H ), and subsequently generalize the observation in this case to arbitrary subsystems A. We first decompose for B = V A, and where H A and H B contain all terms internal to A and B, respectively. We then apply the reductions of section 3.1 to subsystem A. That is, we perform two-spin contractions, as described in section 3.1.1, for any two-spin terms in H A of rank 2 or 3, and perform spin deletions, as described in section 3.1.2, for any single-spin terms in H A . Performing the constraint-induction process of (22)-again on the terms acting on A alone-and then reducing further reductions as necessary, we eventually obtain a HamiltonianH of the form whereÃ ⊆ A is the set of spins remaining after the reduction process, and whereHÃ is a homogeneous and complete Hamiltonian 11 . In other words, we perform a complete tree tensor reduction on subsystem A, until we obtain a Hamiltonian whose restriction to A is homogeneous and complete.
taking the restrictions ofHÃ and H B to their respective subsystemsÃ and B. Letñ = |Ã|: as HÃ is also homogeneous and complete, it has kernel of dimensionñ + 1 by section 4.1.
In the case where A contains multiple components A 1 , A 2 , . . . , A k with respect to the interaction graph of H (where each A j is disconnected from the others but connected internally), we may perform the Hamiltonian reductions of section 3.1 to each component independently. We may further decompose the HamiltonianHÃ obtained in equation (39) as AsHÃ is unfrustrated, each of the sub-HamiltoniansHÃ j is unfrustrated as well, in which case we may write ker(HÃ) ⊆ ker(HÃ 1 ) ⊗ · · · ⊗ ker(HÃ k ), similarly to equation (40). Then, the dimension of ker(HÃ) is bounded by the product of dim ker(HÃ j ) =ñ j + 1 for each subsystem, whereñ j = |Ã j |. Let α j be the number of spins in A j which are adjacent in the spin lattice to spins in B, and letα j α j be the number of such spins inÃ j . 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 for each subsystem A j . We may then bound on the dimension of ker(HÃ) in terms of these 'boundary spins' as where α = α 1 + · · · + α k is the number of spins in A adjacent to elements of B. From this bound on dim ker(HÃ), it follows that any pure state | in the ground space ofH has Schmidt measure [33] at most α/K across the partitionÃ + B and so can support at most this many e-bits of entanglement betweenÃ and B. Because any state vector | ∈ ker(H ) can be obtained from some vector | ∈ ker(H ) by a network of isometries acting only on spins in A, it follows that any state vector | 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 [31]. 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 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 Thus, the amount of entanglement that 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 as 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

Stronger entanglement bounds for contiguous subsystems
The above analysis imposes no additional constraints, beyond 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 general conditions on the subsystem A when it is a contiguous subsystem.

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 dim ker(HÃ) =ñ + 1 (48) forñ = |Ã| by the analysis of section 4.1 (where HÃ is the reduced Hamiltonian acting on the subsystem A described in equation (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 equation (43) to obtain log(dim ker(HÃ)) = log(ñ + 1) log(α/K + 1) c .
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 as 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

Subsystems acted on by many high-rank Hamiltonian terms.
In the above result, we have neglected the difference in the sizes of subsystem A and the reduced subsystemÃ ⊆ A. The difference in their sizes will be precisely the number of isometric reductions performed to obtaiñ A from A. Each isometric reduction corresponds to either an edge contraction in the interaction graph G of the Hamiltonian H or a vertex deletion in G, yielding an interaction graph G for the Hamiltonian H . Two-spin isometries U u:uv represent the reduced state space of the two-spin subsystem {u, v} as the image of a single spin under an isometry: the corresponding reduction may thus be represented as a 'contraction' of two spins into one, as illustrated in figure 2. Singlespin terms h u may be represented by loops on vertices: isometric reductions arising from terms h u,v of rank 3 also yield a loop on the contracted vertex. Spin-removal reductions P u correspond 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. 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 reductions motivates a simple observation. Consider a subsystem A, and consider the Hamiltonian H A together with its interaction graph G A . We may 'color' or 'rank' the edges of G A according 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 3.1.1 required to obtainHÃ correspond to contractions of all heavy edges in G A . As such contractions preserve connectivity, this implies that the interaction graphGÃ corresponding toHÃ has as many vertices as there are connected components in the 'heavy subgraph' of G A . In particular, if the number of 'heavy' connected components (components connected only by heavy edges) is bounded above by some parameter β, we then obtainñ = |Ã| β, so that log(dim ker(HÃ)) log(β + 1).
A consequence of this is that if H is a frustration-free natural Hamiltonian that contains only terms of rank 2 or 3, all edges in G A will be heavy, so that it consists of a single heavy component; we then have log(dim ker(HÃ)) = 1. In 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 Hamiltonians, arising either from rank-1 terms in preceding Hamiltonians or from performing an isometric reduction on terms of rank 3. Such single-spin terms correspond to loops on vertices in the interaction graphs G A of the reduced Hamiltonians H A . In the case of a frustration-free natural Hamiltonian containing such terms, we may show that the Hamiltonian H is non-degenerate with a ground state consisting essentially 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 preferentially performing single-spin deletions in the process of reducing H by isometries: for a natural Hamiltonian, we may easily verify that removal of such a vertex u (i.e. performing the single-spin removal reduction of section 3.1.2) 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-spin isometries that describe a fixed single-qubit state vector |ϕ u . As a result, the entire lattice contains no entanglement, or at most one e-bit if H contained a single rank-3 term giving rise to a 'seed' loop. Any subsystem A that is not acted on by the rank-3 term, therefore, contains no entanglement, nor does it have any entanglement with its environment.
The above exhibits the fragility of the condition of frustration-freeness: it follows, for instance, that any natural Hamiltonian H that contains as many as two terms of 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 reduction, 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 input states. These observations may be used together with the random satisfiability results of Laumann et al [27] to suggest that 'exact' frustration-freeness is likely to be rare in physical systems; small perturbations are likely to cause frustration. This is nothing but a manifestation of a fragility against spontaneous symmetry breaking. However, in section 8, we suggest ways in which systems that differ only slightly from frustration-free systems may be examined using the techniques of sections 5 and 6.

Different models of frustration-free Hamiltonians
In this section, we consider frustration-free Hamiltonians H , but suspend our earlier restriction to natural Hamiltonians (as described in section 4) in order to consider different models 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 6.

Rank-two terms lacking entangled excited states
Any Hamiltonian term h a,b in H which has rank 2 and has only product states orthogonal to its ground space is of the form h a,b = |ϕ ϕ| a ⊗ η b (or the reverse tensor product), where random rank-1 projector for h a,b subject to that constraint on h a,b . Considering only frustrationfree Hamiltonians H constructed under such a model 12 , and subsystems A of the lattice on which H acts, we wish to determine the dimension of the ground state manifold M = ker(H A ). By a similar analysis as in section 6, this will indicate how close the ground states of H come to obeying an entanglement area law.
As we noted in section 4.1, the process of inducing rank-1 constraints, as in equation (22), will yield entangled ('natural') constraints 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 connected only by entangled constraints forms a subsystem for which H A is a natural Hamiltonian, with a kernel of dimension at most |A| + 1. Conversely, we may easily show that for any product term h a,u = |α α| a ⊗ |β β| u , the constraintsh a,v induced by h a,b together with any other constraint h u,v will also be a product term (regardless of whether h u,v is a product term). Thus, such product terms h a,u in the Hamiltonian represent obstacles to the induction of constraints which would yield bounds on entanglement: as we noted in section 7.2 above, the prevalence of product terms in a Hamiltonian H allows for the effective decoupling of large subsystems in the ground state manifold of H , yielding extremely high degeneracy.
These observations suggest an approach to bounding dim(M) using percolation theory [34] to bound the number and size of components connected by entangled edges in a large convex subset (for a review on the applications of percolation theory in quantum information, see [35]). We may consider the worst case scenario in which no additional constraints may be induced between any two subregions A 1 and A 2 that are internally connected by entangling terms, but separated by a barrier of product terms that effectively decouple the subsystems A 1 and A 2 . If the probability p is above the percolation threshold p c of the lattice, we may apply the following results: Both κ p and exp(−η p ) in the propositions above are analytic for p > p c and thus must converge to 0 in the limit p → 1. We may thus describe an upper bound on the dimension of M as follows, for A a large cube containing n 1 spins. For p > p c , there is almost surely a unique maximum-size component A 0 of the lattice that is connected by entangled edges: because the percolation probability θ p is strictly positive (by definition) for p > p c , we will have |A 0 | = θ p n, on average. Each subsystem A 0 , A 1 , . . . , A k ⊆ A that is connected by entangled edges induces a natural Hamiltonian H A j that has a kernel of dimension at most |A j | + 1: we may bound dim(M) by noting that M ⊆ ker(H A 0 ) ⊗ ker(H A 1 ) ⊗ · · · ⊗ ker(H A k ), 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 interesting 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 H U near a realistic one H , where one may show that H U is frustration-free (which may be efficiently verified using the algorithm of [21], as outlined in section 3), and then approximate the ground state of the full Hamiltonian. This approach appears to be particularly suitable for slightly frustrated Hamiltonians reminding us of Shastry-Sutherland-type [36] models, with-in a cubic lattice and a frustration-free Hamiltonian-an additional bond along the main diagonal, which renders the model frustrated.

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 models of spin-1/2 particles on a general lattice. We have seen that the entire ground state manifold can be parameterized 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 behavior can be rigorously proven. It is hoped that the idea of considering 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.
Proposition (lemma 2). Let U: H 2 → H 2 ⊗ H 2 be an isometry that is not a product operator. Let η 0 be an operator on two spins, and η = (U † ⊗ 1 2 )(1 2 ⊗ η)(U ⊗ 1 2 ). If η is not of full rank, then η is a product operator if and only if η is a product operator.