Polynomial PEPS approximation of gapped square-grid ground states

Introduction

A projected entangled-pair state (PEPS) represents a many-body vector by local tensors joined along the edges of its interaction lattice. The central approximation question is whether ground states of gapped local Hamiltonians admit such representations with modest virtual dimensions. An entropy area law controls every bipartite cut, but a global vector approximation requires more: the descriptions at different cuts must be compatible, their errors must remain small through the entire lattice, and the final tensor network must use the prescribed edges.

We prove a uniform polynomial bound for square lattices with arbitrary bounded on-site and nearest-neighbor interactions. The only spectral assumption is a unique ground vector and a positive gap for the full Hamiltonian. We use the entropy area law and the collar estimates of [15], always with their full-system hypotheses. The new argument establishes the stronger representation conclusion.

Statement of the result

For an integer $L \ge2$, let $\Lambda_L = \{1,\ldots,L\}^2$, with the nearest-neighbor square-lattice edge set $E(\Lambda_L)$ and open boundary conditions. Put

A PEPS on this grid is the contraction obtained by placing $\sum_{a=1}^{D_e} |a\rangle\otimes|a\rangle$ on each edge $e$ and applying at each vertex an arbitrary linear map

Its maximum bond dimension is $\max_e D_e$. The maps need not be isometries, and the resulting vector need not be normalized.

Theorem 1.1. Fix an integer $q \ge2$ and constants $J,\Delta> 0$. There are constants $C,c > 0$, depending only on $q,J,\Delta$, with the following property. For every $L \ge2$, consider any Hermitian Hamiltonian

where each summand is supported on its designated site or edge. Suppose that $H$ has a unique normalized ground vector $\Omega$, up to phase, with ground energy $E_0$, and

Then there is a nonzero PEPS $\Phi$ on the same grid with maximum bond dimension at most $CL^c$ such that

Theorem (1) gives an affirmative resolution of the polynomial PEPS approximation assertion for the stated open-square Hamiltonian class. The relation to other formulations of the PEPS conjecture is explained below. No gap for restrictions to subregions, frustration freeness, commutativity, translation invariance, or path from a product state is assumed. The error in (2) concerns the entire vector. The theorem makes no claim that its tensors can be found or contracted efficiently; their entries may depend arbitrarily on $H$ and $\Omega$.

History and significance

Projected entangled-pair states extend matrix product states to higher-dimensional lattices by contracting virtual pairs through local maps [21]. Their local tensor structure makes them natural candidates for describing ground states whose entanglement is concentrated near region boundaries. In one dimension, quantitative Schmidt-tail truncation estimates connect entanglement spectra with global MPS approximation [22]. Hastings established the entropy area law for unique ground states of uniformly gapped spin chains with bounded finite-range interactions and obtained polynomial-dimension MPS approximations [9]. The higher-dimensional question requires a representation compatible across many intersecting cuts.

There are several distinct PEPS approximation questions. The global formulation in [4], Section 2.2, Questions 7–8 asks whether a spectral gap ensures polynomial bond dimension at inverse-polynomial Hilbert-space norm error. That formulation uses translation-invariant finite-range Hamiltonians on a torus. Theorem 1.1 addresses the corresponding assertion for arbitrary bounded on-site and nearest-neighbor interactions on an open square. Local-observable formulations instead control expectations on fixed-size regions; the stronger structural formulation in [19], Conjecture 2 also requires an injective approximant with a gapped parent Hamiltonian. The theorem here concerns the full vector and imposes no such structural requirements on the approximant.

A positive local approximation result is due to Huang [10], Theorem 3. For states satisfying a Rényi entropy area law of fixed order $0 < \alpha< 1$ across every rectangle, he obtains PEPS of bond dimension $e^{O(1/\delta)}$, independent of lattice size, that approximate norm-one local-observable expectations to error $\delta$. The observable support scale is held fixed in this bound. The Rényi hypothesis is stronger than a von Neumann entropy area law, and the conclusion controls local expectations rather than global vector error.

An entropy bound and a representation theorem are different statements. For arbitrary chain-state families, bounded von Neumann block entropy does not ensure efficient global MPS approximation [18]. In higher dimensions, Ge and Eisert constructed families satisfying strong area laws that have no efficient classical tensor-network descriptions [7], Definition 7 and Corollary 9. Their description model includes a bound on the classical complexity of tensor entries; it should be distinguished from a bound on virtual dimension with arbitrary complex entries. These examples concern general states, rather than the uniformly gapped ground-state class in Theorem 1.1. They explain why one must control compatible operations and their accumulated error, in addition to the spectra of separate marginals.

Thermal tensor networks provide another approach. Hastings constructed local spectral filters and tensor-network descriptions of Gibbs operators [8]. Molnár, Schuch, Verstraete, and Cirac refined the thermal approximation and derived quasipolynomial-dimension ground-state PEPS under a gap together with a low-energy density-of-states bound [13], Section III.C. At inverse-polynomial accuracy their bond bound is exponential in a squared logarithm of system size. The low-energy counting condition is an additional hypothesis. A different cooling and postselection construction approximates the unique ground state of a gapped $d$-dimensional local Hamiltonian at the boundary of a $(d+1)$-dimensional PEPS [17], Section V; transferring that representation to the original lattice is a further problem.

Recent tensor-network results also distinguish the dimension and the approximation task. Arad, Firanko, and Jain obtained two-dimensional mutual-information bounds for locally gapped frustration-free systems and polynomial-bond matrix product operator approximations to maximally mixed ground states in one dimension [1], Corollaries 2.7 and 2.9. Such entropy or one-dimensional operator bounds do not provide the simultaneous original-grid PEPS conclusion needed here.

Our analytic inputs are the entropy area law and the stronger collar estimates of the companion [15]. The present proof adapts its nested-filter and two-family ideas, and then develops compatible changes of ownership, compression of a distributed density network, and routing on the prescribed grid. The representation theorem uses the full companion interface recorded in Appendix A, rather than the cutwise entropy inequality alone. Its polynomial bond bound permits arbitrary tensor entries and is an existence statement. The distinction from efficient contraction is substantive: general PEPS contraction has worst-case complexity obstructions [17], although that fact alone does not classify the particular approximants constructed here.

Proof strategy

The main intermediate object is a distribution of the physical registers among auxiliary parties. A party owns a collection of registers and may perform arbitrary linear contractions on them privately. At the start, one party holds the entire vector $\Omega$; at the end, each physical site has its own party. Preparing the initial vector is permitted because we seek a representation, rather than an algorithm for finding it. The constraint is on communication: each party must participate in only boundedly many operations with other parties over its whole lifetime. Each such operation must also have a polynomial-size expansion built from private contractions and normalized bipartite vectors or covectors. The bipartite states themselves may have arbitrarily large Schmidt rank.

Distributing the state while preserving a reference vector. The parties are indexed by dyadic squares. We pass from the ownership assignment on one scale to that on the next by giving each child square its physical registers. Intermediate assignments need not be dyadic, and we use additional independent copies of $\Omega$. Each copy has its own registers and ownership assignment; its reference vector remains $\Omega$ in the original site coordinates, up to the small-patch encodings described below. The operations approximate these reference vectors with an error small enough to sum over the whole construction.

There are two analytic inputs. First, information between regions becomes negligible when their separation tapers linearly toward finitely many points but stays above a sufficiently large multiple of $\log L$ (Proposition 3.4). Small mutual information lets the owner of a separating buffer split its purification approximately into two factors. This implements a change of owner for the interior region, or an exchange of assignments between two copies, using private maps and bipartite resources. To prove the information estimate, an ordered partition into two tile families bounds the conditional-information cost of adding a cell. Gaussian filtering with the full-system gap then resets that cost to an inverse power of $L$ after each annular step.

Second, near a logarithmic-size square we can project with high accuracy onto a sum of mutually orthogonal subspaces. Writing $\mathcal{H}_A=\bigotimes_{v\in A}\mathbb{C}^{q}$ for the registers in $A\subseteq\Lambda_L$, these subspaces have the form

where the $X_j$ are nested square samples within the specified patch and $v_{j\ell}\in\mathcal{H}_{X_j}$ are orthonormal inside vectors (Proposition 4.1). The radius is allowed to depend on $j$. Recording $(j,\ell)$ in a small register and replacing the corresponding inside vector by a fixed product vector gives the encoding used near junctions. The proof combines optimized nested filters, the gap, and the entropy area law; it does not require an accurate truncation at a single prescribed boundary. Section 6 uses these encodings to express suitable changes on small patches in terms of pair states and effects, even when their underlying Hilbert spaces are large.

The remaining geometric problem is to arrange that these operations suffice. Along a long edge, our separation estimate may not justify changing an old owner directly to a new one: a third owner can touch the opposite side of the region being changed. Exchanging part of an auxiliary copy first supplies the surrounding regions needed for successive changes of owner. Small patches handle the finitely many endpoints and junctions, where the separation would otherwise vanish. Section 7 assembles these moves so that a block party participates only in boundedly many nearby repaintings at each of two adjacent scales. Its total participation in operations with other parties is therefore bounded.

Compressing the density and returning to a vector. The distribution initially uses pair resources of unrestricted dimension. Theorem 5.2 replaces them in the density calculation by polynomial sums of product operators. To bound the error from replacing a chosen set of resources, we expand only operations involving the parties that hold their endpoints. Bounded lifetime participation limits the resulting coefficient cost. A Gaussian second-moment bound then controls the trace-norm error independently of all private dimensions and resource Schmidt ranks. This produces an operator tensor network with polynomial virtual dimensions on the party graph. The compression theorem is independent of the planar construction.

The protocol retains norm close to one, so its density and the compressed operator both approximate $|\Omega\rangle\langle\Omega|$ in absolute trace norm. One product-basis column of the compressed operator gives a nonzero ket approximation after normalization. Finally, the links join parties at comparable nearby dyadic scales. Routing them along rows and columns through their dyadic anchors has constant edge congestion, and therefore preserves polynomial bond dimension on the original square grid. Section 8 proves these last steps and the main theorem.

Section 2 fixes the conventions and states the area law. Sections 3 and 4 prove the two analytic inputs above; Appendix A records their exact companion interfaces. All error powers, geometric constants, and polynomial exponents are fixed before taking $L$ large, uniformly in the individual Hamiltonian.

Conventions and entropy inputs

Fix the parameters of Theorem 1.1, and abbreviate $\Lambda= \Lambda_L$. Unsubscripted norms denote the operator norm for maps and the Euclidean norm for vectors; $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ denote trace and Hilbert–Schmidt norms. All Hilbert spaces are finite dimensional. Empty tensor products have dimension one. Constants denoted by $C$, $c$ may change between occurrences. They depend only on $q$, $J$, $\Delta$ and explicitly fixed numerical powers, geometric templates, or tolerances. A polynomial bound means $CL^c$ with this uniform dependence.

Plane distances are sup-norm distances. For a plane set $T$, write $T^{+r}=\{x:\operatorname{dist}_{\infty}(x,T)\le r\}$; sampling always means intersection with the lattice, and physical sampling includes intersection with $\Lambda$. Graph neighborhoods, when used for Hamiltonian locality, are neighborhoods in the induced physical lattice and lie in the corresponding ambient sup-norm enlargement. Distances to the empty set are infinite. Region complements in entropy formulas refer to the full physical system, unless another system is specified. Adjacent region letters denote their disjoint union.

For a normalized state $\omega$ and a region $A$, let $\rho_{\omega,A}$ be its marginal, and set

Logarithms are natural and $0\log0=0$. We omit the state subscript when it is $\Omega$.

The imported area law

The following is the range-one square-domain specialization of the companion theorem. Stating its uniformity here is useful because later auxiliary regions are not themselves assigned gapped Hamiltonians.

Proposition 2.1 (Entropy area law, [15], Theorem 1.1 and Corollary 1.2). Under (1)–(1.2), there is $C_A=C_A(q,J,\Delta)$ such that every $A\subseteq\Lambda$ satisfies

The same constant works for every $L$ and every Hamiltonian in the class.

The companion theorem allows any finite induced square-lattice domain, with one bounded Hermitian summand per designated nonempty support of bounded graph diameter. In (1) the diameter bound is one. Its only spectral assumption is the unique full-system ground vector and (1.2). These are precisely the hypotheses used here. For a physical sample $X$ of a square of radius $r$, this implies $S_\Omega(X)\le C(1+r)$, including squares clipped by the outer boundary.

We also use the companion’s sublinear collar and amplification propositions. Their geometric and exponent requirements are checked explicitly in Proposition 3.1; neither is a PEPS approximation statement. Appendix A records their exact interfaces, including the positive constraints and two-family construction underlying the companion argument. The complete companion is [15].

Finite-dimensional information facts

We use strong subadditivity, the chain rule, monotonicity of mutual information under discarding systems, and, for a pure state on $ABCD$, the duality

These are standard entropy identities and inequalities; see [12] and the preliminary section of [15]. For example $|S(A\mid B)|\le\log\dim\mathcal{H}_A$, so the entropy cost of a bounded set of individual sites is bounded by its cardinality times $\log q$.

For normalized density operators define $F(\rho,\sigma)=\lVert\sqrt{\rho}\sqrt{\sigma}\rVert_1$ and $D(\rho\mathbin\Vert\sigma)=\operatorname{Tr}\rho(\log\rho-\log\sigma)$, with the usual infinite value when the support condition fails.

We recall the relative-entropy estimate through quantum affinity [3] and Uhlmann’s purification-overlap theorem [20]. The finite-dimensional proof below fixes our root-fidelity convention and includes nonminimal purifying spaces.

Lemma 2.2. For normalized finite-dimensional states,

Moreover, after enlarging the purifying space if necessary, an isometry on that space realizes overlap $F(\rho,\sigma)$ between purifications. In particular a mutual-information bound $\delta$ gives purifications of the joint state and product state at vector distance at most $\sqrt{\delta}$, after phase choice, since $2(1-e^{-\delta/2})\le\delta$.

Proof. Write the eigenvalues and orthonormal eigenvectors of $\rho,\sigma$ as $(p_i,|i\rangle)$ and $(q_j,|j'\rangle)$. When $D(\rho\mathbin\Vert\sigma)$ is finite, $w_{ij}=p_i|\langle i|j'\rangle|^2$ is a probability distribution on its positive entries, and

Convexity of the exponential gives

Terms with $p_i=0$ contribute zero. The trace norm is at least the absolute trace, proving the first inequality; the infinite case is immediate. The relative-entropy identity follows by taking the trace against $\log(\rho_A\otimes\rho_C)=\log\rho_A\otimes I+I\otimes\log\rho_C$ on its support.