Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Polynomial PEPS approximation of gapped square-grid ground states

OpenAI

Published Oct 6, 2026• 0 citations
Publisher

Abstract

We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an L\times L square lattice admits a projected entangled-pair state approximation with bond dimension polynomial in L and global vector error at most L−1 after normalization. Only the gap of the full Hamiltonian is assumed. The result is an existence theorem, with constants uniform over Hamiltonians of fixed local dimension, interaction strength, and gap.

DOI
10.83295/d75a-5zeb

Related papers

Approximating Gibbs states of local Hamiltonians efficiently with projected entangled pair states

András Molnár, Norbert Schuch, Frank Verstraete +1 more

We analyze the error of approximating Gibbs states of local quantum spin Hamiltonians on lattices with projected entangled pair states (PEPS) as a function of the bond dimension ($D$), temperature (${\ensuremath{\beta}}^{\ensuremath{-}1}$), and system size ($N$). First, we introduce a compression method in which the bond dimension scales as $D={e}^{O({log}_{2}^{2}(N/\ensuremath{\epsilon}))}$ if $\ensuremath{\beta}<O({log}_{2}N)$. Second, building on the work of Hastings [M. B. Hastings, Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, $D={(N/\ensuremath{\epsilon})}^{O(\ensuremath{\beta})}$. This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local quantum Hamiltonians. From those bounds it also follows that ground states can be approximated with $D={N}^{O({log}_{2}N)}$ whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.

arXiv:1406.2973•Physical Review B•Jan 29, 2015•138 citations

Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians

Alexander M. Dalzell, Fernando G. S. L. Brandão

A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length N of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length- k segments of the chain, the reduced density matrices of our approximations are ϵ -close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like (k/ϵ)1+o(1) , and at the expense of worse but still poly (k,1/ϵ) scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension exp⁡(O(k/ϵ)) , which is exponentially worse, but still independent of N . Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find O(1) -accurate local approximations to the ground state in T(N) time implies the ability to estimate the ground state energy to O(1) precision in O(T(N)log⁡(N)) time.

arXiv:1903.10241•Quantum•Sep 23, 2019•31 citations

PEPS as unique ground states of local Hamiltonians

David Pérez-Garcı́a, Frank Verstraete, J. I. Cirac +1 more

In this paper we consider projected entangled pair states (PEPS) on arbitrary lattices. We construct local parent Hamiltonians for each PEPS and isolate a condition under which the state is the unique ground state of the Hamiltonian. This condition, verified by generic PEPS and examples like the AKLT model, is an injective relation between the boundary and the bulk of any local region. While it implies the existence of an energy gap in the 1D case we will show that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be gapless with a critical ground state. To show this we invoke a mapping between classical and quantum models and prove that in these cases the injectivity relation between boundary and bulk solely depends on the lattice geometry.

Quantum Information and Computation•Jul 1, 2008•76 citations

Peps as unique ground states of local hamiltonians

David Pérez-Garcı́a, Frank Verstraete, Michael M. Wolf +1 more

In this paper we consider projected entangled pair states (PEPS) on arbitrary lattices. We construct local parent Hamiltonians for each PEPS and isolate a condition under which the state is the unique ground state of the Hamiltonian. This condition, verified by generic PEPS and examples like the AKLT model, is an injective relation between the boundary and the bulk of any local region. While it implies the existence of an energy gap in the 1D case we will show that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be gapless with a critical ground state. To show this we invoke a mapping between classical and quantum models and prove that in these cases the injectivity relation between boundary and bulk solely depends on the lattice geometry.

Quantum Information and Computation•Jul 1, 2008•83 citations

PEPS as unique ground states of local Hamiltonians

David Pérez-Garcı́a, Frank Verstraete, J. I. Cirac +1 more

In this paper we consider projected entangled pair states (PEPS) on arbitrary lattices. We construct local parent Hamiltonians for each PEPS and isolate a condition under which the state is the unique ground state of the Hamiltonian. This condition, verified by generic PEPS and examples like the AKLT model, is an injective relation between the boundary and the bulk of any local region. While it implies the existence of an energy gap in the 1D case we will show that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be gapless with a critical ground state. To show this we invoke a mapping between classical and quantum models and prove that in these cases the injectivity relation between boundary and bulk solely depends on the lattice geometry.

arXiv:0707.2260•Jul 16, 2007•1 citation

Simulation of fermionic lattice models in two dimensions with projected entangled-pair states: Next-nearest neighbor Hamiltonians

Philippe Corboz, Jacob Jordan, Guifré Vidal

In a recent contribution [P. Corboz, R. Or\'us, B. Bauer, and G. Vidal, Phys. Rev. B 81, 165104 (2010)] fermionic projected entangled-pair states (PEPSs) were used to approximate the ground state of free and interacting spinless fermion models, as well as the $t\text{\ensuremath{-}}J$ model. This paper revisits these three models in the presence of an additional next-nearest hopping amplitude in the Hamiltonian. First we explain how to account for next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS algorithms based on simulating time evolution. Then we present benchmark calculations for the three models of fermions and compare our results against analytical, mean-field, and variational Monte Carlo results, respectively. Consistent with previous computations restricted to nearest neighbor Hamiltonians, we systematically obtain more accurate (or better converged) results for gapped phases than for gapless ones.

arXiv:1008.3937•Physical Review B•Dec 21, 2010•94 citations
We prove that the unique ground state of a uniformly gapped nearest-neighbor Hamiltonian on an L\times L square lattice admits a projected entangled-pair state approximation with bond dimension polynomial in L and global vector error at most L−1 after normalization. Only the gap of the full Hamiltonian is assumed. The result is an existence theorem, with constants uniform over Hamiltonians of fixed local dimension, interaction strength, and gap.
We analyze the error of approximating Gibbs states of local quantum spin Hamiltonians on lattices with projected entangled pair states (PEPS) as a function of the bond dimension (DDD), temperature (${\ensuremath{\beta}}^{\ensuremath{-}1}$), and system size (NNN). First, we introduce a compression method in which the bond dimension scales as $D={e}^{O({log}_{2}^{2}(N/\ensuremath{\epsilon}))}$ if $\ensuremath{\beta}<O({log}_{2}N)$. Second, building on the work of Hastings [M. B. Hastings, Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, $D={(N/\ensuremath{\epsilon})}^{O(\ensuremath{\beta})}$. This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local quantum Hamiltonians. From those bounds it also follows that ground states can be approximated with D=NO(log2N)D={N}^{O({log}_{2}N)}D=NO(log2​N) whenever the density of states only grows polynomially in the system size. All results hold for any spatial dimension of the lattice.
A key feature of ground states of gapped local 1D Hamiltonians is their relatively low entanglement --- they are well approximated by matrix product states (MPS) with bond dimension scaling polynomially in the length N of the chain, while general states require a bond dimension scaling exponentially. We show that the bond dimension of these MPS approximations can be improved to a constant, independent of the chain length, if we relax our notion of approximation to be more local: for all length- k segments of the chain, the reduced density matrices of our approximations are ϵ -close to those of the exact state. If the state is a ground state of a gapped local Hamiltonian, the bond dimension of the approximation scales like (k/ϵ)1+o(1) , and at the expense of worse but still poly (k,1/ϵ) scaling of the bond dimension, we give an alternate construction with the additional features that it can be generated by a constant-depth quantum circuit with nearest-neighbor gates, and that it applies generally for any state with exponentially decaying correlations. For a completely general state, we give an approximation with bond dimension exp⁡(O(k/ϵ)) , which is exponentially worse, but still independent of N . Then, we consider the prospect of designing an algorithm to find a local approximation for ground states of gapped local 1D Hamiltonians. When the Hamiltonian is translationally invariant, we show that the ability to find O(1) -accurate local approximations to the ground state in T(N) time implies the ability to estimate the ground state energy to O(1) precision in O(T(N)log⁡(N)) time.
In this paper we consider projected entangled pair states (PEPS) on arbitrary lattices. We construct local parent Hamiltonians for each PEPS and isolate a condition under which the state is the unique ground state of the Hamiltonian. This condition, verified by generic PEPS and examples like the AKLT model, is an injective relation between the boundary and the bulk of any local region. While it implies the existence of an energy gap in the 1D case we will show that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be gapless with a critical ground state. To show this we invoke a mapping between classical and quantum models and prove that in these cases the injectivity relation between boundary and bulk solely depends on the lattice geometry.
In this paper we consider projected entangled pair states (PEPS) on arbitrary lattices. We construct local parent Hamiltonians for each PEPS and isolate a condition under which the state is the unique ground state of the Hamiltonian. This condition, verified by generic PEPS and examples like the AKLT model, is an injective relation between the boundary and the bulk of any local region. While it implies the existence of an energy gap in the 1D case we will show that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be gapless with a critical ground state. To show this we invoke a mapping between classical and quantum models and prove that in these cases the injectivity relation between boundary and bulk solely depends on the lattice geometry.
In this paper we consider projected entangled pair states (PEPS) on arbitrary lattices. We construct local parent Hamiltonians for each PEPS and isolate a condition under which the state is the unique ground state of the Hamiltonian. This condition, verified by generic PEPS and examples like the AKLT model, is an injective relation between the boundary and the bulk of any local region. While it implies the existence of an energy gap in the 1D case we will show that in certain cases (e.g., on a 2D hexagonal lattice) the parent Hamiltonian can be gapless with a critical ground state. To show this we invoke a mapping between classical and quantum models and prove that in these cases the injectivity relation between boundary and bulk solely depends on the lattice geometry.
In a recent contribution [P. Corboz, R. Or\'us, B. Bauer, and G. Vidal, Phys. Rev. B 81, 165104 (2010)] fermionic projected entangled-pair states (PEPSs) were used to approximate the ground state of free and interacting spinless fermion models, as well as the $t\text{\ensuremath{-}}J$ model. This paper revisits these three models in the presence of an additional next-nearest hopping amplitude in the Hamiltonian. First we explain how to account for next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS algorithms based on simulating time evolution. Then we present benchmark calculations for the three models of fermions and compare our results against analytical, mean-field, and variational Monte Carlo results, respectively. Consistent with previous computations restricted to nearest neighbor Hamiltonians, we systematically obtain more accurate (or better converged) results for gapped phases than for gapless ones.