The exact contraction of a generic two-dimensional (2D) tensor network state (TNS) is known to be exponentially hard, making simulation of 2D systems difficult. The recently introduced class of isometric TNS (isoTNS) represents a subset of TNS that allows for efficient simulation of such systems on finite square lattices. The isoTNS ansatz requires the identification of an ``orthogonality column'' of tensors, within which one-dimensional matrix product state (MPS) methods can be used for calculation of observables and optimization of tensors. Here we extend isoTNS to infinitely long strip geometries and introduce an infinite version of the Moses Move algorithm for moving the orthogonality column around the network. Using this algorithm, we iteratively transform an infinite MPS representation of a 2D quantum state into a strip isoTNS and investigate the entanglement properties of the resulting state. In addition, we demonstrate that the local observables can be evaluated efficiently. Finally, we introduce an infinite time-evolving block decimation algorithm ( ${\mathrm{iTEBD}}^{2}$ ) and use it to approximate the ground state of the 2D transverse field Ising model on lattices of infinite strip geometry.The exact contraction of a generic two-dimensional (2D) tensor network state (TNS) is known to be exponentially hard, making simulation of 2D systems difficult. The recently introduced class of isometric TNS (isoTNS) represents a subset of TNS that allows for efficient simulation of such systems on finite square lattices. The isoTNS ansatz requires the identification of an ``orthogonality column'' of tensors, within which one-dimensional matrix product state (MPS) methods can be used for calculation of observables and optimization of tensors. Here we extend isoTNS to infinitely long strip geometries and introduce an infinite version of the Moses Move algorithm for moving the orthogonality column around the network. Using this algorithm, we iteratively transform an infinite MPS representation of a 2D quantum state into a strip isoTNS and investigate the entanglement properties of the resulting state. In addition, we demonstrate that the local observables can be evaluated efficiently. Finally, we introduce an infinite time-evolving block decimation algorithm (${\mathrm{iTEBD}}^{2}$) and use it to approximate the ground state of the 2D transverse field Ising model on lattices of infinite strip geometry.

arXiv:2211.14337Physical review. B./Physical review. BJun 9, 20233 citations

Continuous Tensor Network States

Teresa Dimitra Karanikoulaou

Jan 1, 20200 citations

Normal projected entangled pair states generating the same state

Molnár, A., Garre-Rubio, J., Pérez-García, D. +2 more

Tensor networks (TNs) are generated by a set of small rank tensors and define many-body quantum states in a succinct form. The corresponding map is not one-to-one: different sets of tensors may generate the very same state. A fundamental question in the study of TNs naturally arises: what is then the relation between those sets? The answer to this question in one-dimensional setups has found several applications, like the characterization of local and global symmetries, the classification of phases of matter and unitary evolutions, or the determination of the fixed points of renormalization procedures. Here we answer this question for projected entangled pair states in any dimension and lattice geometry (including, for example, the Kagome lattice, hyperbolic lattices, or tree tensor networks), as long as the tensors generating the states are normal, which constitute an important and generic class.Tensor networks (TNs) are generated by a set of small rank tensors and define many-body quantum states in a succinct form. The corresponding map is not one-to-one: different sets of tensors may generate the very same state. A fundamental question in the study of TNs naturally arises: what is then the relation between those sets? The answer to this question in one-dimensional setups has found several applications, like the characterization of local and global symmetries, the classification of phases of matter and unitary evolutions, or the determination of the fixed points of renormalization procedures. Here we answer this question for projected entangled pair states in any dimension and lattice geometry (including, for example, the Kagome lattice, hyperbolic lattices, or tree tensor networks), as long as the tensors generating the states are normal, which constitute an important and generic class.

Nov 12, 201831 citations

{\mathrm{iTEBD}}^{2}

) and use it to approximate the ground state of the 2D transverse field Ising model on lattices of infinite strip geometry.