Optimization at the boundary of the tensor network variety

Abstract

Tensor network states form a variational ansatz class widely used, both analytically and numerically, in the study of quantum many-body systems. It is known that if the underlying graph contains a cycle, e.g. as in projected entangled pair states (PEPS), then the set of tensor network states of given bond dimension is not closed. Its closure is the tensor network variety. Recent work has shown that states on the boundary of this variety can yield more efficient representations for states of physical interest, but it remained unclear how to systematically find and optimize over such representations. We address this issue by defining a new ansatz class of states that includes states at the boundary of the tensor network variety of given bond dimension. We show how to optimize over this class in order to find ground states of local Hamiltonians by only slightly modifying standard algorithms and code for tensor networks. We apply this new method to a different of models and observe favorable energies and runtimes when compared with standard tensor network methods.

arXiv comment20 pages, 6 figures. Close to published version

References in corpus (25)

• The density-matrix renormalization group in the age of matrix product states
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Entanglement renormalization
• A class of quantum many-body states that can be efficiently simulated
• Criticality, the area law, and the computational power of PEPS
• Classical simulation of quantum many-body systems with a tree tensor network
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Accurate determination of tensor network state of quantum lattice models in two dimensions
• Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy)
• The computational complexity of PEPS
• PEPS as ground states: degeneracy and topology
• Tensor renormalization of quantum many-body systems using projected entangled simplex states
• A quantum version of Wielandt's inequality
• Gauge fixing, canonical forms and optimal truncations in tensor networks with closed loops
• Rigorous RG algorithms and area laws for low energy eigenstates in 1D
• Tensor rank is not multiplicative under the tensor product
• Contracting projected entangled pair states is average-case hard
• Exploiting translational invariance in Matrix Product State simulations of spin chains with periodic boundary conditions
• Projected Entangled Pair States: Fundamental analytical and numerical limitations
• On the partially symmetric rank of tensor products of W-states and other symmetric tensors
• Border rank is not multiplicative under the tensor product
• Barriers for Rank Methods in Arithmetic Complexity
• Tensor network representations from the geometry of entangled states
• Quantum version of Wielandt's Inequality revisited
• On the geometry of tensor network states

Cited by in corpus (11)

• On the closedness and geometry of tensor network state sets
• The minimal canonical form of a tensor network
• Dimension of Tensor Network varieties
• The resource theory of tensor networks
• The Linear Span of Uniform Matrix Product States
• Classifying Entanglement by Algebraic Geometry
• Persistent Tensors and Multiqudit Entanglement Transformation
• Partial Degeneration of Tensors
• Border Ranks of Positive and Invariant Tensor Decompositions: Applications to Correlations
• Quantum max-flow in the bridge graph
• The product structure of MPS-under-permutations