On the geometry of tensor network states
By Joseph M. Landsburg, Yang Qi, Ke Ye. 2011
59 citations recorded by the source corpus.
We answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops. Grasedyck's question has a surprising connection to the area of Geometric Complexity Theory, in that the result is equivalent to the statement that the boundary of the Mulmuley-Sohoni type variety associated to matrix multiplication is strictly larger than the projections of matrix multiplication (and re-expressions of matrix multiplication and its projections after changes of bases). Tensor Network States are also related to graphical models in algebraic statistics.
Subfields: Computational Mathematics, Computational Theory and Mathematics, Geometry and Topology
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Citations in the corpus
- A literature survey of low‐rank tensor approximation techniques (2013, neutral): “In general, this statement does not hold for tensor networks containing cycles [178, 179].”
- Changing the topology of Tensor Networks (2012, background): “This leads in the representation to be not stable (see [6]) which results in additional conditions that have to be preserved while performing algorithms.”
- Optimization problems in contracted tensor networks (2011, unknown): “…we finally have u = r′∑ je=1 u1(je)⊗ u2(je) = r′∑ je=1 UG1(û1(je))⊗ UG2(û2(je)) = UG(û), where û := (û1, û2) ∈ Pd,L and r′ ≤ r. ¥ If the tensor network graph G is not a tree (it contains cycles), then the induced tensor network UG is in general not closed, see [12, Landsberg et al., 2011].”