# Tensor surgery and tensor rank

@article{Christandl2018TensorSA, title={Tensor surgery and tensor rank}, author={Matthias Christandl and Jeroen Zuiddam}, journal={computational complexity}, year={2018}, volume={28}, pages={27-56} }

We introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices and edges. We show that tensor surgery is capable of preserving the low rank structure of an initial tensor decomposition and thus allows to prove nontrivial upper bounds on tensor rank, border rank and…

## 10 Citations

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

- Computer Science, Mathematicscomputational complexity
- 2018

An upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on k vertices is presented, showing that the exponent per edge is at most 0.77, outperforming the best known upper bound for matrix multiplication.

Algebraic complexity, asymptotic spectra and entanglement polytopes

- Mathematics, Computer Science
- 2018

The first explicit construction of an infinite family of elements in the asymptotic spectrum of complex k-tensors, based on information theory and entanglement polytopes is given, and it is proved that any polynomial can be efficiently approximated by a width-2 abp.

Border rank non-additivity for higher order tensors

- Mathematics, Computer ScienceSIAM J. Matrix Anal. Appl.
- 2021

This work provides analogs of Schonhage's construction for tensors of order four and higher, and discusses implications of these results for the asymptotic rank of higher order generalizations of the matrix multiplication tensor.

UvA-DARE ( Digital Academic Repository ) Border rank nonadditivity for higher order tensors

- Computer Science, Mathematics
- 2021

This work provides analogues of Sch\" onhage's construction for tensor border rank for tensors of order four and higher, and discusses implications for the asymptotic rank of higher order generalizations of the matrix multiplication tensor.

Probabilistic Tensors and Opportunistic Boolean Matrix Multiplication

- Computer Science, MathematicsSODA
- 2019

These probabilistic extensions satisfy various natural and algorithmically serendipitous properties, such as submultiplicativity under taking of Kronecker products, and enable improvements over their deterministic counterparts for speciﬁc tensors of interest.

Tensor rank and entanglement of pure quantum states

- Computer Science
- 2019

As entanglement of a given quantum state depends on the way the total system is divided into subsystems, the notion of `partitioning rank' of a tensor is introduced, which depends on a way how the entries forming the tensor are treated.

Nondeterministic Quantum Communication Complexity: the Cyclic Equality Game and Iterated Matrix Multiplication

- Computer Science, MathematicsITCS
- 2017

It is employed Strassen's laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem, and how Young flattenings yield nontrivially complexity lower bounds.

Tensor network representations from the geometry of entangled states

- PhysicsSciPost Physics
- 2020

Tensor networks provide descriptions of strongly correlated quantum
systems based on an underlying entanglement structure given by a graph
of entangled states along the edges that identify the…

Quantum entanglement in the triangle network

- Physics, Computer Science
- 2020

This work addresses the questions of which states can be prepared in the so-called triangle network, consisting of three nodes connected pairwise by three sources, and derives necessary criteria for a state to be preparable in such a network.

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