Algebraic complexity, asymptotic spectra and entanglement polytopes
Jeroen Zuiddam
Abstract:
Matrix rank is well-known to be multiplicative under the Kronecker product, additive under the direct sum, normalised on identity matrices and non-increasing under multiplying from the left and from the right by any matrices. In fact, matrix rank is the only real matrix parameter with these four properties. In 1986 Strassen proposed to study the extension to tensors: find all maps from k-tensors to the reals that are multiplicative under the tensor Kronecker product, additive under the direct sum, normalised on "identity tensors", and non-increasing under acting with linear maps on the k tensor factors. Strassen called the collection of these maps the "asymptotic spectrum of k-tensors". He proved that understanding the asymptotic spectrum implies understanding the asymptotic relations among tensors, including the asymptotic subrank and the asymptotic rank. In particular, knowing the asymptotic spectrum means knowing the arithmetic complexity of matrix multiplication, a central problem in algebraic complexity theory.
One of the main results in this dissertation is the first explicit construction of an infinite family of elements in the asymptotic spectrum of complex k-tensors, called the quantum functionals. Our construction is based on information theory and moment polytopes i.e. entanglement polytopes. Moreover, among other things, we study the relation of the recently introduced slice rank to the quantum functionals and find that "asymptotic" slice rank equals the minimum over the quantum functionals. Besides studying the above tensor parameters, we extend the Coppersmith-Winograd method (for obtaining asymptotic combinatorial subrank lower bounds) to higher-order tensors, i.e. order at least 4. We apply this generalisation to obtain new upper bounds on the asymptotic tensor rank of complete graph tensors via the laser method. (Joint work with Christandl and Vrana; QIP 2018, STOC 2018.)
In graph theory, as a new instantiation of the abstract theory of asymptotic spectra we introduce the asymptotic spectrum of graphs. Analogous to the situation for tensors, understanding the asymptotic spectrum of graphs means understanding the Shannon capacity, a graph parameter capturing the zero-error communication complexity of communication channels. In different words: we prove a new duality theorem for Shannon capacity. Some known elements in the asymptotic spectrum of graphs are the LovĂˇsz theta number and the fractional Haemers bounds.
Finally, we study an algebraic model of computation called algebraic branching programs. An algebraic branching program (abp) is the trace of a product of matrices with affine linear forms as matrix entries. The maximum size of the matrices is called the width of the abp. In 1992 Ben-Or and Cleve proved that width-3 abps can compute any polynomial efficiently in the formula size. On the other hand, in 2011 Allender and Wang proved that some polynomials cannot be computed by any width-2 abp. We prove that any polynomial can be efficiently approximated by a width-2 abp, where approximation is defined in the sense of degeneration. (Joint work with Ikenmeyer and Bringmann; CCC 2017, JACM 2018.)