Exact and Approximate Computation of Tensors and Polynomials

This research proposal will address fundamental problems concerning the complexity of computingand manipulating polynomials. For example, consider the following questions:1. What is the complexity of computing the total weight of perfect matchings of a weighted graph?2. Is there an efficient deterministic parallel algorithm that determines whether a graph has a perfect matching?3.

Is approximation much easier than exact computation?4.

How many EPR pairs can we distill from a given quantum state?These seemingly unrelated questions represent some of the most important and challenging open problems in theoretical computer science: the first is the algebraic analog of the famous P vs. NP problem. The second question amounts to asking whether a symbolic matrix associated with the graph has full rank.

Parallel randomized algorithms for computing this rank are known, but not deterministic ones.

This is an instance of the polynomial identity testing (PIT) problem, the most fundamental algebraic derandomization problem.

The third question asks about the relation between a complexity class and its closure, which lies at the heart of the Geometric Complexity Theory (GCT) program. The last question concerns the subrank of a tensor representing the given quantum state.

Problems related to rank of tensors are at the heart of both algebraic complexity and quantum information theory.Recent years have seen tremendous advance in our understanding of algebraic computations with new lower bounds, new PIT algorithms and with increasing connections to other branches of computer science and mathematics discovered.

Results proved by the PI play an important role in all of these advances.This project aims to study these and related problems and to develop new methods for solving them. Making progress on any of these problems will constitute a significant breakthrough.