Matrix Discrepancy from Quantum Communication

TL;DR

This paper connects discrepancy minimization with quantum communication complexity, proving a special case of the Matrix Spencer conjecture using quantum information tools and providing a polynomial-time algorithm.

Contribution

It introduces a novel approach linking discrepancy and quantum communication, resolving a key case of the Matrix Spencer conjecture with an efficient algorithm.

Findings

Established a bound on eigenvalues for certain matrix collections

Developed a polynomial-time algorithm using semidefinite programming

Connected discrepancy minimization to quantum communication complexity

Abstract

We develop a novel connection between discrepancy minimization and (quantum) communication complexity. As an application, we resolve a substantial special case of the Matrix Spencer conjecture. In particular, we show that for every collection of symmetric $n \times n$ matrices $A_1,\ldots,A_n$ with $\|A_i\| \leq 1$ and $\|A_i\|_F \leq n^{1/4}$ there exist signs $x \in \{ \pm 1\}^n$ such that the maximum eigenvalue of $\sum_{i \leq n} x_i A_i$ is at most $O(\sqrt n)$. We give a polynomial-time algorithm based on partial coloring and semidefinite programming to find such $x$. Our techniques open a new avenue to use tools from communication complexity and information theory to study discrepancy. The proof of our main result combines a simple compression scheme for transcripts of repeated (quantum) communication protocols with quantum state purification, the Holevo bound from quantum information, and tools from sketching and dimensionality reduction. Our approach also offers a promising avenue to resolve the Matrix Spencer conjecture completely -- we show it is implied by a natural conjecture in quantum communication complexity.