Lower Bounds on Matrix Rigidity Via a Quantum Argument

Andris Ambainis

Robert Blau

Ronald de Wolf

The rigidity of a matrix measures how many of its entries need to be changed in order to reduce its rank to some value. Good lower bounds on the rigidity of an explicit matrix would imply good lower bounds for arithmetic circuits as well as for communication complexity. Here we reprove the best known bounds on the rigidity of Hadamard matrices, due to Kashin and Razborov, using tools from quantum computing. Our proofs are somewhat simpler than earlier ones (at least for those familiar with quantum) and give slightly better constants. More importantly, they give a new approach to attack this longstanding open problem.

Josh Alman

For a matrix M and a positive integer r, the rank r rigidity of M is the smallest number of entries of M which one must change to make its rank at most r. There are many known applications of rigidity lower bounds to a variety of areas in complexity theory, but fewer known applications of rigidity upper bounds. In this paper, we use rigidity upper bounds to prove new upper bounds in a few different models of computation. Our results include:

Adam R. Brown

Minbo Gao, Chenghua Liu, Guangxu Yang +1 more

We study one-way quantum communication lower bounds for search problems. Unlike decision problems, search problems can have many valid outputs, which pose a fundamental barrier to standard quantum lower-bound techniques. We overcome this by developing a novel method based on matrix discrepancy, which allows us to bound the output measurements of a quantum protocol jointly. As applications of our method, we establish the first tight quantum lower bounds for two fundamental search problems in some natural parameter regimes: collision finding and triangle finding. For collision finding, we prove a tight $Ω(N^{1/4})$ one-way quantum communication lower bound. Previously, the best-known quantum communication lower bound for collision finding was $Ω(N^{1/12})$ due to Göös and Jain (RANDOM 2022), and no stronger bound was known even under the one-way restriction. For triangle finding in graph streams, we prove a one-pass quantum streaming space lower bound of $Ω\left(\sqrt{Δ_V}\right)$ for graphs with $m$ edges, $Θ(m)$ triangles, and constant $Δ_E$, where $Δ_V$ and $Δ_E$ denote the maximum number of triangles sharing a common vertex and edge, respectively, under the condition that $1\le Δ_V\le m^{2/3}$. This constitutes the first nontrivial quantum space lower bound in this regime, matching the classical upper bound of Jayaram and Kallaugher (RANDOM 2021) up to logarithmic factors. Notably, our method also recovers the classical lower bound of Kallaugher and Price (SODA 2017) through an entirely different argument, avoiding their Boolean-Hidden-Matching reduction that breaks down for quantum protocols.