An approximation algorithm for approximation rank

TL;DR

This paper establishes a close relationship between the gamma_2^{alpha} norm and approximation rank, providing a polynomial-time approximation algorithm for the latter and confirming it as a lower bound for quantum communication complexity with entanglement.

Contribution

It proves that the log of gamma_2^{alpha} and the log of approximation rank are essentially equivalent, enabling new bounds and algorithms in quantum communication complexity.

Findings

Log gamma_2^{alpha} and log approximation rank agree up to small factors.

Provides a polynomial-time approximation algorithm for the log of approximate rank.

Confirms the log of approximation rank as a lower bound for quantum communication complexity with entanglement.

Abstract

One of the strongest techniques available for showing lower bounds on quantum communication complexity is the logarithm of the approximation rank of the communication matrix--the minimum rank of a matrix which is entrywise close to the communication matrix. This technique has two main drawbacks: it is difficult to compute, and it is not known to lower bound quantum communication complexity with entanglement. Linial and Shraibman recently introduced a norm, called gamma_2^{alpha}, to quantum communication complexity, showing that it can be used to lower bound communication with entanglement. Here the parameter alpha is a measure of approximation which is related to the allowable error probability of the protocol. This bound can be written as a semidefinite program and gives bounds at least as large as many techniques in the literature, although it is smaller than the corresponding alpha-approximation rank, rk_alpha. We show that in fact log gamma_2^{alpha}(A)$ and log rk_{alpha}(A)$ agree up to small factors. As corollaries we obtain a constant factor polynomial time approximation algorithm to the logarithm of approximate rank, and that the logarithm of approximation rank is a lower bound for quantum communication complexity with entanglement.