An Approximation Algorithm for Approximation Rank

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 close to the communication matrix in ∞ norm. 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 γα 2 , to quantum communication complexity, showing that it can be used to lower bound communication with entanglement. Here α 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 αapproximation rank, rkα. We show that in fact log γα 2 (A) and log rkα(A) agree up to small factors. As corollaries we ...

• Type: Conference
• Year: 2009
• Where: COCO
• Authors: Troy Lee, Adi Shraibman
• Keywords: Approximation Rank, COCO 2009, Lower Bound, Quantum Communication Complexity, Theoretical Computer Science