A Heuristic Framework to Search for Approximate Mutually Unbiased Bases

Abstract

Abstract Mutually Unbiased Bases (MUBs) have varied applications in quantum information. However, obtaining the optimal number of MUBs is a challenging problem for different dimensions. The problem has received serious attention for several decades and still number of questions are unsolved in this domain. As optimal number of MUBs may not always be available for different composite dimensions, Approximate MUBs (AMUBs) received serious attention in recent time. In this paper, we present a heuristic to obtain AMUBs with significantly good parameters. Given a non-prime dimension d, we note the closest prime \(d' > d\) and form \(d'+1\) MUBs through the existing methods. Then our proposed idea is (i) to apply basis reduction techniques (that are well studied in Machine Learning literature) in obtaining the initial solutions, and finally (ii) to exploit the steepest ascent kind of search to achieve further improved results. The efficacy of our technique is shown through construction of AMUBs in dimensions \(d = 6, 10, 46\) from \(d' = 7, 11\) and 47 respectively. Our technique provides a novel framework in construction of AMUBs that can be refined in a case-specific manner. From a more generic view, this approach considers approximately solving a challenging (where efficient deterministic algorithms are not known) mathematical problem in discrete domain through state-of-the-art heuristic ideas.KeywordsMutually Unbiased Bases (MUBs)Approximate Mutually Unbiased Bases (AMUBs)Quantum Key DistributionDimension reductionSingular Value Decomposition (SVD)Gradient ascent