Mutually unbiased bases as a commuting polynomial optimisation problem

TL;DR

This paper formulates the problem of mutually unbiased bases as a polynomial optimization problem and develops methods to explore its solutions, aiming to address the open question in dimension 6 and improve computational tractability.

Contribution

It introduces a novel polynomial optimization framework for MUBs and proposes two optimization-based methods, including a hierarchy of semidefinite programs, to analyze their existence.

Findings

Demonstrates potential to solve the open problem in dimension 6 with finite memory

Proves infeasibility of certain smaller sets in dimension 3

Reduces problem variables by 66%, enhancing tractability

Abstract

We consider the problem of mutually unbiased bases as a polynomial optimization problem over the reals. We heavily reduce it using known symmetries before exploring it using two methods, combining a number of optimization techniques. The first of these is a search for bases using Lagrange-multipliers that converges rapidly in case of MUB existence, whilst the second combines a hierarchy of semidefinite programs with branch-and-bound techniques to perform a global search. We demonstrate that such an algorithm would eventually solve the open question regarding dimension 6 with finite memory, although it still remains intractable. We explore the idea that to show the inexistence of bases, it suffices to search for orthonormal vector sets of certain smaller sizes, rather than full bases. We use our two methods to conjecture the minimum set sizes required to show infeasibility, proving it for dimensions 3. The fact that such sub-problems seem to also be infeasible heavily reduces the number of variables, by 66\% in the case of the open problem, potentially providing an large speedup for other algorithms and bringing them into the realm of tractability.