On Combinatorial Constructions of Mutually Unbiased Bases (MUBs) with Approximations

Abstract

Construction of Mutually Unbiased Bases (MUBs) is a very challenging combinatorial problem in the domain of quantum information theory with several long-standing open questions. The basic definition considers two orthonormal bases in the d-dimensional complex Hilbert space with a fixed specific inner product value among any two vectors from two different bases, which is one by square root of d. Similarly, some k orthonormal bases are called Mutually Unbiased Bases (MUBs) if they are pairwise Mutually Unbiased. Reaching the upper bound on this k is believed to be an extremely challenging problem for more than half a century. Because of the difficulties in constructing a significantly large number of MUBs, the problem is relaxed, and the concept of Approximate Mutually Unbiased Bases (AMUBs) has been introduced. Here, the inner product of two vectors drawn from two different bases is upper bounded by some value instead of being exactly the same, namely one by the square root of d.

In this thesis, we present different construction methods of Approximate MUBs using combinatorial structures such as Hadamard matrices and Resolvable Block Designs. We also propose the concept of Almost Perfect MUBs (APMUB), where we restrict the absolute value of the inner product into two values, one of which is zero. Among several constructions, we also exploit Mutually Orthogonal Latin Squares (MOLS) in this regard. Our results have implications for several existing combinatorial objects, such as Bi-angular vectors.

With the understanding of APMUBs, we revisit a larger class of AMUBs and improve the results further in terms of larger classes for composites that are not prime powers and for both real and complex. The technique is more generalized in terms of exploring novel instances of RBDs that provide improved results.

Finally, a heuristic framework is presented to search for approximate MUBs with significantly good parameters, and experimental outcomes of the computer programs are studied. Our proposed idea considers two parts. First, we apply basis reduction techniques (that are well-studied in Machine Learning literature) in obtaining the initial solutions. Then, we exploit the steepest ascent kind of search to improve the results further. The efficacy of our technique is shown through the construction of AMUBs in dimensions d = 6, 10, 46. From a more generic view, this approach considers approximately solving a challenging (where efficient deterministic algorithms are not known) mathematical problem in a discrete domain through state-of-the-art heuristic ideas.

To summarize, in this thesis, we exploit several involved combinatorial techniques in a disciplined manner and also propose a heuristic to construct approximate MUBs.