Generalized Numerical Construction of MUBs: A Group Theoretical Investigation

TL;DR

This paper introduces a numerical approach to construct Mutually Unbiased Bases (MUBs) in arbitrary dimensions, revealing their algebraic structure and automorphism groups without relying on specific group frameworks.

Contribution

A generalized numerical method for constructing MUBs independent of group structures, with classification based on invariants and automorphism groups.

Findings

All solutions in dimensions 3, 4, and 5 are mutually isomorphic.

Constructed MUBs have automorphism groups matching the Clifford group.

No MUBs found in dimension 6 within the explored parameter space.

Abstract

Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on a priori group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of $d+1$ MUBs in dimension $d$ to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a valid projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions $3$, $4$, and $5$ demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. Though the scope of the search was limited, in dimension $d = 6$ our numerical search yielded no MUBs within explored parameter space.