An Efficient Quantum Circuit Construction Method for Mutually Unbiased Bases in $n$-Qubit Systems

Mutually unbiased bases (MUBs) play a crucial role in numerous applications within quantum information science, such as quantum state tomography, error correction, entanglement detection, and quantum cryptography. Utilizing $2^n + 1$ MUB circuits provides a minimal and optimal measurement strategy for reconstructing all $n$-qubit unknown states. It significantly reduces the number of measurements compared to the traditional $4^n$ Pauli observables, also enhancing the robustness of quantum key distribution (QKD) protocols. Previous circuit designs that rely on a single generator can result in exponential gate costs for some MUB circuits. In this work, we present an efficient algorithm to generate each of the $2^n + 1$ quantum MUB circuits on $n$-qubit systems within $O(n^3)$ time. The algorithm features a three-stage structure, and we have calculated the average number of different gates for random sampling. Additionally, we have identified two linear properties: the entanglement part can be directly defined into $2n - 3$ fixed sub-parts, and the knowledge of $n$ special MUB circuits is sufficient to construct all $2^n + 1$ MUB circuits. This new efficient and simple circuit construction paves the way for the implementation of a complete set of MUBs in diverse quantum information processing tasks on high-dimensional quantum systems.