Construction of mutually unbiased bases with cyclic symmetry for qubit systems

TL;DR

This paper presents a simple, experimentally feasible method for constructing mutually unbiased bases in multi-qubit systems, using only basic quantum gates, with optimal linear scaling in the number of qubits.

Contribution

It introduces a straightforward construction of mutually unbiased bases for qubit registers that is accessible and scalable, unlike previous algebra-heavy methods.

Findings

Construction uses only Hadamard and controlled-phase gates

Number of gates scales linearly with qubits

Method is suitable for practical quantum experiments

Abstract

For the complete estimation of arbitrary unknown quantum states by measurements, the use of mutually unbiased bases has been well-established in theory and experiment for the past 20 years. However, most constructions of these bases make heavy use of abstract algebra and the mathematical theory of finite rings and fields, and no simple and generally accessible construction is available. This is particularly true in the case of a system composed of several qubits, which is arguably the most important case in quantum information science and quantum computation. In this paper, we close this gap by providing a simple and straightforward method for the construction of mutually unbiased bases in the case of a qubit register. We show that our construction is also accessible to experiments, since only Hadamard and controlled-phase gates are needed, which are available in most practical realizations of a quantum computer. Moreover, our scheme possesses the optimal scaling possible, i.e., the number of gates scales only linearly in the number of qubits.