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For the complete estimation of arbitrary unknown quantum states bymeasurements, the use of mutually unbiased bases has been well-established intheory and experiment for the past 20 years. However, most constructions ofthese bases make heavy use of abstract algebra and the mathematical theory offinite rings and fields, and no simple and generally accessible construction isavailable. This is particularly true in the case of a system composed ofseveral qubits, which is arguably the most important case in quantuminformation science and quantum computation. In this paper, we close this gapby providing a simple and straightforward method for the construction ofmutually unbiased bases in the case of a qubit register. We show that ourconstruction is also accessible to experiments, since only Hadamard andcontrolled-phase gates are needed, which are available in most practicalrealizations of a quantum computer. Moreover, our scheme possesses the optimalscaling possible, i.e., the number of gates scales only linearly in the numberof qubits.

Construction of mutually unbiased bases with cyclic symmetry for qubit systems