Variational noise mitigation in quantum circuits: the case of Quantum Fourier Transform

Abstract

Abstract We study a variational quantum algorithm (VQA) approach for improving the implementation of the Quantum Fourier Transform (QFT) under realistic noise conditions. We employ a training strategy based on mutually unbiased bases (MUBs), which probes the action of the noisy variational circuit over a global set of input states and promotes generalization beyond the training ensemble. We numerically study two- and three-qubit QFT implementations under noise models combining incoherent errors with coherent crosstalk, using calibration parameters derived from a quantum device. For the 2-qubit QFT, the optimized variational circuit consistently outperforms the corresponding noisy QFT in scenarios containing coherent noise and exhibits highly reproducible optimization across independent parameter initializations. The variational advantage persists in the direct 3-qubit optimization, although with a stronger dependence on initialization. We further show that reduced MUB training sets can retain most of the performance of the complete set while substantially reducing the number of training states, whereas training only on computational-basis states leads to markedly poorer generalization and a stronger dependence on parameter initialization. Finally, exploiting the recursive structure of the QFT, we demonstrate that an optimized 2-qubit variational block can be embedded into a 3-qubit QFT and improve its noisy implementation without retraining the complete circuit. These results highlight the potential of combining global state-space training with device-specific noise-aware variational optimization to improve small noisy quantum circuits and to explore the transfer of variationally learned noise resilience to larger constructions.