Measuring incompatible observables with quantum neural networks

The expressivity of quantum neural networks (QNNs) is a key issue in quantum machine learning (QML), which determines the capability to learn and represent complex functions. The recently proposed Randomness-Enhanced QNNs [Wu et al., Phys. Rev. Lett. 132, 010602 (2024)] use a mixed-unitary channel to accurately express arbitrary single observables. In this work, we generalize the mixed-unitary channel to a unital channel and prove that the unital channel can accurately express multiple arbitrary observables. We show that by training and implementing a unital quantum channel as a multiple-output QNN, we can accurately measure the expectation values of multiple incompatible observables simultaneously by Pauli-$Z$ measurements on distinct output qubits. We prove this accurate expressivity for multiple observables by computing the complete positivity of the unital quantum channel. An iterative algorithm is proposed to compute the theoretical maximal scaling factor of the QNNs' outcomes. Notably, our theoretical analysis shows that it requires fewer copies of state when measuring some incompatible observables by the multiple-output QNNs, which demonstrates a resource efficiency advantage over separately applying projective measurements.