  The growing demands of remote detection and increasing amount of training
data make distributed machine learning under communication constraints a
critical issue. This work provides a communication-efficient quantum algorithm
that tackles two traditional machine learning problems, the least-square
fitting and softmax regression problem, in the scenario where the data set is
distributed across two parties. Our quantum algorithm finds the model
parameters with a communication complexity of $O(\frac{\log_2(N)}{\epsilon})$,
where $N$ is the number of data points and $\epsilon$ is the bound on parameter
errors. Compared to classical algorithms and other quantum algorithms that
achieve the same output task, our algorithm provides a communication advantage
in the scaling with the data volume. The building block of our algorithm, the
quantum-accelerated estimation of distributed inner product and Hamming
distance, could be further applied to various tasks in distributed machine
learning to accelerate communication.
