  Recurrent neural networks are the foundation of many sequence-to-sequence
models in machine learning, such as machine translation and speech synthesis.
In contrast, applied quantum computing is in its infancy. Nevertheless there
already exist quantum machine learning models such as variational quantum
eigensolvers which have been used successfully e.g. in the context of energy
minimization tasks. In this work we construct a quantum recurrent neural
network (QRNN) with demonstrable performance on non-trivial tasks such as
sequence learning and integer digit classification. The QRNN cell is built from
parametrized quantum neurons, which, in conjunction with amplitude
amplification, create a nonlinear activation of polynomials of its inputs and
cell state, and allow the extraction of a probability distribution over
predicted classes at each step. To study the model's performance, we provide an
implementation in pytorch, which allows the relatively efficient optimization
of parametrized quantum circuits with thousands of parameters. We establish a
QRNN training setup by benchmarking optimization hyperparameters, and analyse
suitable network topologies for simple memorisation and sequence prediction
tasks from Elman's seminal paper (1990) on temporal structure learning. We then
proceed to evaluate the QRNN on MNIST classification, both by feeding the QRNN
each image pixel-by-pixel; and by utilising modern data augmentation as
preprocessing step. Finally, we analyse to what extent the unitary nature of
the network counteracts the vanishing gradient problem that plagues many
existing quantum classifiers and classical RNNs.

Machine Learning;Quantum Physics;Machine Learning