Recurrent neural networks are the foundation of many sequence-to-sequence models in machine learning, such as machine translation and speech synthesis. With applied quantum computing in its infancy, there already exist quantum machine learning models such as variational quantum eigensolvers which have been used e.g. in the context of energy minimization tasks. Yet, to date, no viable recurrent quantum network has been proposed.

In this work we construct the first 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, creates a nonlinear activation of polynomials of its inputs and hidden state, and allows 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 tens of thousands of parameters, and which demonstrates that model does not appear to suffer from the vanishing gradient problem that plagues many exting quantum classifiers and classical RNNs. 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).  We then proceed to evaluate the QRNN on MNIST classification, by feeding the QRNN each image pixel-by-pixel; with a network utilizing only 12 qubits we reach a test set accuracy over 98\% when discriminating between the digits `0' and `1'.