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Quantum Recurrent Neural Networks (QRNNs) are robust candidates to model andpredict future values in multivariate time series. However, the effectiveimplementation of some QRNN models is limited by the need of mid-circuitmeasurements. Those increase the requirements for quantum hardware, which inthe current NISQ era does not allow reliable computations. Emulation arises asthe main near-term alternative to explore the potential of QRNNs, but existingquantum emulators are not dedicated to circuits with multiple intermediatemeasurements. In this context, we design a specific emulation method thatrelies on density matrix formalism. The mathematical development is explicitlyprovided as a compact formulation by using tensor notation. It allows us toshow how the present and past information from a time series is transmittedthrough the circuit, and how to reduce the computational cost in every timestep of the emulated network. In addition, we derive the analytical gradientand the Hessian of the network outputs with respect to its trainableparameters, with an eye on gradient-based training and noisy outputs that wouldappear when using real quantum processors. We finally test the presentedmethods using a novel hardware-efficient ansatz and three diverse datasets thatinclude univariate and multivariate time series. Our results show how QRNNs canmake accurate predictions of future values by capturing non-trivial patterns ofinput series with different complexities.

Density Matrix Emulation of Quantum Recurrent Neural Networks for Multivariate Time Series Prediction