Density Matrix Emulation of Quantum Recurrent Neural Networks for
  Multivariate Time Series Prediction
José Daniel Viqueira Daniel Faílde Mariamo M. Juane Andrés Gómez David Mera
Quantum Recurrent Neural Networks (QRNNs) are robust candidates to model and
predict future values in multivariate time series. However, the effective
implementation of some QRNN models is limited by the need of mid-circuit
measurements. Those increase the requirements for quantum hardware, which in
the current NISQ era does not allow reliable computations. Emulation arises as
the main near-term alternative to explore the potential of QRNNs, but existing
quantum emulators are not dedicated to circuits with multiple intermediate
measurements. In this context, we design a specific emulation method that
relies on density matrix formalism. The mathematical development is explicitly
provided as a compact formulation by using tensor notation. It allows us to
show how the present and past information from a time series is transmitted
through the circuit, and how to reduce the computational cost in every time
step of the emulated network. In addition, we derive the analytical gradient
and the Hessian of the network outputs with respect to its trainable
parameters, with an eye on gradient-based training and noisy outputs that would
appear when using real quantum processors. We finally test the presented
methods using a novel hardware-efficient ansatz and three diverse datasets that
include univariate and multivariate time series. Our results show how QRNNs can
make accurate predictions of future values by capturing non-trivial patterns of
input series with different complexities.