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.

借助密度矩阵形式主义，设计了一种特定的模拟方法，以探索 Quantum Recurrent Neural Networks（QRNNs）的潜力，并且能够通过捕捉不同复杂性的输入序列的非平凡模式，准确预测未来的值。

量子循环神经网络的密度矩阵仿真在多元时间序列预测中的应用

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

Recent advances in quantum computing (QC) and machine learning (ML) have
drawn significant attention to the development of quantum machine learning
(QML). Reinforcement learning (RL) is one of the ML paradigms which can be used
to solve complex sequential decision making problems. Classical RL has been
shown to be capable to solve various challenging tasks. However, RL algorithms
in the quantum world are still in their infancy. One of the challenges yet to
solve is how to train quantum RL in the partially observable environments. In
this paper, we approach this challenge through building QRL agents with quantum
recurrent neural networks (QRNN). Specifically, we choose the quantum long
short-term memory (QLSTM) to be the core of the QRL agent and train the whole
model with deep $Q$-learning. We demonstrate the results via numerical
simulations that the QLSTM-DRQN can solve standard benchmark such as Cart-Pole
with more stable and higher average scores than classical DRQN with similar
architecture and number of model parameters.

本文介绍了使用量子循环神经网络和深度 Q-learning 算法来解决部分可观察环境中的量子强化学习问题，并且在数值模拟中证明了该方法在标准基准测试如 Cart-Pole 中的结果比经典 DRQN 更加稳定和具有更高的平均分数。