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

arXiv: https://arxiv.org/abs/2310.20671 · HTML: https://mlsanity.com/paper/2310.20671/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 61 references that are in the map.
- First posted on arXiv 31 Oct 2023.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Modeling Long- and Short-Term Temporal Patterns with Deep Neural Networks](https://mlsanity.com/paper/1703.07015/) (2017)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [A Critical Review of Recurrent Neural Networks for Sequence Learning](https://mlsanity.com/paper/1506.00019/) (2015)
- [A review on outlier/anomaly detection in time series data](https://mlsanity.com/paper/2002.04236/) (2020)
- [A Deep Neural Network for Unsupervised Anomaly Detection and Diagnosis in Multivariate Time Series Data](https://mlsanity.com/paper/1811.08055/) (2018)
- [Quantum Long Short-Term Memory](https://mlsanity.com/paper/2009.01783/) (2020)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)
- [Recurrent Quantum Neural Networks](https://mlsanity.com/paper/2006.14619/) (2020)
- [Unbiasing Truncated Backpropagation Through Time](https://mlsanity.com/paper/1705.08209/) (2017)
- [Low-depth gradient measurements can improve convergence in variational hybrid quantum-classical algorithms](https://mlsanity.com/paper/1901.05374/) (2019)
- [Quantum Recurrent Neural Networks for Sequential Learning](https://mlsanity.com/paper/2302.03244/) (2023)
- [Quantum deep recurrent reinforcement learning](https://mlsanity.com/paper/2210.14876/) (2022)
- [Rapid training of quantum recurrent neural networks](https://mlsanity.com/paper/2207.00378/) (2022)
- [Path sampling of recurrent neural networks by incorporating known physics](https://mlsanity.com/paper/2203.00597/) (2022)

## Built on by (most cited first, 4 of 4)

- [Recursive QLSTM with Dynamic Variational Quantum Circuit Adaptation](https://mlsanity.com/paper/2606.24932/) (2026)
- [Self-Modulating Quantum Fast-Weight Programmers for Efficient Adaptive Sequential Learning](https://mlsanity.com/paper/2606.24933/) (2026)
- [Hybrid Quantum Temporal Convolutional Networks](https://mlsanity.com/paper/2602.23578/) (2026)
- [Quantum Dynamic Time Warping for Multivariate Time Series Classification](https://mlsanity.com/paper/2606.27815/) (2026)

## Abstract (arXiv)

Quantum Recurrent Neural Networks (QRNNs) are robust candidates for modelling and predicting future values in multivariate time series. However, the effective implementation of some QRNN models is limited by the need for 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. Using a compact tensor notation, we provide the mathematical formulation of the operator-sum representation involved. This 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, which are needed when the outputs have stochastic noise due to hardware errors and a finite number of circuit shots (sampling). We finally test the presented methods using a hardware-efficient ansatz and four diverse datasets that include univariate and multivariate time series, with and without sampling noise. In addition, we compare the model with other existing quantum and classical approaches. Our results show how QRNNs can be trained with numerical and analytical gradients to make accurate predictions of future values by capturing non-trivial patterns of input series with different complexities.