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.

Quantum reservoir computing uses the dynamics of quantum systems to process temporal data, making it particularly well-suited for learning with noisy intermediate-scale quantum devices. Early experimental proposals, such as the restarting and rewinding protocols, relied on repeating previous steps of the quantum map to avoid backaction. However, this approach compromises real-time processing and increases computational overhead. Recent developments have introduced alternative protocols that address these limitations. These include online, mid-circuit measurement, and feedback techniques, which enable real-time computation while preserving the input history. Among these, the feedback protocol stands out for its ability to process temporal information with comparatively fewer components. Despite this potential advantage, the theoretical foundations of feedback-based quantum reservoir computing remain underdeveloped, particularly with regard to the universality and the approximation capabilities of this approach. This paper addresses this issue by presenting a recurrent quantum neural network architecture that extends a class of existing feedforward models to a dynamic, feedback-driven reservoir setting. We provide theoretical guarantees for variational recurrent quantum neural networks, including approximation bounds and universality results. Notably, our analysis demonstrates that the model is universal with linear readouts, making it both powerful and experimentally accessible. These results pave the way for practical and theoretically grounded quantum reservoir computing with real-time processing capabilities.

Additional Feedback: - I am not able to reproduce Equation (2). There are no steps given on how to arrive at this result. I tried to reproduce it but failed. I wrote down the quantum states for the steps of the circuit in Figure 1 left. However, based on my calculations, the state before the measurement is exactly the initial state x 0 0 .

The paper proposes a form of recurrent network built out of quantum neurons. The model is novel and interesting and the paper has a good amount of discussion and experiments and should make good and interesting paper at neurips. One important improvement that should be made for writing the paper is to make it more accessible to people at neurips. The reviewers for this paper ranged from those with very little background in the field to an expert and therefore the reviews provide a lot of comments on what is not clear in the paper and what should be improved. The paper should be understandable to someone that knows neural networks and basic quantum mechanics but nothing about quantum computation.

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.