Quantum Deep Equilibrium Models

Summary

The technique of deep equilibrium models, which were introduced to efficiently handle classical sequential data, is here applied to networks consisting of quantum circuits. The performance is compared to both direct solvers and baseline algorithms (VAE and PCA) for datasets derived from MNIST-4, MNIST, and FashionMNIST. The proposed scheme consistently beats the direct solvers but only sometimes the baselines.

Strengths

Bringing deep equilibrium models to quantum circuits seems promising and this is a significant first step demonstrating a speedup for a concrete example. The article clearly describes the setup and what experiments were done. Also limitations are clearly stated as benchmarking is also performed to reference algorithms and not only more basic architectures.

Weaknesses

All datasets are classical data. It is not clear why one should use quantum circuits for such data except as a first demonstration of the scheme.

Questions

Why were all datasets classical and what advantage might a quantum circuit have in their analysis? Could quantum datasets be tried instead, and would you expect an advantage there?

Rating

6

Confidence

3

Soundness

4

Presentation

4

Contribution

3

Limitations

The limitations are clearly stated in the article.

Summary

In this paper the author present a quantum version of the Deep Equilibrium models. These networks approximate iterative approach through many layers by approximating it with a single set of parameters that would have converged if there were an infinity of layers. The paper is written well and the method is verified on a set of small benchmarks.

Strengths

- The paper presents a quantum generalization of the Deep Equilibrium models for the classification task and implemented through an ensemble methods using the VQA for training.

Weaknesses

- The main disadvantage of the proposed model is the relatively weak result analysis. When looking at the provided results one can conclude that the proposed approach provides in average over one percent accuracy improvement while requiring the double processing time and double memory size. This is when comparing the Implicit solver vs. Single Layer direct learning. Naturally when more layers would be present this could change but with larger dataset the runtime of the implicit solver could change??? Simply based on these observations I wonder about the usefulness. This is because the MNIST dataset is by default the simplest one and is used as a principle demonstrator. So if such a large overhead is present how will it scale for both the accuracy as well as for the requirements (memory and time)?

Questions

It would be interesting to see if and how this approach scales with real datasets such as for instance Cifar.

Rating

4

Confidence

4

Soundness

2

Presentation

2

Contribution

2

Limitations

Scalability and novelty.

Summary

This paper introduces a new paradigm for training quantum machine learning models using Deep Equilibrium Models (DEQ). The authors propose Quantum Deep Equilibrium Models (QDEQ) to enhance the performance of parametrized quantum circuits (PQC) while addressing issues related to circuit depth and parameter scalability. QDEQ uses a root solver to find the fixed points of the network, allowing for the training of shallower circuits with fewer parameters. The paper demonstrates the effectiveness of QDEQ on classification tasks using MNIST-4, MNIST-10, and FashionMNIST-10 datasets, showing competitive performance compared to existing baseline models while requiring fewer resources.

Strengths

The paper introduces the Deep Equilibrium Models to quantum machine learning, a novel approach that has not been explored previously. Using DEQ, the paper shows that it is possible to achieve high performance with significantly shallower quantum circuits, which is crucial for near-term quantum devices where noise is a significant factor. The paper provides a detailed and solid theoretical framework. The presentation of the paper is very easy to understand, with a good mix of equations/theorems and explanations/illustrations. The paper demonstrates experiments on MNIST-4, MNIST-10, and FashionMNIST-10 datasets, showing the practical applicability and competitive performance of QDEQ models, and adding empirical evidence to the claim. The proposed approach is resource efficient and reduces the need for extensive quantum and classical resources, making it more feasible for implementation on current quantum hardware.

Weaknesses

The effectiveness of QDEQ relies on the assumption that the quantum model functions admit fixed points. It is unclear how well this assumption can hold in reality The empirical validation is limited to a few datasets. The reviewer understands the complexity of performing experiments for quantum machine learning, so this point can be ignored. There is only one or two baselines in each benchmark. Additional baselines could help to further validate the result. Although the authors did a lot of work to make the work reproducible, the code is not publicly available yet. The authors promised to post the code after the review, but in the meantime, it would be helpful to have an anonymous repository to host the code.

Questions

How does quantum noise, such as shot noise and other errors, influence the DEQ-based training process on noisy quantum hardware? In practical implementations, how often do the assumptions about the existence of fixed points hold, and what are the implications if they do not? Can the authors comment on how this work is related to the know barrel plateau issue in quantum machine learning?

Rating

6

Confidence

2

Soundness

3

Presentation

3

Contribution

3

Limitations

The limitation and broader impact (which may not exist) are addressed.