Quantum Deep Equilibrium Models

We are grateful for the reviewer’s positive assessment and thoughtful comments on our work. In response to their questions, we offer the following answers.

Classical vs Quantum Data

Firstly, we would like to state that we are not intending this work to make any strong statements about quantum advantages for classical data, and we would agree with the reviewer that utility of quantum models for classical data is still an open question. On one hand, applied hybrid quantum/classical methods are receiving a lot of positive attention, and there are supporting theoretical results showing that provable learning advantages on classical data is possible (under certain complexity-theoretic assumptions) on highly artificial problems related to cryptography or contextuality. On the other hand, some claims of advantage have proven hard to reproduce, and there is well-founded scepticism about whether the structure and bias of quantum models aligns with the needs of real-world datasets. Given the differing viewpoints and conflicting evidence, we believe the question open enough that contributing methods to QML for classical data is still meaningful.

The application of QDEQ to quantum data would be a very interesting avenue of further research. However, generalizing the framework to quantum-state inputs/outputs and contractive (i.e., dissipative) parametrized quantum channels presents several subtle questions. For instance, how can we efficiently measure state-closeness when doing root-finding? What is the best way allow the root finder to move in some (efficiently parametrizable) subset of quantum states? How can we adapt the training algorithm to take advantage of this reduced space rather than objects scaling with the full $2^{2Q}$ size of the input/output space?

Another promising avenue could be considering the outcomes of a preceding quantum experiment as quantum data. This could be an experiment in the sense of a physical experiment, or also in the sense of a preceding quantum algorithm. Either can be seen as an advanced encoding map. For the case of it coming from an algorithm, the final result aka the ML input data if often delivered in amplitude or phase. While this data in amplitude or phase can be transitioned to and from each other by amplitude amplification and quantum phase estimation, such tools may require fault-tolerance and more advanced quantum hardware. In the case of experimental results, in general, we expect this data to be more ``messy'' and it would require a case-by-case basis how to deal with this specifically. Nevertheless, we expect that in the case of data coming from quantum computations and experiments, QDEQ would fare similarly to e.g. variational quantum circuit alternatives on this analysis task, and also similarly inhibit its strengths and weaknesses compared to standard variational circuit approaches.

While the open questions prevented a direct application of QDEQ as presented here to fully quantum data, further research into answering them, and into whether any advantages transfer, does seems promising. Moreover, we are also hopeful that the broader idea of fixed-point search of quantum maps could be a promising algorithmic building block also more generally, i.e., beyond classification tasks. However, due to the subtleties, it is hard to speculate if the advantages of better shallow-circuit performance transfer, and the considerations in general had to be postponed to future work. Some of the above considerations are in some form mentioned very briefly on lines 173-178 of our submission, to be clarified in future revisions.

We sincerely appreciate the reviewer’s constructive feedback and encouraging review. We hope these additional insights clarify any outstanding questions, and we are available to provide more details if necessary.

We appreciate the reviewer’s feedback and constructive critique of our work. In response to their concerns, we have provided the following clarifications and additional information:

Scalability

We agree with the reviewer that scaling is an important question, and one that is very difficult to meaningfully assess using the small-scale experiments available in classical simulations of quantum models. This is also why the MNIST dataset was chosen, since the small size of simulatable circuits makes full 10-class MNIST meaningfully challenging task for quantum models built without significant classical components for assistance, i.e., models we develop in this work and our baselines. In general, MNIST-4 and MNIST-10 are used frequently in quantum ML settings because quantum simulations are expensive. With respect to the observed scaling, it should be noted that the compute-cost of classical simulation reported here is only an imperfect proxy for the effort required to train the models with access to a quantum computer. In the long-term quantum context, the amount of measurements required to evaluate gradients would be a central measure of complexity, since measurement and reset is often slower than logic gates. However, even more important currently is circuit depth, which constitutes a hard limit to the size of implementable models. In this respect, we believe QDEQ to be worthwhile for the model-sizes considered here. As for the measurement requirements, the relative overhead of QDEQ scales with the size of the input space relative to the size of the parameter space. Currently, baselines have these spaces at roughly the same size. However, the impact of scaling models and/or problem sizes in the future is hard to predict -- one could hope for deep-learning-like scaling where parameter-size$\\gg$data-size, but could also end up being limited by poor optimization landscapes (e.g., the barren plateaus discussed in the response to BzP4) to more similar sizes. In the first scenario, the measurement overhead of QDEQ likely becomes insignificant compared to the general effort of estimating gradients wrt. trainable parameters, while the second situation could see more non-negligible overheads.

In other words, for current devices, circuits are still shallow enough that readout constitutes a significant portion of the runtime. However, since noise sets a hard upper limit on the size of implementable models, QDEQ can still bring advantages despite possessing some measurment overhead (related to measuring gradients in data-space). On the other hand, fault tolerant quantum computers may bring much deeper architectures, where the circuit depth dominates over measurement-time, and where the number of trainable parameters is much larger than the size of the data-space. In such a scenario, the measurement overhead of QDEQ (related to the gradients of the now relatively small data-space) has the potential to be insignificantly small compared to the runtime-cost of running twice as deep a circuit. While predicting how quantum models will scale in the future is very difficult, and certain obstacles in trainability would have to be overcome, we believe that the applicability of QDEQ to both near-term and such future scenarios makes it a worthy topic of study, even with the overhead in measurement complexity.

CIFAR dataset

At the reviewer's request, we have conducted experiments on CIFAR-10 [1], which consists of real-world images. We show the results in a table below. For CIFAR-10, we first converted the images to greyscale and resized from 32x32 to 28x28 to adapt them to our setup.

We find that in general, near-term quantum ML models that are amenable to numerical experiments do not perform well on CIFAR-10, which has also been observed in recent prior works [1] (models do better if a lot of classical NN components are added, but we refrained from doing this). Despite this, we find that our QDEQ framework (Implicit + Warmup) still does better on the test set than the direct solver baseline. This can motivate the utility of our method on more realistic datasets.

[1] Baek H. et al., Logarithmic dimension reduction for quantum neural networks. In Proceedings of the 32nd ACM International Conference on Information and Knowledge Management, 2023.

Novelty

We believe our work is indeed novel as we develop a new method enabling the first demonstration of applying deep equilibrium models in a quantum machine learning setting.

We are grateful for the reviewer’s evaluation of our study. We hope our responses address the raised concerns and provide further clarity. We are available to offer additional details if needed.

We appreciate the reviewer’s insightful comments and are thankful for the positive feedback. In response to the reviewer’s questions, we have provided the following answers:

Influence of noise

We agree that this is a very interesting and relevant question that we opted to defer to further research -- the emphasis being, noise is a very wide and subtle topic and we would like to spend more time on this question in a thorough manner in a follow-up work. Furthermore, accurately modelling it requires code modifications beyond what was feasible in a week of rebuttal.

The main noise-sensitive component of QDEQ is that gradients need to be retrieved through sampling, which makes the training potentially sensitive to both shot noise and other noise sources. In this sense, we do not expect QDEQ to be significantly different to standard variational algorithms, with similar error mitigation strategies likely being applicable here.

Note that some noise sources, e.g., amplitude damping noise, might aid the existence of fixed points. However, in the presence of strong noise, these fixed points tend to be trivial (maximally mixed state) and thus likely less useful. As such, a closer investigation is required on whether the impact of intermediate noise levels would be beneficial.

Fixed points in practice

We acknowledge that the question of fixed points is important and not fully closed by our work. We consider the presence of fixed points in the architectures we investigated as promising empirical evidence. Contractions are guaranteed to have fixed points, thus Appendix A of our manuscript aims to provide guidelines regarding which properties are important for ensuring this property. The two most important factors are the encoding and the readout. Encoding should not amplify differences between two data points, i.e., neighbouring data points should encode to similar quantum states. The amplitude encoding we use satisfies this if the classical vector to be encoded is not very small. This is supported by input injection in our approach, so this should be satisfied for nontrivial datasets. Other widely-used encodings, such as angle encoding, have the same property. With respect to readout, similar non-amplification is needed. This implies that the norm of the measured operator should not be too large and that there is an upper bound on how well it distinguishes encoded states, in the sense of how states are aligned with the measurement operator's eigenbasis , cf. Eq. (9) or (12). The norm is easy to calculate and relatively small for single-qubit Paulis and basis-state projections, as common in quantum classifiers. The distinguishing-power is more difficult to bound analytically. Based on this and the empirics, we expect fixed points to be common for quantum classifier architectures. Further work on a formal proof, e.g. using arXiv:2006.08591, is left as future work.

If the map completely lacks fixed points, we expect training of the model to fail; the essential implicit differentiation formalism breaks down. We briefly investigated this by trying to fit our model to a Fourier Series -- there is no fixed point in this setting and thus QDEQ performed well below the baseline model.

Benchmarks

We agree that baselines are vital to establish a context for the presented results. We expanded our search for admissible baselines and have added a new baseline by arXiv:2309.09424 to the MNIST-10 and FashionMNIST-10 benchmarks. This reference provides evidence of performance of $\\sim$80-85\% on MNIST-10 and $\\sim$75\% on FashionMNIST-10 using a slightly modified amplitude encoding and a similar number of qubits. Additionally, the baseline we already included in the submission for FashionMNIST-10 contains more experiments in its appendix. These experiments show performance of $77-82\\%$, and use non-approximate amplitude encoding, similar to our strategy. It should be noted that in both cases, the high performance is likely attributable to significantly deeper circuits with more trainable parameters (about 10x and 3x the parameters of our architecture, respectively). We see no limitations to scaling QDEQ to similar sizes, and leave this as future work.

Beyond these references, existing work is dominated by binary classification, convolutional architectures, or significant initial compression of the images using deep classical NNs. We are concerned that adding baselines with very different architectures or tasks would not support the main focus of the paper -- comparing QDEQ to other training of similar architectures -- and that extended discussions of architectural differences risks reducing the clarity of the paper. As we see it, comparisons between the direct solver and the implicit solver should be the focal point of the discussion, as this most cleanly reflects the effects of QDEQ. We hope that the reviewer will find these additions an reasonable trade-off between context and focus.

Barren plateaus

The training (and evaluation) of the models still relies on gradients evaluated by measurement. As such, we would expect the barren plateau issues of the model function to be identical to those of other parametrized quantum circuits, presenting similar challenges and being amenable to several known mitigation strategies. One such mitigation approach, local observables, is present in our experiments. We expect the need to evaluate gradients with respect to $z$ to behave similarly, especially in the case of angle encoding. As we see it, the main bearing of QDEQ on barren plateau issues would be as a potential way of pushing the performance of shallow trainable models further, to some extend mitigating the need to increase circuit depth to hard-to-train levels.

Once again, we thank the reviewer for their thoughtful feedback and positive evaluation of our work. We hope these additional comments address any remaining questions and are happy to provide further details if needed.

Dear reviewer, the AC gave permission to share this link to the repository now:

https://anonymous.4open.science/r/qdeq-neurips-F57D/

Best, Authors