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Quantum Deep Equilibrium Models
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Applies deep equilibrium model (DEQ) implicit differentiation to variational quantum circuits (QDEQ), finding shallow PQC classifiers match or beat deeper explicitly-unrolled circuits on MNIST, FashionMNIST and CIFAR-10.
Problem
Variational quantum classifiers need deep parametrized circuits for expressivity, but deep circuits accumulate errors and the parameter count drives measurement overhead for gradients; the paper asks whether implicit (equilibrium) training lets much shallower circuits perform well.
Approach
Treat a weight-tied quantum model layer (encoding, PQC, Pauli-Z measurement, upsampling, input injection) as the layer of a DEQ, find its fixed point with a Broyden root solver, and train with implicit differentiation, optionally after a 2-layer direct warm-up.
Key results
- On MNIST-4 with amplitude encoding, the IMPLICIT+WARMUP solver reaches 93.4% test accuracy versus 85.3% for the QML circuit baseline and 91.8% for the 10-layer DIRECT model.
- On MNIST-10, IMPLICIT reaches 73.68% versus 72.78% for the 1-layer DIRECT model and 71.18% for the 10-layer DIRECT model, with DIRECT-10 using about 45x more memory (1233.3 MB vs 27.3 MB).
- On CIFAR-10, IMPLICIT+WARMUP attains 25.45% test accuracy, the best among the reported variants, slightly above the DIRECT solvers (23.71-24.90%).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| implicit (DEQ) differentiation instead of explicit unrolled layers (ablation) | 4-qubit PQC classifier · MNIST-4 angle encoding | test accuracy: IMPLICIT 85.8 vs DIRECT 10 layers 84.8 | Helped |
| implicit (DEQ) differentiation instead of explicit unrolled layers (ablation) | 4-qubit PQC classifier · MNIST-4 amplitude encoding | test accuracy: IMPLICIT+WARMUP 93.4 vs DIRECT 1 layer 93.5 | Hurt |
| implicit (DEQ) differentiation instead of explicit unrolled layers (ablation) | 10-qubit PQC classifier · MNIST-10 | test accuracy: IMPLICIT 73.68 vs best DIRECT 72.78 (1 layer) | Helped |
| implicit (DEQ) differentiation instead of explicit unrolled layers (ablation) | 10-qubit PQC classifier · FashionMNIST-10 | test accuracy: IMPLICIT 72.11 vs best DIRECT 71.83 (10 layers) | Helped |
| implicit (DEQ) differentiation with warm-up instead of direct explicit training (ablation) | 10-qubit PQC classifier · CIFAR-10 | test accuracy: IMPLICIT+WARMUP 25.45 vs best DIRECT 24.90 (2 layers) | Helped |
| warm-up pretraining with shallow explicit model before implicit training (ablation) | implicit DEQ training · MNIST-4 angle encoding | test accuracy: 86.7 with warm-up vs 85.8 without | Helped |
| warm-up pretraining with shallow explicit model before implicit training (ablation) | implicit DEQ training · MNIST-4 amplitude encoding | test accuracy: 93.4 with warm-up vs 92.9 without | Helped |
| warm-up pretraining with shallow explicit model before implicit training (ablation) | implicit DEQ training · MNIST-10 and FashionMNIST-10 | test accuracy: MNIST-10 73.33 vs 73.68; FashionMNIST-10 71.17 vs 72.11 | Hurt |
| amplitude encoding instead of angle encoding (ablation) | 4-qubit PQC classifier · MNIST-4 | test accuracy: IMPLICIT+WARMUP 93.4 (amplitude) vs 86.7 (angle) | Helped |
| increasing explicit depth from 1 to 10 layers (DIRECT solver) (ablation) | 10-qubit PQC classifier · MNIST-10 and FashionMNIST-10 | test accuracy: MNIST-10 72.78 (1 layer) vs 71.18 (10 layers); FashionMNIST-10 71.05 (1) vs 71.83 (10) | Hurt |
| increasing explicit depth from 1 to 10 layers (DIRECT solver) (ablation) | 4-qubit PQC classifier · MNIST-4 amplitude encoding | test accuracy: 93.5 (1 layer) vs 91.8 (10 layers) | Hurt |
| increasing explicit depth from 1 to 10 layers (DIRECT solver) (ablation) | 4-qubit PQC classifier · MNIST-4 angle encoding | test accuracy: 83.9 (1 layer) vs 84.8 (10 layers) | Helped |
Surprising findings
- For amplitude encoding on MNIST-4, shallow DIRECT models (1 layer) performed best, and deeper DIRECT models suffered, though implicit training did not share this issue on other datasets.
- The best FashionMNIST-10 model (72.11%) is still below a larger cited baseline reporting 74.5%, and the MNIST-10 VAE baseline reaches 89.80%, outperforming QDEQ.
Limitations
- Experiments are simulations in torchquantum; shot noise and hardware noise are not studied.
- Theoretical fixed-point/contractiveness arguments rely on assumptions and are partly conjectural, notably for amplitude encoding with renormalization.
- Measurement overhead of Broyden Jacobian estimation on quantum hardware is not quantified; memory is used as a proxy.
Experimental setup
Datasets: MNIST FashionMNIST CIFAR-10
Benchmarks: MNIST-4 MNIST-10 FashionMNIST-10 CIFAR-10
Metrics: test accuracy GPU memory runtime residual
Baselines: QML Circuit (torchquantum amplitude encoding example) QML Circuit (VAE) compression baseline QML Circuit (PCA) compression baseline DIRECT solver with 1, 2, 5, 10 layers
Builds on: deep equilibrium models torchquantum parameter-shift rule Broyden's method
Closest papers by meaning
- Rethink the Role of Deep Learning towards Large-scale Quantum Systems (ICML 2025 · similarity 0.88)
- TensorRL-QAS: Reinforcement learning with tensor networks for improved quantum architecture search (NeurIPS 2025 · similarity 0.86)
- PALQO: Physics-informed model for Accelerating Large-scale Quantum Optimization (NeurIPS 2025 · similarity 0.86)
- Q-MAML: Quantum Model-Agnostic Meta-Learning for Variational Quantum Algorithms (AAAI 2025 · similarity 0.86)
- MG-Net: Learn to Customize QAOA with Circuit Depth Awareness (NeurIPS 2024 · similarity 0.85)
- Training-Free Quantum Architecture Search (AAAI 2024 · similarity 0.85)
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