# Sustained Performance and Energy Accounting for Nonlinear Forecasting Across Classical and Simulated Quantum Models

arXiv: https://arxiv.org/abs/2510.25183 · HTML: https://mlsanity.com/paper/2510.25183/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 31 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Green AI](https://mlsanity.com/paper/1907.10597/) (2019)
- [Carbon Emissions and Large Neural Network Training](https://mlsanity.com/paper/2104.10350/) (2021)
- [Quantifying the Carbon Emissions of Machine Learning](https://mlsanity.com/paper/1910.09700/) (2019)
- [Towards the Systematic Reporting of the Energy and Carbon Footprints of Machine Learning](https://mlsanity.com/paper/2002.05651/) (2020)
- [Carbontracker: Tracking and Predicting the Carbon Footprint of Training Deep Learning Models](https://mlsanity.com/paper/2007.03051/) (2020)
- [Measuring the Carbon Intensity of AI in Cloud Instances](https://mlsanity.com/paper/2206.05229/) (2022)
- [Robust quantum reservoir computers for forecasting chaotic dynamics: generalized synchronization and stability](https://mlsanity.com/paper/2506.22335/) (2025)
- [Sustainable Quantum Computing: Opportunities and Challenges of Benchmarking Carbon in the Quantum Computing Lifecycle](https://mlsanity.com/paper/2408.05679/) (2024)

## Built on by (most cited first, 2 of 2)

- [Quantum Reservoir Computing: Recent Advances and Future Directions](https://mlsanity.com/paper/2607.18552/) (2026)
- [Variational Quantum Conditional Boltzmann Machines for Time-Series Forecasting: Architectures, Symmetric Hyperparameter Evaluation, and a Nonlinear Benchmark](https://mlsanity.com/paper/2607.24065/) (2026)

## Abstract (arXiv)

Energy-efficient AI should be evaluated across the full application pipeline, not only by lowest error or shortest training time. We study this through nonlinear time-series forecasting using simulated quantum reservoir computing (QRC) as an emerging-computing case study. Our evaluation spans 33 forecasting configurations and 825 completed runs on NARMA-10, NARMA-20, Mackey--Glass, Lorenz-63, and Santa Fe laser data. The core benchmark includes 775 fully instrumented runs across statistical, linear, reservoir, neural, continuous-variable Gaussian QRC, and gate-based statevector QRC models, with 50 additional variational QNN runs extending the trainable-quantum comparison. We measure NRMSE, RMSE, MAE, wall time, inference latency, peak CPU/GPU memory, parameter count, operational energy, and carbon. Since a fixed QRC encoder can generate reusable features for multiple readouts, we report both cold-start and amortized costs. We also use a Sustainable Forecasting Score (SFS), a diagnostic geometric mean of normalized predictive skill and log-scaled carbon efficiency, while retaining raw measurements and Pareto analyses. The best mean NRMSE is achieved by continuous-variable QRC with a Transformer readout (0.332), followed closely by classical TCN (0.341) and QRC+TCN (0.340). However, ESN and ridge-lag deliver the strongest sustained efficiency, with average amortized carbon of 0.014 and 0.012 gCO2 per run and SFS values of 0.814 and 0.805. Larger QRC feature maps increase cost without improving average accuracy, while gate-based statevector simulation is not competitive. These results support three practices for emerging AI systems: expose stage-level energy, use reuse-aware accounting boundaries, and co-design the feature generator or accelerator with the classical readout.