# Kernel-based optimization of measurement operators for quantum reservoir computers

arXiv: https://arxiv.org/abs/2602.14677 · HTML: https://mlsanity.com/paper/2602.14677/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 78 references that are in the map.
- First posted on arXiv 16 Feb 2026.

## Builds on (most cited first, 21 of 21)

- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Robust data encodings for quantum classifiers](https://mlsanity.com/paper/2003.01695/) (2020)
- [The Inductive Bias of Quantum Kernels](https://mlsanity.com/paper/2106.03747/) (2021)
- [Bandwidth Enables Generalization in Quantum Kernel Models](https://mlsanity.com/paper/2206.06686/) (2022)
- [Structural risk minimization for quantum linear classifiers](https://mlsanity.com/paper/2105.05566/) (2021)
- [On fundamental aspects of quantum extreme learning machines](https://mlsanity.com/paper/2312.15124/) (2023)
- [Reservoir Computing meets Recurrent Kernels and Structured Transforms](https://mlsanity.com/paper/2006.07310/) (2020)
- [Learning to Measure Quantum Neural Networks](https://mlsanity.com/paper/2501.05663/) (2025)
- [A Unified Framework for Trace-induced Quantum Kernels](https://mlsanity.com/paper/2311.13552/) (2023)
- [Optimal training of finitely-sampled quantum reservoir computers for forecasting of chaotic dynamics](https://mlsanity.com/paper/2409.01394/) (2024)
- [Infinite-dimensional next-generation reservoir computing](https://mlsanity.com/paper/2412.09800/) (2024)
- [Double descent in quantum kernel methods](https://mlsanity.com/paper/2501.10077/) (2025)
- [Adaptive Non-local Observable on Quantum Neural Networks](https://mlsanity.com/paper/2504.13414/) (2025)
- [Learning to Program Quantum Measurements for Machine Learning](https://mlsanity.com/paper/2505.13525/) (2025)
- [Quantum Bayesian Computation](https://mlsanity.com/paper/2208.08068/) (2022)
- [You Only Measure Once: On Designing Single-Shot Quantum Machine Learning Models](https://mlsanity.com/paper/2509.20090/) (2025)

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

- [Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach](https://mlsanity.com/paper/2602.18377/) (2026)
- [Quantum Reservoir Computing: Recent Advances and Future Directions](https://mlsanity.com/paper/2607.18552/) (2026)

## Abstract (arXiv)

Finding optimal measurement operators is crucial for the performance of quantum reservoir computers (QRCs), since they employ a fixed quantum feature map. We formulate the training of both stateless (quantum extreme learning machines, QELMs) and stateful (memory dependent) QRCs in the framework of kernel ridge regression. We thus extend the kernel viewpoint of supervised quantum models to recurrent QRCs by deriving an exact Hilbert--Schmidt kernel representation of the optimal readout observable on history space. This approach renders an optimal measurement operator that minimizes prediction error for a given reservoir and training dataset. For large qubit numbers, this method is more efficient than the conventional training of QRCs. We discuss efficiency and practical implementation strategies, including Pauli basis decomposition and operator diagonalization, to adapt the optimal observable to hardware constraints. To demonstrate the effectiveness of this approach, we present numerical experiments on image classification and time series prediction tasks, including chaotic and strongly non-Markovian systems. The developed method can also be applied to other quantum machine learning models.