# Adaptive Non-local Observable on Quantum Neural Networks

arXiv: https://arxiv.org/abs/2504.13414 · HTML: https://mlsanity.com/paper/2504.13414/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 45 references that are in the map.
- First posted on arXiv 18 Apr 2025.

## Builds on (most cited first, 18 of 18)

- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Variational Quantum Circuits for Deep Reinforcement Learning](https://mlsanity.com/paper/1907.00397/) (2019)
- [Quantum machine learning beyond kernel methods](https://mlsanity.com/paper/2110.13162/) (2021)
- [Quantum Convolutional Neural Networks for High Energy Physics Data Analysis](https://mlsanity.com/paper/2012.12177/) (2020)
- [The Dawn of Quantum Natural Language Processing](https://mlsanity.com/paper/2110.06510/) (2021)
- [An end-to-end trainable hybrid classical-quantum classifier](https://mlsanity.com/paper/2102.02416/) (2021)
- [Variational Quantum Reinforcement Learning via Evolutionary Optimization](https://mlsanity.com/paper/2109.00540/) (2021)
- [When BERT Meets Quantum Temporal Convolution Learning for Text Classification in Heterogeneous Computing](https://mlsanity.com/paper/2203.03550/) (2022)
- [Quantum machine learning of large datasets using randomized measurements](https://mlsanity.com/paper/2108.01039/) (2021)
- [QTN-VQC: An End-to-End Learning framework for Quantum Neural Networks](https://mlsanity.com/paper/2110.03861/) (2021)
- [PQLM -- Multilingual Decentralized Portable Quantum Language Model for Privacy Protection](https://mlsanity.com/paper/2210.03221/) (2022)
- [Learning to Program Variational Quantum Circuits with Fast Weights](https://mlsanity.com/paper/2402.17760/) (2024)
- [Applying QNLP to sentiment analysis in finance](https://mlsanity.com/paper/2307.11788/) (2023)
- [Federated Quantum Long Short-term Memory (FedQLSTM)](https://mlsanity.com/paper/2312.14309/) (2023)
- [Learning to Measure Quantum Neural Networks](https://mlsanity.com/paper/2501.05663/) (2025)
- [Quantum Gradient Class Activation Map for Model Interpretability](https://mlsanity.com/paper/2408.05899/) (2024)

## Built on by (most cited first, 7 of 7)

- [Quantum Reinforcement Learning by Adaptive Non-local Observables](https://mlsanity.com/paper/2507.19629/) (2025)
- [Quantum Super-resolution by Adaptive Non-local Observables](https://mlsanity.com/paper/2601.14433/) (2026)
- [Kernel-based optimization of measurement operators for quantum reservoir computers](https://mlsanity.com/paper/2602.14677/) (2026)
- [Quantum Hierarchical Reinforcement Learning via Variational Quantum Circuits](https://mlsanity.com/paper/2605.03434/) (2026)
- [Multivariate Time Series Forecasting with Adaptive Non-Local Observables](https://mlsanity.com/paper/2607.24399/) (2026)
- [Diagonal Adaptive Non-local Observables on Quantum Neural Networks](https://mlsanity.com/paper/2605.15410/) (2026)
- [Learning to Program Adaptive Non-Local Observables for Machine Learning](https://mlsanity.com/paper/2609.18655/) (2026)

## Abstract (arXiv)

Conventional Variational Quantum Circuits (VQCs) for Quantum Machine Learning typically rely on a fixed Hermitian observable, often built from Pauli operators. Inspired by the Heisenberg picture, we propose an adaptive non-local measurement framework that substantially increases the model complexity of the quantum circuits. Our introduction of dynamical Hermitian observables with evolving parameters shows that optimizing VQC rotations corresponds to tracing a trajectory in the observable space. This viewpoint reveals that standard VQCs are merely a special case of the Heisenberg representation. Furthermore, we show that properly incorporating variational rotations with non-local observables enhances qubit interaction and information mixture, admitting flexible circuit designs. Two non-local measurement schemes are introduced, and numerical simulations on classification tasks confirm that our approach outperforms conventional VQCs, yielding a more powerful and resource-efficient approach as a Quantum Neural Network.