# Enabling Non-Linear Quantum Operations through Variational Quantum Splines

arXiv: https://arxiv.org/abs/2303.04788 · HTML: https://mlsanity.com/paper/2303.04788/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 25 references that are in the map.
- First posted on arXiv 8 Mar 2023.

## Builds on (most cited first, 4 of 4)

- [Quantum Neuron: an elementary building block for machine learning on quantum computers](https://mlsanity.com/paper/1711.11240/) (2017)
- [Quantum activation functions for quantum neural networks](https://mlsanity.com/paper/2201.03700/) (2022)
- [GCS-Q: Quantum Graph Coalition Structure Generation](https://mlsanity.com/paper/2212.11372/) (2022)
- [BILP-Q: Quantum Coalition Structure Generation](https://mlsanity.com/paper/2204.13802/) (2022)

## Built on by (most cited first, 3 of 3)

- [Quantum Variational Activation Functions Empower Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2509.14026/) (2025)
- [Quantum Supervised Learning](https://mlsanity.com/paper/2407.17161/) (2024)
- [Hybrid Quantum Neural Networks for Efficient Protein-Ligand Binding Affinity Prediction](https://mlsanity.com/paper/2509.11046/) (2025)

## Abstract (arXiv)

The postulates of quantum mechanics impose only unitary transformations on quantum states, which is a severe limitation for quantum machine learning algorithms. Quantum Splines (QSplines) have recently been proposed to approximate quantum activation functions to introduce non-linearity in quantum algorithms. However, QSplines make use of the HHL as a subroutine and require a fault-tolerant quantum computer to be correctly implemented. This work proposes the Generalised Hybrid Quantum Splines (GHQSplines), a novel method for approximating non-linear quantum activation functions using hybrid quantum-classical computation. The GHQSplines overcome the highly demanding requirements of the original QSplines in terms of quantum hardware and can be implemented using near-term quantum computers. Furthermore, the proposed method relies on a flexible problem representation for non-linear approximation and it is suitable to be embedded in existing quantum neural network architectures. In addition, we provide a practical implementation of the GHQSplines using Pennylane and show that our model outperforms the original QSplines in terms of quality of fitting.