# An exponentially-growing family of universal quantum circuits

arXiv: https://arxiv.org/abs/2212.00736 · HTML: https://mlsanity.com/paper/2212.00736/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 47 references that are in the map.
- First posted on arXiv 1 Dec 2022.

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

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [The effect of data encoding on the expressive power of variational quantum machine learning models](https://mlsanity.com/paper/2008.08605/) (2020)
- [Supervised quantum machine learning models are kernel methods](https://mlsanity.com/paper/2101.11020/) (2021)
- [Noise-Induced Barren Plateaus in Variational Quantum Algorithms](https://mlsanity.com/paper/2007.14384/) (2020)
- [Layerwise learning for quantum neural networks](https://mlsanity.com/paper/2006.14904/) (2020)
- [Analyzing the barren plateau phenomenon in training quantum neural networks with the ZX-calculus](https://mlsanity.com/paper/2102.01828/) (2021)
- [Hybrid quantum neural network for drug response prediction](https://mlsanity.com/paper/2211.05777/) (2022)
- [Exponentially Many Local Minima in Quantum Neural Networks](https://mlsanity.com/paper/2110.02479/) (2021)
- [Benchmarking simulated and physical quantum processing units using quantum and hybrid algorithms](https://mlsanity.com/paper/2211.15631/) (2022)
- [Hybrid quantum ResNet for car classification and its hyperparameter optimization](https://mlsanity.com/paper/2205.04878/) (2022)
- [Practical application-specific advantage through hybrid quantum computing](https://mlsanity.com/paper/2205.04858/) (2022)
- [Fock State-enhanced Expressivity of Quantum Machine Learning Models](https://mlsanity.com/paper/2107.05224/) (2021)
- [Quantum Machine Learning for Radio Astronomy](https://mlsanity.com/paper/2112.02655/) (2021)

## Built on by (most cited first, 17 of 17)

- [Quantum machine learning for image classification](https://mlsanity.com/paper/2304.09224/) (2023)
- [Hybrid quantum physics-informed neural networks for simulating computational fluid dynamics in complex shapes](https://mlsanity.com/paper/2304.11247/) (2023)
- [Hybrid quantum image classification and federated learning for hepatic steatosis diagnosis](https://mlsanity.com/paper/2311.02402/) (2023)
- [Quantum Conformal Prediction for Reliable Uncertainty Quantification in Quantum Machine Learning](https://mlsanity.com/paper/2304.03398/) (2023)
- [Forecasting steam mass flow in power plants using the parallel hybrid network](https://mlsanity.com/paper/2307.09483/) (2023)
- [Hybrid quantum cycle generative adversarial network for small molecule generation](https://mlsanity.com/paper/2402.00014/) (2023)
- [A supervised hybrid quantum machine learning solution to the emergency escape routing problem](https://mlsanity.com/paper/2307.15682/) (2023)
- [Photovoltaic power forecasting using quantum machine learning](https://mlsanity.com/paper/2312.16379/) (2023)
- [Method for noise-induced regularization in quantum neural networks](https://mlsanity.com/paper/2410.19921/) (2024)
- [Spectral invariance and maximality properties of the frequency spectrum of quantum neural networks](https://mlsanity.com/paper/2402.14515/) (2024)
- [Predictive control of blast furnace temperature in steelmaking with hybrid depth-infused quantum neural networks](https://mlsanity.com/paper/2504.12389/) (2025)
- [Superposed parameterised quantum circuits](https://mlsanity.com/paper/2506.08749/) (2025)
- [Certifiably Robust Encoding Schemes](https://mlsanity.com/paper/2408.01200/) (2024)
- [Mitigating Exponential Mixed Frequency Growth through Frequency Selection](https://mlsanity.com/paper/2508.10533/) (2025)
- [Hybrid quantum recurrent neural network for remaining useful life prediction of turbofan engines](https://mlsanity.com/paper/2504.20823/) (2025)
- [Shot-based quantum encoding: a data-loading paradigm for quantum neural networks](https://mlsanity.com/paper/2604.06135/) (2026)
- [Soft-Quantum Algorithms](https://mlsanity.com/paper/2604.06523/) (2026)

## Abstract (arXiv)

Quantum machine learning has become an area of growing interest but has certain theoretical and hardware-specific limitations. Notably, the problem of vanishing gradients, or barren plateaus, renders the training impossible for circuits with high qubit counts, imposing a limit on the number of qubits that data scientists can use for solving problems. Independently, angle-embedded supervised quantum neural networks were shown to produce truncated Fourier series with a degree directly dependent on two factors: the depth of the encoding and the number of parallel qubits the encoding applied to. The degree of the Fourier series limits the model expressivity. This work introduces two new architectures whose Fourier degrees grow exponentially: the sequential and parallel exponential quantum machine learning architectures. This is done by efficiently using the available Hilbert space when encoding, increasing the expressivity of the quantum encoding. Therefore, the exponential growth allows staying at the low-qubit limit to create highly expressive circuits avoiding barren plateaus. Practically, parallel exponential architecture was shown to outperform the existing linear architectures by reducing their final mean square error value by up to 44.7% in a one-dimensional test problem. Furthermore, the feasibility of this technique was also shown on a trapped ion quantum processing unit.