# Dimension reduction with structure-aware quantum circuits for hybrid machine learning

arXiv: https://arxiv.org/abs/2508.00048 · HTML: https://mlsanity.com/paper/2508.00048/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 31 Jul 2025.

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

- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [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)
- [To prune, or not to prune: exploring the efficacy of pruning for model compression](https://mlsanity.com/paper/1710.01878/) (2017)
- [Power of data in quantum machine learning](https://mlsanity.com/paper/2011.01938/) (2020)
- [Predicting Parameters in Deep Learning](https://mlsanity.com/paper/1306.0543/) (2013)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Barren Plateaus in Variational Quantum Computing](https://mlsanity.com/paper/2405.00781/) (2024)
- [Towards Quantum Machine Learning with Tensor Networks](https://mlsanity.com/paper/1803.11537/) (2018)
- [Methods for Pruning Deep Neural Networks](https://mlsanity.com/paper/2011.00241/) (2020)
- [Universal Approximation Property of Quantum Machine Learning Models in Quantum-Enhanced Feature Spaces](https://mlsanity.com/paper/2009.00298/) (2020)
- [Tensor networks for quantum machine learning](https://mlsanity.com/paper/2303.11735/) (2023)
- [Tensor networks for interpretable and efficient quantum-inspired machine learning](https://mlsanity.com/paper/2311.11258/) (2023)
- [Classical and Quantum Algorithms for Tensor Principal Component Analysis](https://mlsanity.com/paper/1907.12724/) (2019)
- [Variational Quantum Algorithms for Dimensionality Reduction and Classification](https://mlsanity.com/paper/1910.12164/) (2019)
- [Efficient Quantum Feature Extraction for CNN-based Learning](https://mlsanity.com/paper/2201.01246/) (2022)
- [Dimension reduction and redundancy removal through successive Schmidt decompositions](https://mlsanity.com/paper/2302.04801/) (2023)
- [Iterative Quantum Feature Maps](https://mlsanity.com/paper/2506.19461/) (2025)
- [Obtaining A Linear Combination of the Principal Components of a Matrix on Quantum Computers](https://mlsanity.com/paper/1512.02109/) (2015)
- [Learnable quantum spectral filters for hybrid graph neural networks](https://mlsanity.com/paper/2507.05640/) (2025)

## Abstract (arXiv)

Schmidt decomposition of a vector can be understood as writing the singular value decomposition (SVD) in vector form. A vector can be written as a linear combination of tensor product of two dimensional vectors by recursively applying Schmidt decompositions via SVD to all subsystems. Given a vector expressed as a linear combination of tensor products, using only the k principal terms yields a k-rank approximation of the vector. Therefore, writing a vector in this reduced form allows to retain most important parts of the vector while removing small noises from it, analogous to SVD-based denoising. In this paper, we show that quantum circuits designed based on a value k (determined from the tensor network decomposition of the mean vector of the training sample) can approximate the reduced-form representations of entire datasets. We then employ this circuit ansatz with a classical neural network head to construct a hybrid machine learning model. Since the output of the quantum circuit for an 2n dimensional vector is an n dimensional probability vector, this provides an exponential compression of the input and potentially can reduce the number of learnable parameters for training large-scale models. We use datasets provided in the Python scikit-learn module for the experiments. The results confirm the quantum circuit is able to compress data successfully to provide effective k-rank approximations to the classical processing component.