# Variational Quantum Singular Value Decomposition

arXiv: https://arxiv.org/abs/2006.02336 · HTML: https://mlsanity.com/paper/2006.02336/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jun 2020.

## Built on by (most cited first, 9 of 9)

- [Escaping from the Barren Plateau via Gaussian Initializations in Deep Variational Quantum Circuits](https://mlsanity.com/paper/2203.09376/) (2022)
- [Mitigating barren plateaus of variational quantum eigensolvers](https://mlsanity.com/paper/2205.13539/) (2022)
- [Quantum algorithms for SVD-based data representation and analysis](https://mlsanity.com/paper/2104.08987/) (2021)
- [Toward Neural Network Simulation of Variational Quantum Algorithms](https://mlsanity.com/paper/2211.02929/) (2022)
- [Towards Quantum Tensor Decomposition in Biomedical Applications](https://mlsanity.com/paper/2502.13140/) (2025)
- [A Hybrid Quantum-Classical Hamiltonian Learning Algorithm](https://mlsanity.com/paper/2103.01061/) (2021)
- [Fundamental limitations on optimization in variational quantum algorithms](https://mlsanity.com/paper/2205.05056/) (2022)
- [Statistical Analysis of Quantum State Learning Process in Quantum Neural Networks](https://mlsanity.com/paper/2309.14980/) (2023)
- [QIANets: Quantum-Integrated Adaptive Networks for Reduced Latency and Improved Inference Times in CNN Models](https://mlsanity.com/paper/2410.10318/) (2024)

## Abstract (arXiv)

Singular value decomposition is central to many problems in engineering and scientific fields. Several quantum algorithms have been proposed to determine the singular values and their associated singular vectors of a given matrix. Although these algorithms are promising, the required quantum subroutines and resources are too costly on near-term quantum devices. In this work, we propose a variational quantum algorithm for singular value decomposition (VQSVD). By exploiting the variational principles for singular values and the Ky Fan Theorem, we design a novel loss function such that two quantum neural networks (or parameterized quantum circuits) could be trained to learn the singular vectors and output the corresponding singular values. Furthermore, we conduct numerical simulations of VQSVD for random matrices as well as its applications in image compression of handwritten digits. Finally, we discuss the applications of our algorithm in recommendation systems and polar decomposition. Our work explores new avenues for quantum information processing beyond the conventional protocols that only works for Hermitian data, and reveals the capability of matrix decomposition on near-term quantum devices.