# Quantum machine learning with subspace states

arXiv: https://arxiv.org/abs/2202.00054 · HTML: https://mlsanity.com/paper/2202.00054/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2022.

## Built on by (most cited first, 12 of 12)

- [Quantum Vision Transformers](https://mlsanity.com/paper/2209.08167/) (2022)
- [Quantum Deep Hedging](https://mlsanity.com/paper/2303.16585/) (2023)
- [Improved Financial Forecasting via Quantum Machine Learning](https://mlsanity.com/paper/2306.12965/) (2023)
- [Training-efficient density quantum machine learning](https://mlsanity.com/paper/2405.20237/) (2024)
- [QuIC: Quantum-Inspired Compound Adapters for Parameter Efficient Fine-Tuning](https://mlsanity.com/paper/2502.06916/) (2025)
- [Quantum Convolutional Neural Networks with Interaction Layers for Classification of Classical Data](https://mlsanity.com/paper/2307.11792/) (2023)
- [Bayesian Quantum Orthogonal Neural Networks for Anomaly Detection](https://mlsanity.com/paper/2504.18103/) (2025)
- [On sampling determinantal and Pfaffian point processes on a quantum computer](https://mlsanity.com/paper/2305.15851/) (2023)
- [Scalable On-Hardware Training of Quantum Neural Networks and Application to Clinical Data Imputation](https://mlsanity.com/paper/2606.03517/) (2026)
- [Quantum Machine Learning for Industrial Applications](https://mlsanity.com/paper/2606.14822/) (2026)
- [Scalable Message-Passing Quantum Graph Neural Networks in the Weisfeiler-Leman Hierarchy](https://mlsanity.com/paper/2606.26873/) (2026)
- [Quantum Topological Data Encoding](https://mlsanity.com/paper/2607.13847/) (2026)

## Abstract (arXiv)

We introduce a new approach for quantum linear algebra based on quantum subspace states and present three new quantum machine learning algorithms. The first is a quantum determinant sampling algorithm that samples from the distribution Pr[S]= det(XSXST) for |S|=d using O(nd) gates and with circuit depth O(dlog n). The state of art classical algorithm for the task requires O(d3) operations citederezinski2019minimax. The second is a quantum singular value estimation algorithm for compound matrices Ak, the speedup for this algorithm is potentially exponential. It decomposes a binomnk dimensional vector of order-k correlations into a linear combination of subspace states corresponding to k-tuples of singular vectors of A. The third algorithm reduces exponentially the depth of circuits used in quantum topological data analysis from O(n) to O(log n). Our basic tool are quantum subspace states, defined as |Col(X)rangle = sumSsubset [n], |S|=d det(XS) |Srangle for matrices X ∈ ℝn × d such that XT X = Id, that encode d-dimensional subspaces of ℝn. We develop two efficient state preparation techniques, the first using Givens circuits uses the representation of a subspace as a sequence of Givens rotations, while the second uses efficient implementations of unitaries Gamma(x) = sumi xi Zotimes (i-1) otimes X otimes In-i with O(log n) depth circuits that we term Clifford loaders.