{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/quantum-machine-learning-with-subspace-states","title":"Quantum machine learning with subspace states","arxiv_id":"2202.00054","date":"2022-01-31","proceeding":null,"authors":["Iordanis Kerenidis","Anupam Prakash"],"abstract":"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 $\\Pr[S]= det(X_{S}X_{S}^{T})$ for $|S|=d$ using $O(nd)$ gates and with circuit depth $O(d\\log n)$. The state of art classical algorithm for the task requires $O(d^{3})$ operations \\cite{derezinski2019minimax}. The second is a quantum singular value estimation algorithm for compound matrices $\\mathcal{A}^{k}$, the speedup for this algorithm is potentially exponential. It decomposes a $\\binom{n}{k}$ 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 $|Col(X)\\rangle = \\sum_{S\\subset [n], |S|=d} det(X_{S}) |S\\rangle$ for matrices $X \\in \\mathbb{R}^{n \\times d}$ such that $X^{T} X = I_{d}$, that encode $d$-dimensional subspaces of $\\mathbb{R}^{n}$. 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 $\\Gamma(x) = \\sum_{i} x_{i} Z^{\\otimes (i-1)} \\otimes X \\otimes I^{n-i}$ with $O(\\log n)$ depth circuits that we term Clifford loaders.","url_abs":"https://arxiv.org/abs/2202.00054v2","url_pdf":"https://arxiv.org/pdf/2202.00054v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"quantum-machine-learning-with-subspace-states","repo_url":"https://github.com/mrfanuel/dpps-with-clifford-loaders","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}