{"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/q-means-a-quantum-algorithm-for-unsupervised","title":"q-means: A quantum algorithm for unsupervised machine learning","arxiv_id":"1812.03584","date":"2018-12-10","proceeding":"NeurIPS 2019 12","authors":["Iordanis Kerenidis","Jonas Landman","Alessandro Luongo","Anupam Prakash"],"abstract":"Quantum machine learning is one of the most promising applications of a\nfull-scale quantum computer. Over the past few years, many quantum machine\nlearning algorithms have been proposed that can potentially offer considerable\nspeedups over the corresponding classical algorithms. In this paper, we\nintroduce q-means, a new quantum algorithm for clustering which is a canonical\nproblem in unsupervised machine learning. The $q$-means algorithm has\nconvergence and precision guarantees similar to $k$-means, and it outputs with\nhigh probability a good approximation of the $k$ cluster centroids like the\nclassical algorithm. Given a dataset of $N$ $d$-dimensional vectors $v_i$ (seen\nas a matrix $V \\in \\mathbb{R}^{N \\times d})$ stored in QRAM, the running time\nof q-means is $\\widetilde{O}\\left( k d \\frac{\\eta}{\\delta^2}\\kappa(V)(\\mu(V) +\nk \\frac{\\eta}{\\delta}) + k^2 \\frac{\\eta^{1.5}}{\\delta^2} \\kappa(V)\\mu(V)\n\\right)$ per iteration, where $\\kappa(V)$ is the condition number, $\\mu(V)$ is\na parameter that appears in quantum linear algebra procedures and $\\eta =\n\\max_{i} ||v_{i}||^{2}$. For a natural notion of well-clusterable datasets, the\nrunning time becomes $\\widetilde{O}\\left( k^2 d \\frac{\\eta^{2.5}}{\\delta^3} +\nk^{2.5} \\frac{\\eta^2}{\\delta^3} \\right)$ per iteration, which is linear in the\nnumber of features $d$, and polynomial in the rank $k$, the maximum square norm\n$\\eta$ and the error parameter $\\delta$. Both running times are only\npolylogarithmic in the number of datapoints $N$. Our algorithm provides\nsubstantial savings compared to the classical $k$-means algorithm that runs in\ntime $O(kdN)$ per iteration, particularly for the case of large datasets.","url_abs":"http://arxiv.org/abs/1812.03584v2","url_pdf":"http://arxiv.org/pdf/1812.03584v2.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":"abstracts"},"code_links":[{"paper_slug":"q-means-a-quantum-algorithm-for-unsupervised","repo_url":"https://github.com/JonasLandman/quantum_kmeans_NeurIPS_2019","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"q-means-a-quantum-algorithm-for-unsupervised","repo_url":"https://github.com/Morcu/q-means","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"Apache-2.0"}}],"tasks":[{"task_slug":"machine-learning","task_name":"BIG-bench Machine Learning"},{"task_slug":"clustering","task_name":"Clustering"},{"task_slug":"quantum-machine-learning","task_name":"Quantum Machine Learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1812.03584","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1812.03584"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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