{"paper":{"title":"Fast Quantum Algorithms for Learning Linear Threshold Functions","arxiv_id":"2609.40331","primary_cat":"quant-ph","submitted_at":"2026-09-30T17:56:41+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"Linear threshold functions are $f_{w,\\theta}(x)=\\text{sign}(\\langle x,w \\rangle -\\theta)$, where the weight vector $w\\in\\mathbb{R}^n$ is a unit vector, $\\theta\\in\\mathbb R$ is a threshold, and typically $x\\in\\mathbb{R}^n$ or $x\\in\\{-1,1\\}^n$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.05,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.14,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"When $\\theta=0$, the LTF is called homogeneous, and we write $f_w:=f_{w,0}$.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.2,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.17,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":false,"role":"background","text":"Such functions are among the most important objects in machine learning, since they serve to linearly discriminate positive and negative examples.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":1.0,"id":"s3","own":true,"role":"contribution","text":"We give three positive results about learning LTFs: 1.","hedge":{"p":0.95,"label":"asserted"},"own_p":0.88,"main_claim_p":0.27,"quantitative":false,"quantitative_p":0.43,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":1.0,"interpretation":0.0}},{"i":4,"p":0.55,"id":"s4","own":false,"role":"method","text":"Suppose we can make real-domain queries, meaning we can compute $f_{w,\\theta}(x)$ at any $x\\in\\mathbb{R}^n$ of our choice.","hedge":{"p":0.59,"label":"asserted"},"own_p":0.18,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.55,"result":0.0,"background":0.34,"limitation":0.0,"future_work":0.0,"contribution":0.1,"interpretation":0.0}},{"i":5,"p":0.56,"id":"s5","own":true,"role":"contribution","text":"We give a quantum algorithm that learns $f_{w,\\theta}$ up to Euclidean error~$\\epsilon$ using $O(\\log(n/\\epsilon))$ membership queries and $\\widetilde{O}(n)$ other gates.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.3,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.44,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.56,"interpretation":0.0}},{"i":6,"p":0.77,"id":"s6","own":true,"role":"result","text":"Then we have also learned $f_{w,\\theta}$ up to error $O(\\epsilon)$ when $x$ is Gaussian.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.82,"main_claim_p":0.1,"quantitative":true,"quantitative_p":0.9,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.77,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.22}},{"i":7,"p":0.51,"id":"s7","own":false,"role":"result","text":"Classical algorithms need $\\Omega(n\\log(1/\\epsilon))$ queries. 2.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.28,"main_claim_p":0.09,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.28,"method":0.0,"result":0.51,"background":0.15,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.04}},{"i":8,"p":0.95,"id":"s8","own":false,"role":"background","text":"A homogeneous LTF $f_w$ on domain $\\{-1,1\\}^n$ where $w$ has only $k$ nonzero entries of the same value, is the Majority function on the support of $w$.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.15,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.26,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.01,"background":0.95,"limitation":0.0,"future_work":0.0,"contribution":0.03,"interpretation":0.01}},{"i":9,"p":0.98,"id":"s9","own":false,"role":"background","text":"Belovs gave a bounded-error quantum algorithm that identifies the hidden support exactly (and hence learns $f_w$) using $O(k^{1/4})$ queries.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.97,"role_probabilities":{"gap":0.01,"method":0.0,"result":0.01,"background":0.98,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":10,"p":0.72,"id":"s10","own":true,"role":"result","text":"We give an exponential improvement, using $O(\\log k)$ queries. 3.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.17,"quantitative":true,"quantitative_p":0.98,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.72,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.28,"interpretation":0.0}},{"i":11,"p":0.6,"id":"s11","own":false,"role":"method","text":"Suppose we have a unitary $U$ that can produce (discretized) \\emph{quantum examples} under Gaussian measure, corresponding to $\\int_x \\sqrt{\\gamma_n(x)}|x\\rangle |f_w(x)\\rangle dx$.","hedge":{"p":0.47,"label":"asserted"},"own_p":0.28,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.6,"result":0.0,"background":0.26,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.0}},{"i":12,"p":0.55,"id":"s12","own":false,"role":"background","text":"This is a weaker access model than membership queries.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.35,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.72,"role_probabilities":{"gap":0.0,"method":0.07,"result":0.0,"background":0.55,"limitation":0.14,"future_work":0.0,"contribution":0.01,"interpretation":0.23}},{"i":13,"p":0.82,"id":"s13","own":true,"role":"contribution","text":"We give a quantum algorithm based on the efficient \\emph{Hermite transform} of Jain et al.\\ to learn homogeneous LTFs~$f_w$ with error~$\\epsilon$ under the Gaussian distribution, using $O(n^{1/4}/\\sqrt{\\epsilon})$ applications of $U$ and $U^\\dagger$ and 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