Approximate Cosine Similarity Estimation via an Angle-Encoding Hadamard Test

Quantum Physics [Submitted on 17 Apr 2026] Approximate Cosine Similarity Estimation via an Angle-Encoding Hadamard Test Hiroshi Ohno The Hadamard test is a standard quantum primitive for estimating inner products and expectation values, but in data-processing settings its practical utility is often limited by the cost of preparing amplitude-encoded quantum states. In this study, we investigate an angle-encoding variant of the Hadamard test for estimating cosine similarity between normalized real-valued vectors. The proposed method decomposes the similarity computation into elementwise two-qubit Hadamard-test circuits that can, in principle, be executed in parallel, resulting in constant circuit depth with respect to the vector dimension at the expense of a larger qubit footprint and classical post-processing. Because the resulting estimator is approximate, we analyze the induced bias and show that it is non-negative under the approximation used in our derivation. Numerical experiments on random normalized vectors show that, in the tested setting, the estimation error decreases as the vector dimension increases. We further illustrate a possible application to cosine-attention-based Transformer models. These results suggest that the angle-encoding Hadamard test may provide a useful design point for near-term similarity estimation when shallow circuit depth is preferred over compact qubit usage. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2604.15867 [quant-ph]   (or arXiv:2604.15867v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.15867 Focus to learn more Submission history From: Hiroshi Ohno [view email] [v1] Fri, 17 Apr 2026 09:16:36 UTC (119 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)