{"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-amplitude-amplification-and","title":"Quantum Amplitude Amplification and Estimation","arxiv_id":"quant-ph/0005055","date":"2000-05-15","proceeding":null,"authors":["Gilles Brassard","Peter Hoyer","Michele Mosca","Alain Tapp"],"abstract":"Consider a Boolean function $\\chi: X \\to \\{0,1\\}$ that partitions set $X$ between its good and bad elements, where $x$ is good if $\\chi(x)=1$ and bad otherwise. Consider also a quantum algorithm $\\mathcal A$ such that $A |0\\rangle= \\sum_{x\\in X} \\alpha_x |x\\rangle$ is a quantum superposition of the elements of $X$, and let $a$ denote the probability that a good element is produced if $A |0\\rangle$ is measured. If we repeat the process of running $A$, measuring the output, and using $\\chi$ to check the validity of the result, we shall expect to repeat $1/a$ times on the average before a solution is found. *Amplitude amplification* is a process that allows to find a good $x$ after an expected number of applications of $A$ and its inverse which is proportional to $1/\\sqrt{a}$, assuming algorithm $A$ makes no measurements. This is a generalization of Grover's searching algorithm in which $A$ was restricted to producing an equal superposition of all members of $X$ and we had a promise that a single $x$ existed such that $\\chi(x)=1$. Our algorithm works whether or not the value of $a$ is known ahead of time. In case the value of $a$ is known, we can find a good $x$ after a number of applications of $A$ and its inverse which is proportional to $1/\\sqrt{a}$ even in the worst case. We show that this quadratic speedup can also be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform amplitude estimation, a process that allows to estimate the value of $a$. We apply amplitude estimation to the problem of *approximate counting*, in which we wish to estimate the number of $x\\in X$ such that $\\chi(x)=1$. We obtain optimal quantum algorithms in a variety of settings.","url_abs":"https://arxiv.org/abs/quant-ph/0005055v1","url_pdf":"https://arxiv.org/pdf/quant-ph/0005055v1.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-amplitude-amplification-and","repo_url":"https://github.com/MonitSharma/Quantum-Finance-and-Numerical-Methods/blob/main/Quantum_Amplitude_Estimation.ipynb","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=quant-ph%2F0005055","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}