{"paper":{"title":"Quantum Amplitude Amplification and Estimation","license":"","headline":"Amplitude amplification finds a good element after a number of steps proportional to one over the square root of its initial probability.","cross_cats":[],"primary_cat":"quant-ph","authors_text":"(2) BRICS University of Aarhus, (3) CACR University of Waterloo), Alain Tapp (3) ((1) DIRO Universite de Montreal, Gilles Brassard (1), Michele Mosca (3), Peter Hoyer (2)","submitted_at":"2000-05-15T18:19:59Z","abstract_excerpt":"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 solutio"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Amplitude amplification finds a good x after expected applications of A and inverse proportional to 1/sqrt(a), generalizing Grover's algorithm.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Algorithm A makes no measurements and is unitary.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Amplitude amplification finds solutions quadratically faster than classical methods and enables quantum estimation of solution counts.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Amplitude amplification finds a good element after a number of steps proportional to one over the square root of its initial probability.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e6b198b377f4eb56d0a1fee57edc7c51dcac51ee778ad307baddc6cd8ca3aa67"},"source":{"id":"quant-ph/0005055","kind":"arxiv","version":1},"verdict":{"id":"805e2622-963c-444d-be6a-4401743b2195","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T03:39:06.064896Z","strongest_claim":"Amplitude amplification finds a good x after expected applications of A and inverse proportional to 1/sqrt(a), generalizing Grover's algorithm.","one_line_summary":"Amplitude amplification finds solutions quadratically faster than classical methods and enables quantum estimation of solution counts.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Algorithm A makes no measurements and is unitary.","pith_extraction_headline":"Amplitude amplification finds a good element after a number of steps proportional to one over the square root of its initial probability."},"references":{"count":15,"sample":[{"doi":"","year":1998,"title":"Quantum lower bounds by polynomials","work_id":"88db342f-2783-446b-b3f3-2d1d1daf77d5","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1988,"title":"Notes on the history of reversible computatio n","work_id":"40545b08-cdbe-4001-a34a-571c88f8cbb5","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"Tight bounds on quantum searching","work_id":"607ede23-cae8-44dd-9b38-ac66ece911bd","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"An exact quantum polynomial- time algorithm for Simon’s problem","work_id":"8003070e-59eb-4ca5-8ff6-093fbdc75095","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"Quantum count- ing","work_id":"c489dd43-7c82-4ea8-9fc2-4821d54d562d","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":15,"snapshot_sha256":"c34ae57493df0a2548a89c43f1abef05d49d9cbd6be30cc15068a40f290710b5","internal_anchors":1},"formal_canon":{"evidence_count":2,"snapshot_sha256":"6b096659011d0abc58c79920dc230039088f57c2a4c10898f9c8e6c794c69bec"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}