{"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/hamiltonian-simulation-by-qubitization","title":"Hamiltonian Simulation by Qubitization","arxiv_id":"1610.06546","date":"2016-10-20","proceeding":null,"authors":["Guang Hao Low","Isaac L. Chuang"],"abstract":"We present the problem of approximating the time-evolution operator $e^{-i\\hat{H}t}$ to error $\\epsilon$, where the Hamiltonian $\\hat{H}=(\\langle G|\\otimes\\hat{\\mathcal{I}})\\hat{U}(|G\\rangle\\otimes\\hat{\\mathcal{I}})$ is the projection of a unitary oracle $\\hat{U}$ onto the state $|G\\rangle$ created by another unitary oracle. Our algorithm solves this with a query complexity $\\mathcal{O}\\big(t+\\log({1/\\epsilon})\\big)$ to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are $d$-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where $\\hat{H}$ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any $\\hat{H}$ in an invariant $\\text{SU}(2)$ subspace. A large class of operator functions of $\\hat{H}$ can then be computed with optimal query complexity, of which $e^{-i\\hat{H}t}$ is a special case.","url_abs":"http://arxiv.org/abs/1610.06546v3","url_pdf":"http://arxiv.org/pdf/1610.06546v3.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":"hamiltonian-simulation-by-qubitization","repo_url":"https://github.com/goodchemistryco/Tangelo/blob/main/tangelo/toolboxes/circuits/qsp.py","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1610.06546","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}