{"work":{"id":"f2227e7e-b502-4e3a-9f3c-fcd66cc5ab07","openalex_id":"https://openalex.org/W2761673598","doi":"10.1137/16m1087072","arxiv_id":"1511.02306","raw_key":null,"title":"Quantum algorithm for systems of linear equations with exponentially improved dependence on precision","authors":[{"given":"Andrew M.","family":"Childs","sequence":"first","affiliation":[]},{"given":"Robin","family":"Kothari","sequence":"additional","affiliation":[]},{"given":"Rolando D.","family":"Somma","sequence":"additional","affiliation":[]}],"authors_text":"A","year":1920,"venue":"quant-ph","abstract":"Harrow, Hassidim, and Lloyd showed that for a suitably specified $N \\times N$ matrix $A$ and $N$-dimensional vector $\\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations $A\\vec{x}=\\vec{b}$. If $A$ is sparse and well-conditioned, their algorithm runs in time $\\mathrm{poly}(\\log N, 1/\\epsilon)$, where $\\epsilon$ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in $\\log(1/\\epsilon)$, exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on $\\epsilon$ is prohibitive.","external_url":"https://doi.org/10.1137/16m1087072","cited_by_count":505,"metadata_source":"doi_reference","metadata_fetched_at":"2026-07-10T12:07:03.365777+00:00","pith_arxiv_id":"1511.02306","created_at":"2026-05-08T17:13:36.926310+00:00","updated_at":"2026-07-11T11:50:26.030339+00:00","title_quality_ok":true,"display_title":"Quantum algorithm for systems of linear equations with exponentially improved dependence on precision","render_title":"Quantum algorithm for systems of linear equations with exponentially improved dependence on precision"},"hub":{"state":{"work_id":"f2227e7e-b502-4e3a-9f3c-fcd66cc5ab07","tier":"hub","tier_reason":"10+ Pith inbound or 1,000+ external citations","pith_inbound_count":23,"external_cited_by_count":505,"distinct_field_count":5,"first_pith_cited_at":"2024-06-17T20:54:11+00:00","last_pith_cited_at":"2026-07-09T07:10:01+00:00","author_build_status":"not_needed","summary_status":"needed","contexts_status":"needed","graph_status":"needed","ask_index_status":"not_needed","reader_status":"not_needed","recognition_status":"not_needed","updated_at":"2026-08-22T02:59:44.573683+00:00","tier_text":"hub"},"tier":"hub","role_counts":[{"context_role":"background","n":8},{"context_role":"method","n":1}],"polarity_counts":[{"context_polarity":"background","n":8},{"context_polarity":"use_method","n":1}],"runs":{},"summary":{},"graph":{},"authors":[]}}