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We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (e.g. energies per unit cell), as is often the goal for condensed-phase systems. In this context, simulations of the Hubbard and plane-wave electronic structure models with <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>N<\/mml:mi><mml:mo>&lt;<\/mml:mo><mml:msup><mml:mn>10<\/mml:mn><mml:mn>5<\/mml:mn><\/mml:msup><\/mml:math> fermionic modes can be performed with roughly <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">O<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msup><mml:mi>N<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> T complexities. We perform numerics revealing tradeoffs between the error and gate complexity of a Trotter step; e.g., we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates and assuming error rates of one part per thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with a few hundred thousand physical qubits.<\/jats:p>","DOI":"10.22331\/q-2020-07-16-296","type":"journal-article","created":{"date-parts":[[2020,7,16]],"date-time":"2020-07-16T11:53:37Z","timestamp":1594900417000},"page":"296","source":"Crossref","is-referenced-by-count":139,"title":["Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization"],"prefix":"10.22331","volume":"4","author":[{"given":"Ian D.","family":"Kivlichan","sequence":"first","affiliation":[{"name":"Google Research, Venice, CA 90291, USA"},{"name":"Department of Physics, Harvard University, Cambridge, MA 02138, USA"}]},{"given":"Craig","family":"Gidney","sequence":"additional","affiliation":[{"name":"Google Research, Santa Barbara, CA 93117, USA"}]},{"given":"Dominic W.","family":"Berry","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2113, Australia"}]},{"given":"Nathan","family":"Wiebe","sequence":"additional","affiliation":[{"name":"Institute for Nuclear Theory, University of Washington, Seattle, WA 98195, USA"}]},{"given":"Jarrod","family":"McClean","sequence":"additional","affiliation":[{"name":"Google Research, Venice, CA 90291, USA"}]},{"given":"Wei","family":"Sun","sequence":"additional","affiliation":[{"name":"Google Research, Mountain View, CA 94043, USA"}]},{"given":"Zhang","family":"Jiang","sequence":"additional","affiliation":[{"name":"Google Research, Venice, CA 90291, USA"}]},{"given":"Nicholas","family":"Rubin","sequence":"additional","affiliation":[{"name":"Google Research, Venice, CA 90291, USA"}]},{"given":"Austin","family":"Fowler","sequence":"additional","affiliation":[{"name":"Google Research, Santa Barbara, CA 93117, USA"}]},{"given":"Al\u00e1n","family":"Aspuru-Guzik","sequence":"additional","affiliation":[{"name":"Department of Chemistry, University of Toronto, Toronto, Ontario M5G 1Z8, Canada"},{"name":"Department of Computer Science, University of Toronto, Toronto, Ontario M5G 1Z8, Canada"}]},{"given":"Hartmut","family":"Neven","sequence":"additional","affiliation":[{"name":"Google Research, Venice, CA 90291, USA"}]},{"given":"Ryan","family":"Babbush","sequence":"additional","affiliation":[{"name":"Google Research, Venice, CA 90291, USA"}]}],"member":"9598","published-online":{"date-parts":[[2020,7,16]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"R. 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