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We consider Schr\u00f6dinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter&#x210F;<\/mml:mi><\/mml:math>, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schr\u00f6dinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of&#x210F;<\/mml:mi><\/mml:math>and the precision&#x03B5;<\/mml:mi><\/mml:math>are obtained. It is found that the number of required qubits,m<\/mml:mi><\/mml:math>, scales only logarithmically with respect to&#x210F;<\/mml:mi><\/mml:math>. When the solution has bounded derivatives up to order&#x2113;<\/mml:mi><\/mml:math>, the symmetric Trotting method has gate complexityO<\/mml:mi><\/mml:mrow>(<\/mml:mo><\/mml:mrow>(<\/mml:mo>&#x03B5;<\/mml:mi>&#x210F;<\/mml:mi>)<\/mml:mo>&#x2212;<\/mml:mo>1<\/mml:mn>2<\/mml:mn><\/mml:mfrac><\/mml:mrow><\/mml:msup>p<\/mml:mi>o<\/mml:mi>l<\/mml:mi>y<\/mml:mi>l<\/mml:mi>o<\/mml:mi>g<\/mml:mi><\/mml:mrow>(<\/mml:mo>&#x03B5;<\/mml:mi>&#x2212;<\/mml:mo>3<\/mml:mn>2<\/mml:mn>&#x2113;<\/mml:mi><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:msup>&#x210F;<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn>&#x2212;<\/mml:mo>1<\/mml:mn>2<\/mml:mn>&#x2113;<\/mml:mi><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow>,<\/mml:mo><\/mml:math>provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented withp<\/mml:mi>o<\/mml:mi>l<\/mml:mi>y<\/mml:mi><\/mml:mrow>(<\/mml:mo>m<\/mml:mi>)<\/mml:mo><\/mml:math>operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of&#x210F;<\/mml:mi><\/mml:math>. The gate complexity in this case is reduced toO<\/mml:mi><\/mml:mrow>(<\/mml:mo><\/mml:mrow>&#x03B5;<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn>2<\/mml:mn><\/mml:mfrac><\/mml:mrow><\/mml:msup>p<\/mml:mi>o<\/mml:mi>l<\/mml:mi>y<\/mml:mi>l<\/mml:mi>o<\/mml:mi>g<\/mml:mi><\/mml:mrow>(<\/mml:mo>&#x03B5;<\/mml:mi>&#x2212;<\/mml:mo>3<\/mml:mn>2<\/mml:mn>&#x2113;<\/mml:mi><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:msup>&#x210F;<\/mml:mi>&#x2212;<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msup>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow>,<\/mml:mo><\/mml:math>with&#x2113;<\/mml:mi><\/mml:math>again indicating the smoothness of the solution.<\/jats:p>","DOI":"10.22331\/q-2022-06-17-739","type":"journal-article","created":{"date-parts":[[2022,6,17]],"date-time":"2022-06-17T08:07:03Z","timestamp":1655453223000},"page":"739","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":11,"title":["Quantum simulation in the semi-classical regime"],"prefix":"10.22331","volume":"6","author":[{"given":"Shi","family":"Jin","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai, China"}]},{"given":"Xiantao","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA"}]},{"given":"Nana","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute of Natural Sciences, University of Michigan-Shanghai Jiao Tong University Joint Institute, MOE-LSEC, Shanghai Jiao Tong University, Shanghai, China"}]}],"member":"9598","published-online":{"date-parts":[[2022,6,17]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Shi Jin, Peter Markowich, and Christof Sparber. ``Mathematical and computational methods for semiclassical Schr\u00f6dinger equations''. 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