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We introduce a \"periodized stationary phase method\" to discrete Wigner functions of systems with odd prime dimension and show that the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mfrac><mml:mi>\u03c0<\/mml:mi><mml:mn>8<\/mml:mn><\/mml:mfrac><\/mml:math> gate is the discrete analog of the Airy function. We then establish a relationship between the stabilizer rank of states and the number of quadratic Gauss sums necessary in the periodized stationary phase method. This allows us to develop a classical strong simulation of a single qutrit marginal on <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>t<\/mml:mi><\/mml:math> qutrit <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mfrac><mml:mi>\u03c0<\/mml:mi><mml:mn>8<\/mml:mn><\/mml:mfrac><\/mml:math> gates that are followed by Clifford evolution, and show that this only requires <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mn>3<\/mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mfrac><mml:mi>t<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:mfrac><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math> quadratic Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo>\u223c<\/mml:mo><mml:mspace width=\"-3pt\" \/><mml:msup><mml:mn>3<\/mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mn>0.8<\/mml:mn><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:math> for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mn>10<\/mml:mn><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>\u2212<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math> precision) for any number of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mfrac><mml:mi>\u03c0<\/mml:mi><mml:mn>8<\/mml:mn><\/mml:mfrac><\/mml:math> gates to full precision.<\/jats:p>","DOI":"10.22331\/q-2021-07-05-494","type":"journal-article","created":{"date-parts":[[2021,7,5]],"date-time":"2021-07-05T14:34:17Z","timestamp":1625495657000},"page":"494","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits"],"prefix":"10.22331","volume":"5","author":[{"given":"Lucas","family":"Kocia","sequence":"first","affiliation":[{"name":"National Institute of Standards and Technology, Gaithersburg, Maryland, 20899, U.S.A."}]},{"given":"Peter","family":"Love","sequence":"additional","affiliation":[{"name":"Department of Physics, Tufts University, Medford, Massachusetts 02155, U.S.A."}]}],"member":"9598","published-online":{"date-parts":[[2021,7,5]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Greg Kuperberg. 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