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For a given polynomial <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>f<\/mml:mi><\/mml:math>, the parameters (called phase factors) can be obtained by solving an optimization problem. However, the cost function is non-convex, and has a very complex energy landscape with numerous global and local minima. It is therefore surprising that the solution can be robustly obtained in practice, starting from a fixed initial guess <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi mathvariant=\"normal\">&amp;#x03A6;<\/mml:mi><mml:mn>0<\/mml:mn><\/mml:msup><\/mml:math> that contains no information of the input polynomial. To investigate this phenomenon, we first explicitly characterize all the global minima of the cost function. We then prove that one particular global minimum (called the maximal solution) belongs to a neighborhood of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi mathvariant=\"normal\">&amp;#x03A6;<\/mml:mi><mml:mn>0<\/mml:mn><\/mml:msup><\/mml:math>, on which the cost function is strongly convex under the condition <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mrow><mml:mo symmetric=\"true\">&amp;#x2016;<\/mml:mo><mml:mi>f<\/mml:mi><mml:mo symmetric=\"true\">&amp;#x2016;<\/mml:mo><\/mml:mrow><\/mml:mrow><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"normal\">&amp;#x221E;<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><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>d<\/mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>&amp;#x2212;<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> with <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mo>=<\/mml:mo><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mi mathvariant=\"normal\">d<\/mml:mi><mml:mi mathvariant=\"normal\">e<\/mml:mi><mml:mi mathvariant=\"normal\">g<\/mml:mi><\/mml:mrow><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>f<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>. Our result provides a partial explanation of the aforementioned success of optimization algorithms.<\/jats:p>","DOI":"10.22331\/q-2022-11-03-850","type":"journal-article","created":{"date-parts":[[2022,11,3]],"date-time":"2022-11-03T13:22:07Z","timestamp":1667481727000},"page":"850","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":27,"title":["On the energy landscape of symmetric quantum signal processing"],"prefix":"10.22331","volume":"6","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1321-2649","authenticated-orcid":false,"given":"Jiasu","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California, Berkeley, CA 94720, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0577-2475","authenticated-orcid":false,"given":"Yulong","family":"Dong","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, Berkeley, CA 94720, USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6860-9566","authenticated-orcid":false,"given":"Lin","family":"Lin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, Berkeley, CA 94720, USA."},{"name":"Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA"},{"name":"Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2022,11,3]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"D. 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