{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,17]],"date-time":"2026-03-17T00:17:58Z","timestamp":1773706678313,"version":"3.50.1"},"reference-count":26,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2024,12,10]],"date-time":"2024-12-10T00:00:00Z","timestamp":1733788800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/quantum","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-AC02-05CH11231"],"award-info":[{"award-number":["DE-AC02-05CH11231"]}],"id":[{"id":"10.13039\/quantum","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/challenge","name":"National Science Foundation","doi-asserted-by":"publisher","award":["OMA-2016245"],"award-info":[{"award-number":["OMA-2016245"]}],"id":[{"id":"10.13039\/challenge","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"<jats:p>Quantum signal processing (QSP) represents a real scalar polynomial of degree <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><\/mml:math> using a product of unitary matrices of size <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>2<\/mml:mn><mml:mo>&amp;#x00D7;<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math>, parameterized by <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>d<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> real numbers called the phase factors. This innovative representation of polynomials has a wide range of applications in quantum computation. When the polynomial of interest is obtained by truncating an infinite polynomial series, a natural question is whether the phase factors have a well defined limit as the degree <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mo stretchy=\"false\">&amp;#x2192;<\/mml:mo><mml:mi mathvariant=\"normal\">&amp;#x221E;<\/mml:mi><\/mml:math>. While the phase factors are generally not unique, we find that there exists a consistent choice of parameterization so that the limit is well defined in the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>&amp;#x2113;<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msup><\/mml:math> space. This generalization of QSP, called the infinite quantum signal processing, can be used to represent a large class of non-polynomial functions. Our analysis reveals a surprising connection between the regularity of the target function and the decay properties of the phase factors. Our analysis also inspires a very simple and efficient algorithm to approximately compute the phase factors in the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>&amp;#x2113;<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msup><\/mml:math> space. The algorithm uses only double precision arithmetic operations, and provably converges when the <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>&amp;#x2113;<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:msup><\/mml:math> norm of the Chebyshev coefficients of the target function is upper bounded by a constant that is independent of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><\/mml:math>. This is also the first numerically stable algorithm for finding phase factors with provable performance guarantees in the limit <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mo stretchy=\"false\">&amp;#x2192;<\/mml:mo><mml:mi mathvariant=\"normal\">&amp;#x221E;<\/mml:mi><\/mml:math>.<\/jats:p>","DOI":"10.22331\/q-2024-12-10-1558","type":"journal-article","created":{"date-parts":[[2024,12,10]],"date-time":"2024-12-10T16:47:37Z","timestamp":1733849257000},"page":"1558","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":10,"title":["Infinite quantum signal processing"],"prefix":"10.22331","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0577-2475","authenticated-orcid":false,"given":"Yulong","family":"Dong","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California, Berkeley, CA 94720, USA."}]},{"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":"Applied Mathematics and Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA"},{"name":"Challenge Institute for Quantum Computation, University of California, Berkeley, CA 94720, USA"}]},{"given":"Hongkang","family":"Ni","sequence":"additional","affiliation":[{"name":"Institute for Computational and Mathematical Engineering (ICME), Stanford University, Stanford, CA 94305, USA."}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1321-2649","authenticated-orcid":false,"given":"Jiasu","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of California, Berkeley, CA 94720, USA."}]}],"member":"9598","published-online":{"date-parts":[[2024,12,10]]},"reference":[{"key":"0","unstructured":"M. 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