• Jara Juani Bermejo Vega

    Jara Juani Bermejo Vega

    Licenciada en Física, Ingeniera Ténica Informática de Sistemas

    ORCID

    SCOPUS

    Web of Science

    Email

    Sitio Web

    X (antes Twitter)

    Linkedin

    YouTube

    Centros de trabajo

    Facultad de Ciencias de la Universidad de Granada, Departamento de Física de la Materia, Área de Materia

    Biografía:

    PhD in physics and computer science (specialization on quantum computing) from the Technical University of Munich. Ramón y Cajal Researcher and HORIZON RIA Principal Investigator at University of Granada. Marie Curie - Athenea3i at the University of Granada, Spain (2019-2022). Postdoctoral researcher at the Free University of Berlin, Germany (2016-2019). Predoctoral researcher at the Max Planck Institute for Quantum Optics Munich, Germany (2010-2015). She has a double degree in Physics and Technical Engineering in Computer Science from the University of Salamanca, Spain (2005-2010). Juani Bermejo-Vega is an activist for rights, equality and inclusion in science. She is co-founder and co-organizer of the Q-turn inclusive quantum information conference (2018-2020) and the Equal Opportunity Group of the Max-Planck PhDnet (2014-2017). https://es.wikipedia.org/wiki/Juani_Bermejo_Vega

    Publicaciones y Citas Destacadas (Google Scholar)

    Ver más en Google Scholar

    Desde Zotero

    6397418 Bermejo-Vega 1 vancouver 20 date asc 1 1 1 71201463 https://quantumgranada.com/wp-content/plugins/zotpress/
    %7B%22status%22%3A%22success%22%2C%22updateneeded%22%3Afalse%2C%22instance%22%3Afalse%2C%22meta%22%3A%7B%22request_last%22%3A0%2C%22request_next%22%3A0%2C%22used_cache%22%3Atrue%7D%2C%22data%22%3A%5B%7B%22key%22%3A%22IIZYGAEY%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Hangleiter%20et%20al.%22%2C%22numChildren%22%3A1%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BHangleiter%20D%2C%20Bermejo-Vega%20J%2C%20Schwarz%20M%2C%20Eisert%20J.%20Anti-concentration%20theorems%20for%20schemes%20showing%20a%20quantum%20computational%20speedup.%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DIIZYGAEY%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Anti-concentration%20theorems%20for%20schemes%20showing%20a%20quantum%20computational%20speedup%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22D.%22%2C%22lastName%22%3A%22Hangleiter%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22J.%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22M.%22%2C%22lastName%22%3A%22Schwarz%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22J.%22%2C%22lastName%22%3A%22Eisert%22%7D%5D%2C%22abstractNote%22%3A%22%22%2C%22date%22%3A%22%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22%22%2C%22collections%22%3A%5B%22S952MD25%22%5D%2C%22dateModified%22%3A%222025-10-17T07%3A43%3A36Z%22%2C%22tags%22%3A%5B%5D%7D%7D%2C%7B%22key%22%3A%22ZRHASS2Q%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20et%20al.%22%2C%22numChildren%22%3A1%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Hangleiter%20D%2C%20Schwarz%20M%2C%20Raussendorf%20R%2C%20Eisert%20J.%20Architectures%20for%20quantum%20simulation%20showing%20a%20quantum%20speedup.%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DZRHASS2Q%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Architectures%20for%20quantum%20simulation%20showing%20a%20quantum%20speedup%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22J.%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22D.%22%2C%22lastName%22%3A%22Hangleiter%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22M.%22%2C%22lastName%22%3A%22Schwarz%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22R.%22%2C%22lastName%22%3A%22Raussendorf%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22J.%22%2C%22lastName%22%3A%22Eisert%22%7D%5D%2C%22abstractNote%22%3A%22%22%2C%22date%22%3A%22%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22%22%2C%22collections%22%3A%5B%22S952MD25%22%5D%2C%22dateModified%22%3A%222025-10-17T07%3A43%3A36Z%22%2C%22tags%22%3A%5B%5D%7D%7D%2C%7B%22key%22%3A%22Z2ZJATSD%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%22%2C%22parsedDate%22%3A%222011%22%2C%22numChildren%22%3A0%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J.%20Classical%20simulations%20of%20non-abelian%20quantum%20Fourier%20transforms.%20Master%20Thesis.%20%5Bphd%5D.%20Technical%20University%20of%20Munich%3B%202011.%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DZ2ZJATSD%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22thesis%22%2C%22title%22%3A%22Classical%20simulations%20of%20non-abelian%20quantum%20Fourier%20transforms.%20Master%20Thesis.%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%5D%2C%22abstractNote%22%3A%22%22%2C%22thesisType%22%3A%22phd%22%2C%22university%22%3A%22Technical%20University%20of%20Munich%22%2C%22date%22%3A%222011%22%2C%22language%22%3A%22%22%2C%22url%22%3A%22%22%2C%22collections%22%3A%5B%22IYETWNLW%22%5D%2C%22dateModified%22%3A%222025-10-17T08%3A02%3A20Z%22%2C%22tags%22%3A%5B%5D%7D%7D%2C%7B%22key%22%3A%228XM6V5XP%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20and%20Nest%22%2C%22parsedDate%22%3A%222013-10%22%2C%22numChildren%22%3A2%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Nest%20MV%20den.%20Classical%20simulations%20of%20Abelian-group%20normalizer%20circuits%20with%20intermediate%20measurements.%20arXiv%3A12103637%20%5Bquant-ph%5D%20%5BInternet%5D.%202013%20Oct%20%5Bcited%202021%20Aug%2013%5D%3B%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1210.3637%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1210.3637%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3D8XM6V5XP%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Classical%20simulations%20of%20Abelian-group%20normalizer%20circuits%20with%20intermediate%20measurements%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maarten%20Van%20den%22%2C%22lastName%22%3A%22Nest%22%7D%5D%2C%22abstractNote%22%3A%22Quantum%20normalizer%20circuits%20were%20recently%20introduced%20as%20generalizations%20of%20Clifford%20circuits%20%5BarXiv%3A1201.4867%5D%3A%20a%20normalizer%20circuit%20over%20a%20finite%20Abelian%20group%20%24G%24%20is%20composed%20of%20the%20quantum%20Fourier%20transform%20%28QFT%29%20over%20G%2C%20together%20with%20gates%20which%20compute%20quadratic%20functions%20and%20automorphisms.%20In%20%5BarXiv%3A1201.4867%5D%20it%20was%20shown%20that%20every%20normalizer%20circuit%20can%20be%20simulated%20efficiently%20classically.%20This%20result%20provides%20a%20nontrivial%20example%20of%20a%20family%20of%20quantum%20circuits%20that%20cannot%20yield%20exponential%20speed-ups%20in%20spite%20of%20usage%20of%20the%20QFT%2C%20the%20latter%20being%20a%20central%20quantum%20algorithmic%20primitive.%20Here%20we%20extend%20the%20aforementioned%20result%20in%20several%20ways.%20Most%20importantly%2C%20we%20show%20that%20normalizer%20circuits%20supplemented%20with%20intermediate%20measurements%20can%20also%20be%20simulated%20efficiently%20classically%2C%20even%20when%20the%20computation%20proceeds%20adaptively.%20This%20yields%20a%20generalization%20of%20the%20Gottesman-Knill%20theorem%20%28valid%20for%20n-qubit%20Clifford%20operations%20%5Bquant-ph%5C%2F9705052%2C%20quant-ph%5C%2F9807006%5D%20to%20quantum%20circuits%20described%20by%20arbitrary%20finite%20Abelian%20groups.%20Moreover%2C%20our%20simulations%20are%20twofold%3A%20we%20present%20efficient%20classical%20algorithms%20to%20sample%20the%20measurement%20probability%20distribution%20of%20any%20adaptive-normalizer%20computation%2C%20as%20well%20as%20to%20compute%20the%20amplitudes%20of%20the%20state%20vector%20in%20every%20step%20of%20it.%20Finally%20we%20develop%20a%20generalization%20of%20the%20stabilizer%20formalism%20%5Bquant-ph%5C%2F9705052%2C%20quant-ph%5C%2F9807006%5D%20relative%20to%20arbitrary%20finite%20Abelian%20groups%3A%20for%20example%20we%20characterize%20how%20to%20update%20stabilizers%20under%20generalized%20Pauli%20measurements%20and%20provide%20a%20normal%20form%20of%20the%20amplitudes%20of%20generalized%20stabilizer%20states%20using%20quadratic%20functions%20and%20subgroup%20cosets.%22%2C%22date%22%3A%222013-10%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1210.3637%22%2C%22collections%22%3A%5B%22N8KUR9ZB%22%5D%2C%22dateModified%22%3A%222021-08-13T18%3A13%3A20Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Computer%20Science%20-%20Computational%20Complexity%22%7D%2C%7B%22tag%22%3A%22Quantum%20Physics%22%7D%5D%7D%7D%2C%7B%22key%22%3A%22A2NHLJZJ%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20and%20Nest%22%2C%22parsedDate%22%3A%222013-10-22%22%2C%22numChildren%22%3A2%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Nest%20MV%20den.%20Classical%20simulations%20of%20Abelian-group%20normalizer%20circuits%20with%20intermediate%20measurements.%20arXiv%3A12103637%20%5Bquant-ph%5D%20%5BInternet%5D.%202013%20Oct%2022%20%5Bcited%202021%20Aug%2013%5D%3B%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1210.3637%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1210.3637%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DA2NHLJZJ%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Classical%20simulations%20of%20Abelian-group%20normalizer%20circuits%20with%20intermediate%20measurements%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maarten%20Van%20den%22%2C%22lastName%22%3A%22Nest%22%7D%5D%2C%22abstractNote%22%3A%22Quantum%20normalizer%20circuits%20were%20recently%20introduced%20as%20generalizations%20of%20Clifford%20circuits%20%5BarXiv%3A1201.4867%5D%3A%20a%20normalizer%20circuit%20over%20a%20finite%20Abelian%20group%20%24G%24%20is%20composed%20of%20the%20quantum%20Fourier%20transform%20%28QFT%29%20over%20G%2C%20together%20with%20gates%20which%20compute%20quadratic%20functions%20and%20automorphisms.%20In%20%5BarXiv%3A1201.4867%5D%20it%20was%20shown%20that%20every%20normalizer%20circuit%20can%20be%20simulated%20efficiently%20classically.%20This%20result%20provides%20a%20nontrivial%20example%20of%20a%20family%20of%20quantum%20circuits%20that%20cannot%20yield%20exponential%20speed-ups%20in%20spite%20of%20usage%20of%20the%20QFT%2C%20the%20latter%20being%20a%20central%20quantum%20algorithmic%20primitive.%20Here%20we%20extend%20the%20aforementioned%20result%20in%20several%20ways.%20Most%20importantly%2C%20we%20show%20that%20normalizer%20circuits%20supplemented%20with%20intermediate%20measurements%20can%20also%20be%20simulated%20efficiently%20classically%2C%20even%20when%20the%20computation%20proceeds%20adaptively.%20This%20yields%20a%20generalization%20of%20the%20Gottesman-Knill%20theorem%20%28valid%20for%20n-qubit%20Clifford%20operations%20%5Bquant-ph%5C%2F9705052%2C%20quant-ph%5C%2F9807006%5D%20to%20quantum%20circuits%20described%20by%20arbitrary%20finite%20Abelian%20groups.%20Moreover%2C%20our%20simulations%20are%20twofold%3A%20we%20present%20efficient%20classical%20algorithms%20to%20sample%20the%20measurement%20probability%20distribution%20of%20any%20adaptive-normalizer%20computation%2C%20as%20well%20as%20to%20compute%20the%20amplitudes%20of%20the%20state%20vector%20in%20every%20step%20of%20it.%20Finally%20we%20develop%20a%20generalization%20of%20the%20stabilizer%20formalism%20%5Bquant-ph%5C%2F9705052%2C%20quant-ph%5C%2F9807006%5D%20relative%20to%20arbitrary%20finite%20Abelian%20groups%3A%20for%20example%20we%20characterize%20how%20to%20update%20stabilizers%20under%20generalized%20Pauli%20measurements%20and%20provide%20a%20normal%20form%20of%20the%20amplitudes%20of%20generalized%20stabilizer%20states%20using%20quadratic%20functions%20and%20subgroup%20cosets.%22%2C%22date%22%3A%222013-10-22%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1210.3637%22%2C%22collections%22%3A%5B%22Y84PES7X%22%5D%2C%22dateModified%22%3A%222021-08-13T15%3A40%3A48Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Computer%20Science%20-%20Computational%20Complexity%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Quantum%20Physics%22%2C%22type%22%3A1%7D%5D%7D%7D%2C%7B%22key%22%3A%22XC5PEQR9%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20and%20Van%20Den%20Nest%22%2C%22parsedDate%22%3A%222014%22%2C%22numChildren%22%3A0%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Van%20Den%20Nest%20M.%20Classical%20simulations%20of%20Abelian-group%20normalizer%20circuits%20with%20intermediate%20measurements.%20Quantum%20Info%20Comput.%202014%3B14%283%26%23x2013%3B4%29.%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DXC5PEQR9%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Classical%20simulations%20of%20Abelian-group%20normalizer%20circuits%20with%20intermediate%20measurements%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maarten%22%2C%22lastName%22%3A%22Van%20Den%20Nest%22%7D%5D%2C%22abstractNote%22%3A%22%22%2C%22date%22%3A%222014%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22%22%2C%22collections%22%3A%5B%22IYETWNLW%22%5D%2C%22dateModified%22%3A%222025-10-17T07%3A59%3A06Z%22%2C%22tags%22%3A%5B%5D%7D%7D%2C%7B%22key%22%3A%22FCHBPJDJ%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20et%20al.%22%2C%22parsedDate%22%3A%222014-09%22%2C%22numChildren%22%3A2%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Lin%20CYY%2C%20Nest%20MV%20den.%20The%20computational%20power%20of%20normalizer%20circuits%20over%20black-box%20groups.%20arXiv%3A14094800%20%5Bquant-ph%5D%20%5BInternet%5D.%202014%20Sept%20%5Bcited%202021%20Aug%2013%5D%3B%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.4800%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.4800%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DFCHBPJDJ%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22The%20computational%20power%20of%20normalizer%20circuits%20over%20black-box%20groups%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Cedric%20Yen-Yu%22%2C%22lastName%22%3A%22Lin%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maarten%20Van%20den%22%2C%22lastName%22%3A%22Nest%22%7D%5D%2C%22abstractNote%22%3A%22This%20work%20presents%20a%20precise%20connection%20between%20Clifford%20circuits%2C%20Shor%26%23039%3Bs%20factoring%20algorithm%20and%20several%20other%20famous%20quantum%20algorithms%20with%20exponential%20quantum%20speed-ups%20for%20solving%20Abelian%20hidden%20subgroup%20problems.%20We%20show%20that%20all%20these%20different%20forms%20of%20quantum%20computation%20belong%20to%20a%20common%20new%20restricted%20model%20of%20quantum%20operations%20that%20we%20call%20%5C%5Ctextbackslashemph%7Bblack-box%20normalizer%20circuits%7D.%20To%20define%20these%2C%20we%20extend%20the%20previous%20model%20of%20normalizer%20circuits%20%5BarXiv%3A1201.4867v1%2CarXiv%3A1210.3637%2CarXiv%3A1409.3208%5D%2C%20which%20are%20built%20of%20quantum%20Fourier%20transforms%2C%20group%20automorphism%20and%20quadratic%20phase%20gates%20associated%20to%20an%20Abelian%20group%20%24G%24.%20In%20previous%20works%2C%20the%20group%20%24G%24%20is%20always%20given%20in%20an%20explicitly%20decomposed%20form.%20In%20our%20model%2C%20we%20remove%20this%20assumption%20and%20allow%20%24G%24%20to%20be%20a%20black-box%20group.%20While%20standard%20normalizer%20circuits%20were%20shown%20to%20be%20efficiently%20classically%20simulable%20%5BarXiv%3A1201.4867v1%2CarXiv%3A1210.3637%2CarXiv%3A1409.3208%5D%2C%20we%20find%20that%20normalizer%20circuits%20are%20powerful%20enough%20to%20factorize%20and%20solve%20classically-hard%20problems%20in%20the%20black-box%20setting.%20We%20further%20set%20upper%20limits%20to%20their%20computational%20power%20by%20showing%20that%20decomposing%20finite%20Abelian%20groups%20is%20complete%20for%20the%20associated%20complexity%20class.%20In%20particular%2C%20solving%20this%20problem%20renders%20black-box%20normalizer%20circuits%20efficiently%20classically%20simulable%20by%20exploiting%20the%20generalized%20stabilizer%20formalism%20in%20%5BarXiv%3A1201.4867v1%2CarXiv%3A1210.3637%2CarXiv%3A1409.3208%5D.%20Lastly%2C%20we%20employ%20our%20connection%20to%20draw%20a%20few%20practical%20implications%20for%20quantum%20algorithm%20design%3A%20namely%2C%20we%20give%20a%20no-go%20theorem%20for%20finding%20new%20quantum%20algorithms%20with%20black-box%20normalizer%20circuits%2C%20a%20universality%20result%20for%20low-depth%20normalizer%20circuits%2C%20and%20identify%20two%20other%20complete%20problems.%22%2C%22date%22%3A%222014-09%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.4800%22%2C%22collections%22%3A%5B%22N8KUR9ZB%22%5D%2C%22dateModified%22%3A%222021-08-13T18%3A13%3A36Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Computer%20Science%20-%20Computational%20Complexity%22%7D%2C%7B%22tag%22%3A%22Quantum%20Physics%22%7D%5D%7D%7D%2C%7B%22key%22%3A%22UKV8FBRS%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20et%20al.%22%2C%22parsedDate%22%3A%222014-09-16%22%2C%22numChildren%22%3A2%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Lin%20CYY%2C%20Nest%20MV%20den.%20The%20computational%20power%20of%20normalizer%20circuits%20over%20black-box%20groups.%20arXiv%3A14094800%20%5Bquant-ph%5D%20%5BInternet%5D.%202014%20Sept%2016%20%5Bcited%202021%20Aug%2013%5D%3B%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.4800%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.4800%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DUKV8FBRS%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22The%20computational%20power%20of%20normalizer%20circuits%20over%20black-box%20groups%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Cedric%20Yen-Yu%22%2C%22lastName%22%3A%22Lin%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maarten%20Van%20den%22%2C%22lastName%22%3A%22Nest%22%7D%5D%2C%22abstractNote%22%3A%22This%20work%20presents%20a%20precise%20connection%20between%20Clifford%20circuits%2C%20Shor%26%23039%3Bs%20factoring%20algorithm%20and%20several%20other%20famous%20quantum%20algorithms%20with%20exponential%20quantum%20speed-ups%20for%20solving%20Abelian%20hidden%20subgroup%20problems.%20We%20show%20that%20all%20these%20different%20forms%20of%20quantum%20computation%20belong%20to%20a%20common%20new%20restricted%20model%20of%20quantum%20operations%20that%20we%20call%20%5C%5Cemph%7Bblack-box%20normalizer%20circuits%7D.%20To%20define%20these%2C%20we%20extend%20the%20previous%20model%20of%20normalizer%20circuits%20%5BarXiv%3A1201.4867v1%2CarXiv%3A1210.3637%2CarXiv%3A1409.3208%5D%2C%20which%20are%20built%20of%20quantum%20Fourier%20transforms%2C%20group%20automorphism%20and%20quadratic%20phase%20gates%20associated%20to%20an%20Abelian%20group%20%24G%24.%20In%20previous%20works%2C%20the%20group%20%24G%24%20is%20always%20given%20in%20an%20explicitly%20decomposed%20form.%20In%20our%20model%2C%20we%20remove%20this%20assumption%20and%20allow%20%24G%24%20to%20be%20a%20black-box%20group.%20While%20standard%20normalizer%20circuits%20were%20shown%20to%20be%20efficiently%20classically%20simulable%20%5BarXiv%3A1201.4867v1%2CarXiv%3A1210.3637%2CarXiv%3A1409.3208%5D%2C%20we%20find%20that%20normalizer%20circuits%20are%20powerful%20enough%20to%20factorize%20and%20solve%20classically-hard%20problems%20in%20the%20black-box%20setting.%20We%20further%20set%20upper%20limits%20to%20their%20computational%20power%20by%20showing%20that%20decomposing%20finite%20Abelian%20groups%20is%20complete%20for%20the%20associated%20complexity%20class.%20In%20particular%2C%20solving%20this%20problem%20renders%20black-box%20normalizer%20circuits%20efficiently%20classically%20simulable%20by%20exploiting%20the%20generalized%20stabilizer%20formalism%20in%20%5BarXiv%3A1201.4867v1%2CarXiv%3A1210.3637%2CarXiv%3A1409.3208%5D.%20Lastly%2C%20we%20employ%20our%20connection%20to%20draw%20a%20few%20practical%20implications%20for%20quantum%20algorithm%20design%3A%20namely%2C%20we%20give%20a%20no-go%20theorem%20for%20finding%20new%20quantum%20algorithms%20with%20black-box%20normalizer%20circuits%2C%20a%20universality%20result%20for%20low-depth%20normalizer%20circuits%2C%20and%20identify%20two%20other%20complete%20problems.%22%2C%22date%22%3A%222014-09-16%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.4800%22%2C%22collections%22%3A%5B%22Y84PES7X%22%5D%2C%22dateModified%22%3A%222021-08-13T17%3A55%3A39Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Computer%20Science%20-%20Computational%20Complexity%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Quantum%20Physics%22%2C%22type%22%3A1%7D%5D%7D%7D%2C%7B%22key%22%3A%22E55ZK789%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20et%20al.%22%2C%22parsedDate%22%3A%222015-01%22%2C%22numChildren%22%3A2%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Lin%20CYY%2C%20Nest%20MV%20den.%20Normalizer%20circuits%20and%20a%20Gottesman-Knill%20theorem%20for%20infinite-dimensional%20systems.%20arXiv%3A14093208%20%5Bmath-ph%2C%20physics%3Aquant-ph%5D%20%5BInternet%5D.%202015%20Jan%20%5Bcited%202021%20Aug%2013%5D%3B%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.3208%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.3208%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DE55ZK789%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Normalizer%20circuits%20and%20a%20Gottesman-Knill%20theorem%20for%20infinite-dimensional%20systems%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Cedric%20Yen-Yu%22%2C%22lastName%22%3A%22Lin%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maarten%20Van%20den%22%2C%22lastName%22%3A%22Nest%22%7D%5D%2C%22abstractNote%22%3A%22%24%5C%5Ctextbackslashtextit%7BNormalizer%20circuits%7D%24%20%5B1%2C2%5D%20are%20generalized%20Clifford%20circuits%20that%20act%20on%20arbitrary%20finite-dimensional%20systems%20%24%5C%5Ctextbackslashmathcal%7BH%7D_%7Bd_1%7D%5C%5Ctextbackslashotimes%20...%20%5C%5Ctextbackslashotimes%20%5C%5Ctextbackslashmathcal%7BH%7D_%7Bd_n%7D%24%20with%20a%20standard%20basis%20labeled%20by%20the%20elements%20of%20a%20finite%20Abelian%20group%20%24G%3D%5C%5Ctextbackslashmathbb%7BZ%7D_%7Bd_1%7D%5C%5Ctextbackslashtimes...%20%5C%5Ctextbackslashtimes%20%5C%5Ctextbackslashmathbb%7BZ%7D_%7Bd_n%7D%24.%20Normalizer%20gates%20implement%20operations%20associated%20with%20the%20group%20%24G%24%20and%20can%20be%20of%20three%20types%3A%20quantum%20Fourier%20transforms%2C%20group%20automorphism%20gates%20and%20quadratic%20phase%20gates.%20In%20this%20work%2C%20we%20extend%20the%20normalizer%20formalism%20%5B1%2C2%5D%20to%20infinite%20dimensions%2C%20by%20allowing%20normalizer%20gates%20to%20act%20on%20systems%20of%20the%20form%20%24%5C%5Ctextbackslashmathcal%7BH%7D_%5C%5Ctextbackslashmathbb%7BZ%7D%5Cu02c6%7B%5C%5Ctextbackslashotimes%20a%7D%24%3A%20each%20factor%20%24%5C%5Ctextbackslashmathcal%7BH%7D_%5C%5Ctextbackslashmathbb%7BZ%7D%24%20has%20a%20standard%20basis%20labeled%20by%20%24%5C%5Ctextbackslashtextit%7Bintegers%7D%24%20%24%5C%5Ctextbackslashmathbb%7BZ%7D%24%2C%20and%20a%20Fourier%20basis%20labeled%20by%20%24%5C%5Ctextbackslashtextit%7Bangles%7D%24%2C%20elements%20of%20the%20circle%20group%20%24%5C%5Ctextbackslashmathbb%7BT%7D%24.%20Normalizer%20circuits%20become%20hybrid%20quantum%20circuits%20acting%20both%20on%20continuous-%20and%20discrete-variable%20systems.%20We%20show%20that%20infinite-dimensional%20normalizer%20circuits%20can%20be%20efficiently%20simulated%20classically%20with%20a%20generalized%20%24%5C%5Ctextbackslashtextit%7Bstabilizer%20formalism%7D%24%20for%20Hilbert%20spaces%20associated%20with%20groups%20of%20the%20form%20%24%5C%5Ctextbackslashmathbb%7BZ%7D%5Cu02c6a%5C%5Ctextbackslashtimes%20%5C%5Ctextbackslashmathbb%7BT%7D%5Cu02c6b%20%5C%5Ctextbackslashtimes%20%5C%5Ctextbackslashmathbb%7BZ%7D_%7Bd_1%7D%5C%5Ctextbackslashtimes...%5C%5Ctextbackslashtimes%20%5C%5Ctextbackslashmathbb%7BZ%7D_%7Bd_n%7D%24.%20We%20develop%20new%20techniques%20to%20track%20stabilizer-groups%20based%20on%20normal%20forms%20for%20group%20automorphisms%20and%20quadratic%20functions.%20We%20use%20our%20normal%20forms%20to%20reduce%20the%20problem%20of%20simulating%20normalizer%20circuits%20to%20that%20of%20finding%20general%20solutions%20of%20systems%20of%20mixed%20real-integer%20linear%20equations%20%5B3%5D%20and%20exploit%20this%20fact%20to%20devise%20a%20robust%20simulation%20algorithm%3A%20the%20latter%20remains%20efficient%20even%20in%20pathological%20cases%20where%20stabilizer%20groups%20become%20infinite%2C%20uncountable%20and%20non-compact.%20The%20techniques%20developed%20in%20this%20paper%20might%20find%20applications%20in%20the%20study%20of%20fault-tolerant%20quantum%20computation%20with%20superconducting%20qubits%20%5B4%2C5%5D.%22%2C%22date%22%3A%222015-01%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.3208%22%2C%22collections%22%3A%5B%22N8KUR9ZB%22%5D%2C%22dateModified%22%3A%222021-08-13T18%3A13%3A34Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Computer%20Science%20-%20Computational%20Complexity%22%7D%2C%7B%22tag%22%3A%22Mathematical%20Physics%22%7D%2C%7B%22tag%22%3A%22Quantum%20Physics%22%7D%5D%7D%7D%2C%7B%22key%22%3A%22IRWGBZZU%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20et%20al.%22%2C%22parsedDate%22%3A%222015-01-20%22%2C%22numChildren%22%3A2%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Lin%20CYY%2C%20Nest%20MV%20den.%20Normalizer%20circuits%20and%20a%20Gottesman-Knill%20theorem%20for%20infinite-dimensional%20systems.%20arXiv%3A14093208%20%5Bmath-ph%2C%20physics%3Aquant-ph%5D%20%5BInternet%5D.%202015%20Jan%2020%20%5Bcited%202021%20Aug%2013%5D%3B%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.3208%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.3208%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DIRWGBZZU%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Normalizer%20circuits%20and%20a%20Gottesman-Knill%20theorem%20for%20infinite-dimensional%20systems%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Cedric%20Yen-Yu%22%2C%22lastName%22%3A%22Lin%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maarten%20Van%20den%22%2C%22lastName%22%3A%22Nest%22%7D%5D%2C%22abstractNote%22%3A%22%24%5C%5Ctextit%7BNormalizer%20circuits%7D%24%20%5B1%2C2%5D%20are%20generalized%20Clifford%20circuits%20that%20act%20on%20arbitrary%20finite-dimensional%20systems%20%24%5C%5Cmathcal%7BH%7D_%7Bd_1%7D%5C%5Cotimes%20...%20%5C%5Cotimes%20%5C%5Cmathcal%7BH%7D_%7Bd_n%7D%24%20with%20a%20standard%20basis%20labeled%20by%20the%20elements%20of%20a%20finite%20Abelian%20group%20%24G%3D%5C%5Cmathbb%7BZ%7D_%7Bd_1%7D%5C%5Ctimes...%20%5C%5Ctimes%20%5C%5Cmathbb%7BZ%7D_%7Bd_n%7D%24.%20Normalizer%20gates%20implement%20operations%20associated%20with%20the%20group%20%24G%24%20and%20can%20be%20of%20three%20types%3A%20quantum%20Fourier%20transforms%2C%20group%20automorphism%20gates%20and%20quadratic%20phase%20gates.%20In%20this%20work%2C%20we%20extend%20the%20normalizer%20formalism%20%5B1%2C2%5D%20to%20infinite%20dimensions%2C%20by%20allowing%20normalizer%20gates%20to%20act%20on%20systems%20of%20the%20form%20%24%5C%5Cmathcal%7BH%7D_%5C%5Cmathbb%7BZ%7D%5E%7B%5C%5Cotimes%20a%7D%24%3A%20each%20factor%20%24%5C%5Cmathcal%7BH%7D_%5C%5Cmathbb%7BZ%7D%24%20has%20a%20standard%20basis%20labeled%20by%20%24%5C%5Ctextit%7Bintegers%7D%24%20%24%5C%5Cmathbb%7BZ%7D%24%2C%20and%20a%20Fourier%20basis%20labeled%20by%20%24%5C%5Ctextit%7Bangles%7D%24%2C%20elements%20of%20the%20circle%20group%20%24%5C%5Cmathbb%7BT%7D%24.%20Normalizer%20circuits%20become%20hybrid%20quantum%20circuits%20acting%20both%20on%20continuous-%20and%20discrete-variable%20systems.%20We%20show%20that%20infinite-dimensional%20normalizer%20circuits%20can%20be%20efficiently%20simulated%20classically%20with%20a%20generalized%20%24%5C%5Ctextit%7Bstabilizer%20formalism%7D%24%20for%20Hilbert%20spaces%20associated%20with%20groups%20of%20the%20form%20%24%5C%5Cmathbb%7BZ%7D%5Ea%5C%5Ctimes%20%5C%5Cmathbb%7BT%7D%5Eb%20%5C%5Ctimes%20%5C%5Cmathbb%7BZ%7D_%7Bd_1%7D%5C%5Ctimes...%5C%5Ctimes%20%5C%5Cmathbb%7BZ%7D_%7Bd_n%7D%24.%20We%20develop%20new%20techniques%20to%20track%20stabilizer-groups%20based%20on%20normal%20forms%20for%20group%20automorphisms%20and%20quadratic%20functions.%20We%20use%20our%20normal%20forms%20to%20reduce%20the%20problem%20of%20simulating%20normalizer%20circuits%20to%20that%20of%20finding%20general%20solutions%20of%20systems%20of%20mixed%20real-integer%20linear%20equations%20%5B3%5D%20and%20exploit%20this%20fact%20to%20devise%20a%20robust%20simulation%20algorithm%3A%20the%20latter%20remains%20efficient%20even%20in%20pathological%20cases%20where%20stabilizer%20groups%20become%20infinite%2C%20uncountable%20and%20non-compact.%20The%20techniques%20developed%20in%20this%20paper%20might%20find%20applications%20in%20the%20study%20of%20fault-tolerant%20quantum%20computation%20with%20superconducting%20qubits%20%5B4%2C5%5D.%22%2C%22date%22%3A%222015-01-20%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.3208%22%2C%22collections%22%3A%5B%22Y84PES7X%22%5D%2C%22dateModified%22%3A%222021-08-13T17%3A55%3A30Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Computer%20Science%20-%20Computational%20Complexity%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Mathematical%20Physics%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Quantum%20Physics%22%2C%22type%22%3A1%7D%5D%7D%7D%2C%7B%22key%22%3A%2257P7D7V9%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20et%20al.%22%2C%22parsedDate%22%3A%222015-01-20%22%2C%22numChildren%22%3A2%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Lin%20CYY%2C%20Nest%20MV%20den.%20Normalizer%20circuits%20and%20a%20Gottesman-Knill%20theorem%20for%20infinite-dimensional%20systems%20%5BInternet%5D.%20arXiv%3B%202015%20%5Bcited%202025%20Oct%2017%5D.%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.3208%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.3208%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3D57P7D7V9%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22preprint%22%2C%22title%22%3A%22Normalizer%20circuits%20and%20a%20Gottesman-Knill%20theorem%20for%20infinite-dimensional%20systems%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Cedric%20Yen-Yu%22%2C%22lastName%22%3A%22Lin%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maarten%20Van%20den%22%2C%22lastName%22%3A%22Nest%22%7D%5D%2C%22abstractNote%22%3A%22%24%5C%5Ctextit%7BNormalizer%20circuits%7D%24%20%5B1%2C2%5D%20are%20generalized%20Clifford%20circuits%20that%20act%20on%20arbitrary%20finite-dimensional%20systems%20%24%5C%5Cmathcal%7BH%7D_%7Bd_1%7D%5C%5Cotimes%20...%20%5C%5Cotimes%20%5C%5Cmathcal%7BH%7D_%7Bd_n%7D%24%20with%20a%20standard%20basis%20labeled%20by%20the%20elements%20of%20a%20finite%20Abelian%20group%20%24G%3D%5C%5Cmathbb%7BZ%7D_%7Bd_1%7D%5C%5Ctimes...%20%5C%5Ctimes%20%5C%5Cmathbb%7BZ%7D_%7Bd_n%7D%24.%20Normalizer%20gates%20implement%20operations%20associated%20with%20the%20group%20%24G%24%20and%20can%20be%20of%20three%20types%3A%20quantum%20Fourier%20transforms%2C%20group%20automorphism%20gates%20and%20quadratic%20phase%20gates.%20In%20this%20work%2C%20we%20extend%20the%20normalizer%20formalism%20%5B1%2C2%5D%20to%20infinite%20dimensions%2C%20by%20allowing%20normalizer%20gates%20to%20act%20on%20systems%20of%20the%20form%20%24%5C%5Cmathcal%7BH%7D_%5C%5Cmathbb%7BZ%7D%5E%7B%5C%5Cotimes%20a%7D%24%3A%20each%20factor%20%24%5C%5Cmathcal%7BH%7D_%5C%5Cmathbb%7BZ%7D%24%20has%20a%20standard%20basis%20labeled%20by%20%24%5C%5Ctextit%7Bintegers%7D%24%20%24%5C%5Cmathbb%7BZ%7D%24%2C%20and%20a%20Fourier%20basis%20labeled%20by%20%24%5C%5Ctextit%7Bangles%7D%24%2C%20elements%20of%20the%20circle%20group%20%24%5C%5Cmathbb%7BT%7D%24.%20Normalizer%20circuits%20become%20hybrid%20quantum%20circuits%20acting%20both%20on%20continuous-%20and%20discrete-variable%20systems.%20We%20show%20that%20infinite-dimensional%20normalizer%20circuits%20can%20be%20efficiently%20simulated%20classically%20with%20a%20generalized%20%24%5C%5Ctextit%7Bstabilizer%20formalism%7D%24%20for%20Hilbert%20spaces%20associated%20with%20groups%20of%20the%20form%20%24%5C%5Cmathbb%7BZ%7D%5Ea%5C%5Ctimes%20%5C%5Cmathbb%7BT%7D%5Eb%20%5C%5Ctimes%20%5C%5Cmathbb%7BZ%7D_%7Bd_1%7D%5C%5Ctimes...%5C%5Ctimes%20%5C%5Cmathbb%7BZ%7D_%7Bd_n%7D%24.%20We%20develop%20new%20techniques%20to%20track%20stabilizer-groups%20based%20on%20normal%20forms%20for%20group%20automorphisms%20and%20quadratic%20functions.%20We%20use%20our%20normal%20forms%20to%20reduce%20the%20problem%20of%20simulating%20normalizer%20circuits%20to%20that%20of%20finding%20general%20solutions%20of%20systems%20of%20mixed%20real-integer%20linear%20equations%20%5B3%5D%20and%20exploit%20this%20fact%20to%20devise%20a%20robust%20simulation%20algorithm%3A%20the%20latter%20remains%20efficient%20even%20in%20pathological%20cases%20where%20stabilizer%20groups%20become%20infinite%2C%20uncountable%20and%20non-compact.%20The%20techniques%20developed%20in%20this%20paper%20might%20find%20applications%20in%20the%20study%20of%20fault-tolerant%20quantum%20computation%20with%20superconducting%20qubits%20%5B4%2C5%5D.%22%2C%22genre%22%3A%22%22%2C%22repository%22%3A%22arXiv%22%2C%22archiveID%22%3A%22arXiv%3A1409.3208%22%2C%22date%22%3A%222015-01-20%22%2C%22DOI%22%3A%2210.48550%5C%2FarXiv.1409.3208%22%2C%22citationKey%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1409.3208%22%2C%22language%22%3A%22%22%2C%22collections%22%3A%5B%5D%2C%22dateModified%22%3A%222025-10-17T07%3A41%3A31Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Computer%20Science%20-%20Computational%20Complexity%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Mathematical%20Physics%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Mathematics%20-%20Mathematical%20Physics%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Quantum%20Physics%22%2C%22type%22%3A1%7D%5D%7D%7D%2C%7B%22key%22%3A%22GJ3BG9ZR%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20and%20Zatloukal%22%2C%22parsedDate%22%3A%222015-09%22%2C%22numChildren%22%3A1%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Zatloukal%20KC.%20Abelian%20Hypergroups%20and%20Quantum%20Computation.%20arXiv%3A150905806%20%5Bmath-ph%2C%20physics%3Aquant-ph%5D%20%5BInternet%5D.%202015%20Sept%20%5Bcited%202021%20Aug%2013%5D%3B%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1509.05806%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1509.05806%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DGJ3BG9ZR%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Abelian%20Hypergroups%20and%20Quantum%20Computation%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Kevin%20C.%22%2C%22lastName%22%3A%22Zatloukal%22%7D%5D%2C%22abstractNote%22%3A%22Motivated%20by%20a%20connection%2C%20described%20here%20for%20the%20first%20time%2C%20between%20the%20hidden%20normal%20subgroup%20problem%20%28HNSP%29%20and%20abelian%20hypergroups%20%28algebraic%20objects%20that%20model%20collisions%20of%20physical%20particles%29%2C%20we%20develop%20a%20stabilizer%20formalism%20using%20abelian%20hypergroups%20and%20an%20associated%20classical%20simulation%20theorem%20%28a%20la%20Gottesman-Knill%29.%20Using%20these%20tools%2C%20we%20develop%20the%20first%20provably%20efficient%20quantum%20algorithm%20for%20finding%20hidden%20subhypergroups%20of%20nilpotent%20abelian%20hypergroups%20and%2C%20via%20the%20aforementioned%20connection%2C%20a%20new%2C%20hypergroup-based%20algorithm%20for%20the%20HNSP%20on%20nilpotent%20groups.%20We%20also%20give%20efficient%20methods%20for%20manipulating%20non-unitary%2C%20non-monomial%20stabilizers%20and%20an%20adaptive%20Fourier%20sampling%20technique%20of%20general%20interest.%22%2C%22date%22%3A%222015-09%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1509.05806%22%2C%22collections%22%3A%5B%22N8KUR9ZB%22%5D%2C%22dateModified%22%3A%222021-08-13T18%3A13%3A19Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%2220N20%22%7D%2C%7B%22tag%22%3A%2243A62%22%7D%2C%7B%22tag%22%3A%22B.8.1%22%7D%2C%7B%22tag%22%3A%22Computer%20Science%20-%20Computational%20Complexity%22%7D%2C%7B%22tag%22%3A%22F.1.1%22%7D%2C%7B%22tag%22%3A%22F.1.2%22%7D%2C%7B%22tag%22%3A%22F.1.3%22%7D%2C%7B%22tag%22%3A%22F.2.1%22%7D%2C%7B%22tag%22%3A%22Mathematical%20Physics%22%7D%2C%7B%22tag%22%3A%22Quantum%20Physics%22%7D%5D%7D%7D%2C%7B%22key%22%3A%22GSZXPY8J%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20and%20Zatloukal%22%2C%22parsedDate%22%3A%222015-09-18%22%2C%22numChildren%22%3A1%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Zatloukal%20KC.%20Abelian%20Hypergroups%20and%20Quantum%20Computation.%20arXiv%3A150905806%20%5Bmath-ph%2C%20physics%3Aquant-ph%5D%20%5BInternet%5D.%202015%20Sept%2018%20%5Bcited%202021%20Aug%2013%5D%3B%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1509.05806%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1509.05806%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DGSZXPY8J%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Abelian%20Hypergroups%20and%20Quantum%20Computation%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Kevin%20C.%22%2C%22lastName%22%3A%22Zatloukal%22%7D%5D%2C%22abstractNote%22%3A%22Motivated%20by%20a%20connection%2C%20described%20here%20for%20the%20first%20time%2C%20between%20the%20hidden%20normal%20subgroup%20problem%20%28HNSP%29%20and%20abelian%20hypergroups%20%28algebraic%20objects%20that%20model%20collisions%20of%20physical%20particles%29%2C%20we%20develop%20a%20stabilizer%20formalism%20using%20abelian%20hypergroups%20and%20an%20associated%20classical%20simulation%20theorem%20%28a%20la%20Gottesman-Knill%29.%20Using%20these%20tools%2C%20we%20develop%20the%20first%20provably%20efficient%20quantum%20algorithm%20for%20finding%20hidden%20subhypergroups%20of%20nilpotent%20abelian%20hypergroups%20and%2C%20via%20the%20aforementioned%20connection%2C%20a%20new%2C%20hypergroup-based%20algorithm%20for%20the%20HNSP%20on%20nilpotent%20groups.%20We%20also%20give%20efficient%20methods%20for%20manipulating%20non-unitary%2C%20non-monomial%20stabilizers%20and%20an%20adaptive%20Fourier%20sampling%20technique%20of%20general%20interest.%22%2C%22date%22%3A%222015-09-18%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1509.05806%22%2C%22collections%22%3A%5B%22Y84PES7X%22%5D%2C%22dateModified%22%3A%222021-08-13T15%3A40%3A08Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%2243A62%2C%2020N20%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22B.8.1%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Computer%20Science%20-%20Computational%20Complexity%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22F.1.1%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22F.1.2%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22F.1.3%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22F.2.1%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Mathematical%20Physics%22%2C%22type%22%3A1%7D%2C%7B%22tag%22%3A%22Quantum%20Physics%22%2C%22type%22%3A1%7D%5D%7D%7D%2C%7B%22key%22%3A%22WL8NAGQH%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%22%2C%22parsedDate%22%3A%222016%22%2C%22numChildren%22%3A1%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J.%20Normalizer%20circuits%20and%20quantum%20computation.%20CoRR%20%5BInternet%5D.%202016%3Babs%5C%2F1611.09274.%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1611.09274%26%23039%3B%26gt%3Bhttp%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1611.09274%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DWL8NAGQH%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Normalizer%20circuits%20and%20quantum%20computation%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%5D%2C%22abstractNote%22%3A%22%22%2C%22date%22%3A%222016%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22http%3A%5C%2F%5C%2Farxiv.org%5C%2Fabs%5C%2F1611.09274%22%2C%22collections%22%3A%5B%22IYETWNLW%22%5D%2C%22dateModified%22%3A%222025-10-17T08%3A05%3A29Z%22%2C%22tags%22%3A%5B%5D%7D%7D%2C%7B%22key%22%3A%22RTU53RUC%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Bermejo-Vega%20et%20al.%22%2C%22parsedDate%22%3A%222016-04%22%2C%22numChildren%22%3A0%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBermejo-Vega%20J%2C%20Lin%20CYY%2C%20Van%20Den%20Nest%20M.%20Normalizer%20circuits%20and%20a%20Gottesman-Knill%20theorem%20for%20infinite-dimensional%20systems.%20Quantum%20Info%20Comput.%202016%20Apr%3B16%285%26%23x2013%3B6%29%3A361%26%23x2013%3B422.%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DRTU53RUC%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Normalizer%20circuits%20and%20a%20Gottesman-Knill%20theorem%20for%20infinite-dimensional%20systems%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juan%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Cedric%20Yen-Yu%22%2C%22lastName%22%3A%22Lin%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maarten%22%2C%22lastName%22%3A%22Van%20Den%20Nest%22%7D%5D%2C%22abstractNote%22%3A%22Normalizer%20circuits%20%5B1%2C%202%5D%20are%20generalized%20Clifford%20circuits%20that%20act%20on%20arbitrary%20finite-dimensional%20systems%20Hd1%20%5Cu2297%20.%20.%20.%20%5Cu2297%20Hdn%20with%20a%20standard%20basis%20labeled%20by%20the%20elements%20of%20a%20finite%20Abelian%20group%20G%20%3D%20Zd1%20%5Cu00d7%20.%20.%20.%20%5Cu00d7%20Zdn.%20Normalizer%20gates%20implement%20operations%20associated%20with%20the%20group%20G%20and%20can%20be%20of%20three%20types%3A%20quantum%20Fourier%20transforms%2C%20group%20automorphism%20gates%20and%20quadratic%20phase%20gates.%20In%20this%20work%2C%20we%20extend%20the%20normalizer%20formalism%20%5B1%2C%202%5D%20to%20infinite%20dimensions%2C%20by%20allowing%20normalizer%20gates%20to%20act%20on%20systems%20of%20the%20form%20HZ%5Cu2297a%3A%20each%20factor%20HZ%20has%20a%20standard%20basis%20labeled%20by%20integers%20Z%2C%20and%20a%20Fourier%20basis%20labeled%20by%20angles%2C%20elements%20of%20the%20circle%20group%20T.%20Normalizer%20circuits%20become%20hybrid%20quantum%20circuits%20acting%20both%20on%20continuous-%20and%20discrete-variable%20systems.%20We%20show%20that%20infinite-dimensional%20normalizer%20circuits%20can%20be%20efficiently%20simulated%20classically%20with%20a%20generalized%20stabilizer%20formalism%20for%20Hilbert%20spaces%20associated%20with%20groups%20of%20the%20form%20Za%5Cu00d7TZb%5Cu00d7Zd1%20%5Cu00d7%20.%20.%20.%20%5Cu00d7%20Zdn.%20We%20develop%20new%20techniques%20to%20track%20stabilizer-groups%20based%20on%20normal%20forms%20for%20group%20automorphisms%20and%20quadratic%20functions.%20We%20use%20our%20normal%20forms%20to%20reduce%20the%20problem%20of%20simulating%20normalizer%20circuits%20to%20that%20of%20finding%20general%20solutions%20of%20systems%20of%20mixed%20real-integer%20linear%20equations%20%5B3%5D%20and%20exploit%20this%20fact%20to%20devise%20a%20robust%20simulation%20algorithm%3A%20the%20latter%20remains%20efficient%20even%20in%20pathological%20cases%20where%20stabilizer%20groups%20become%20infinite%2C%20uncountable%20and%20non-compact.%20The%20techniques%20developed%20in%20this%20paper%20might%20find%20applications%20in%20the%20study%20of%20fault-tolerant%20quantum%20computation%20with%20superconducting%20qubits%20%5B4%2C%205%5D.%22%2C%22date%22%3A%222016-04%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%221533-7146%22%2C%22url%22%3A%22%22%2C%22collections%22%3A%5B%22IYETWNLW%22%5D%2C%22dateModified%22%3A%222025-10-17T08%3A02%3A07Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22classical%20simulations%22%7D%2C%7B%22tag%22%3A%22infinite-dimensional%20quantum%20systems%22%7D%2C%7B%22tag%22%3A%22normalizer%20circuits%22%7D%2C%7B%22tag%22%3A%22stabilizer%20formalism%22%7D%5D%7D%7D%2C%7B%22key%22%3A%22QQ7CQQCZ%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Stephen%20et%20al.%22%2C%22parsedDate%22%3A%222019-05-20%22%2C%22numChildren%22%3A1%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BStephen%20DT%2C%20Nautrup%20HP%2C%20Bermejo-Vega%20J%2C%20Eisert%20J%2C%20Raussendorf%20R.%20Subsystem%20symmetries%2C%20quantum%20cellular%20automata%2C%20and%20computational%20phases%20of%20quantum%20matter.%202019%20May%2020%20%5Bcited%202025%20Oct%2017%5D%3B%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttps%3A%5C%2F%5C%2Fquantum-journal.org%5C%2Fpapers%5C%2Fq-2019-05-20-142%5C%2F%26%23039%3B%26gt%3Bhttps%3A%5C%2F%5C%2Fquantum-journal.org%5C%2Fpapers%5C%2Fq-2019-05-20-142%5C%2F%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DQQ7CQQCZ%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Subsystem%20symmetries%2C%20quantum%20cellular%20automata%2C%20and%20computational%20phases%20of%20quantum%20matter%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22David%20T.%22%2C%22lastName%22%3A%22Stephen%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Hendrik%20Poulsen%22%2C%22lastName%22%3A%22Nautrup%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juani%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Jens%22%2C%22lastName%22%3A%22Eisert%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Robert%22%2C%22lastName%22%3A%22Raussendorf%22%7D%5D%2C%22abstractNote%22%3A%22David%20T.%20Stephen%2C%20Hendrik%20Poulsen%20Nautrup%2C%20Juani%20Bermejo-Vega%2C%20Jens%20Eisert%2C%20and%20Robert%20Raussendorf%2C%5CnQuantum%203%2C%20142%20%282019%29.%5CnQuantum%20phases%20of%20matter%20are%20resources%20for%20notions%20of%20quantum%20computation.%20In%20this%20work%2C%20we%20establish%20a%20new%20link%20between%20concepts%20of%20quantum%20information%20theory%20and%20condensed%20matter%20physics%20b%5Cu2026%22%2C%22date%22%3A%222019-05-20%22%2C%22language%22%3A%22en-GB%22%2C%22DOI%22%3A%22%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22https%3A%5C%2F%5C%2Fquantum-journal.org%5C%2Fpapers%5C%2Fq-2019-05-20-142%5C%2F%22%2C%22collections%22%3A%5B%22IYETWNLW%22%5D%2C%22dateModified%22%3A%222025-10-17T07%3A30%3A07Z%22%2C%22tags%22%3A%5B%5D%7D%7D%2C%7B%22key%22%3A%22AWZFZ4PP%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Raussendorf%20et%20al.%22%2C%22parsedDate%22%3A%222020-01%22%2C%22numChildren%22%3A3%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BRaussendorf%20R%2C%20Bermejo-Vega%20J%2C%20Tyhurst%20E%2C%20Okay%20C%2C%20Zurel%20M.%20Phase-space-simulation%20method%20for%20quantum%20computation%20with%20magic%20states%20on%20qubits.%20Physical%20Review%20A%20%5BInternet%5D.%202020%20Jan%20%5Bcited%202021%20July%209%5D%3B101%281%29%3A012350.%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttps%3A%5C%2F%5C%2Flink.aps.org%5C%2Fdoi%5C%2F10.1103%5C%2FPhysRevA.101.012350%26%23039%3B%26gt%3Bhttps%3A%5C%2F%5C%2Flink.aps.org%5C%2Fdoi%5C%2F10.1103%5C%2FPhysRevA.101.012350%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DAWZFZ4PP%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Phase-space-simulation%20method%20for%20quantum%20computation%20with%20magic%20states%20on%20qubits%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Robert%22%2C%22lastName%22%3A%22Raussendorf%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juani%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Emily%22%2C%22lastName%22%3A%22Tyhurst%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Cihan%22%2C%22lastName%22%3A%22Okay%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Michael%22%2C%22lastName%22%3A%22Zurel%22%7D%5D%2C%22abstractNote%22%3A%22We%20propose%20a%20method%20for%20classical%20simulation%20of%20finite-dimensional%20quantum%20systems%2C%20based%20on%20sampling%20from%20a%20quasiprobability%20distribution%2C%20i.e.%2C%20a%20generalized%20Wigner%20function.%20Our%20construction%20applies%20to%20all%20finite%20dimensions%2C%20with%20the%20most%20interesting%20case%20being%20that%20of%20qubits.%20For%20multiple%20qubits%2C%20we%20find%20that%20quantum%20computation%20by%20Clifford%20gates%20and%20Pauli%20measurements%20on%20magic%20states%20can%20be%20efficiently%20classically%20simulated%20if%20the%20quasiprobability%20distribution%20of%20the%20magic%20states%20is%20non-negative.%20This%20provides%20the%20so%20far%20missing%20qubit%20counterpart%20of%20the%20corresponding%20result%20%5BV.%20Veitch%20et%20al.%2C%20New%20J.%20Phys.%2014%2C%20113011%20%282012%29%5D%20applying%20only%20to%20odd%20dimension.%20Our%20approach%20is%20more%20general%20than%20previous%20ones%20based%20on%20mixtures%20of%20stabilizer%20states.%20Namely%2C%20all%20mixtures%20of%20stabilizer%20states%20can%20be%20efficiently%20simulated%2C%20but%20for%20any%20number%20of%20qubits%20there%20also%20exist%20efficiently%20simulable%20states%20outside%20the%20stabilizer%20polytope.%20Further%2C%20our%20simulation%20method%20extends%20to%20negative%20quasiprobability%20distributions%2C%20where%20it%20provides%20probability%20estimation.%20The%20simulation%20cost%20is%20then%20proportional%20to%20a%20robustness%20measure%20squared.%20For%20all%20quantum%20states%2C%20this%20robustness%20is%20smaller%20than%20or%20equal%20to%20robustness%20of%20magic.%22%2C%22date%22%3A%222020-01%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%2210.1103%5C%2FPhysRevA.101.012350%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22https%3A%5C%2F%5C%2Flink.aps.org%5C%2Fdoi%5C%2F10.1103%5C%2FPhysRevA.101.012350%22%2C%22collections%22%3A%5B%22N8KUR9ZB%22%5D%2C%22dateModified%22%3A%222021-08-13T18%3A12%3A15Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Quantum%20Physics%22%7D%5D%7D%7D%2C%7B%22key%22%3A%225VAL2MZI%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Raussendorf%20et%20al.%22%2C%22parsedDate%22%3A%222020-01%22%2C%22numChildren%22%3A3%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BRaussendorf%20R%2C%20Bermejo-Vega%20J%2C%20Tyhurst%20E%2C%20Okay%20C%2C%20Zurel%20M.%20Phase-space-simulation%20method%20for%20quantum%20computation%20with%20magic%20states%20on%20qubits.%20Physical%20Review%20A%20%5BInternet%5D.%202020%20Jan%20%5Bcited%202021%20July%209%5D%3B101%281%29%3A012350.%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttps%3A%5C%2F%5C%2Flink.aps.org%5C%2Fdoi%5C%2F10.1103%5C%2FPhysRevA.101.012350%26%23039%3B%26gt%3Bhttps%3A%5C%2F%5C%2Flink.aps.org%5C%2Fdoi%5C%2F10.1103%5C%2FPhysRevA.101.012350%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3D5VAL2MZI%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Phase-space-simulation%20method%20for%20quantum%20computation%20with%20magic%20states%20on%20qubits%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Robert%22%2C%22lastName%22%3A%22Raussendorf%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juani%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Emily%22%2C%22lastName%22%3A%22Tyhurst%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Cihan%22%2C%22lastName%22%3A%22Okay%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Michael%22%2C%22lastName%22%3A%22Zurel%22%7D%5D%2C%22abstractNote%22%3A%22We%20propose%20a%20method%20for%20classical%20simulation%20of%20finite-dimensional%20quantum%20systems%2C%20based%20on%20sampling%20from%20a%20quasiprobability%20distribution%2C%20i.e.%2C%20a%20generalized%20Wigner%20function.%20Our%20construction%20applies%20to%20all%20finite%20dimensions%2C%20with%20the%20most%20interesting%20case%20being%20that%20of%20qubits.%20For%20multiple%20qubits%2C%20we%20find%20that%20quantum%20computation%20by%20Clifford%20gates%20and%20Pauli%20measurements%20on%20magic%20states%20can%20be%20efficiently%20classically%20simulated%20if%20the%20quasiprobability%20distribution%20of%20the%20magic%20states%20is%20non-negative.%20This%20provides%20the%20so%20far%20missing%20qubit%20counterpart%20of%20the%20corresponding%20result%20%5BV.%20Veitch%20et%20al.%2C%20New%20J.%20Phys.%2014%2C%20113011%20%282012%29%5D%20applying%20only%20to%20odd%20dimension.%20Our%20approach%20is%20more%20general%20than%20previous%20ones%20based%20on%20mixtures%20of%20stabilizer%20states.%20Namely%2C%20all%20mixtures%20of%20stabilizer%20states%20can%20be%20efficiently%20simulated%2C%20but%20for%20any%20number%20of%20qubits%20there%20also%20exist%20efficiently%20simulable%20states%20outside%20the%20stabilizer%20polytope.%20Further%2C%20our%20simulation%20method%20extends%20to%20negative%20quasiprobability%20distributions%2C%20where%20it%20provides%20probability%20estimation.%20The%20simulation%20cost%20is%20then%20proportional%20to%20a%20robustness%20measure%20squared.%20For%20all%20quantum%20states%2C%20this%20robustness%20is%20smaller%20than%20or%20equal%20to%20robustness%20of%20magic.%22%2C%22date%22%3A%222020-01%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%2210.1103%5C%2FPhysRevA.101.012350%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22https%3A%5C%2F%5C%2Flink.aps.org%5C%2Fdoi%5C%2F10.1103%5C%2FPhysRevA.101.012350%22%2C%22collections%22%3A%5B%22N8KUR9ZB%22%5D%2C%22dateModified%22%3A%222021-08-13T18%3A12%3A53Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Quantum%20Physics%22%7D%5D%7D%7D%2C%7B%22key%22%3A%22AMADTE66%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Raussendorf%20et%20al.%22%2C%22parsedDate%22%3A%222020-01-31%22%2C%22numChildren%22%3A3%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BRaussendorf%20R%2C%20Bermejo-Vega%20J%2C%20Tyhurst%20E%2C%20Okay%20C%2C%20Zurel%20M.%20Phase-space-simulation%20method%20for%20quantum%20computation%20with%20magic%20states%20on%20qubits.%20Phys%20Rev%20A%20%5BInternet%5D.%202020%20Jan%2031%20%5Bcited%202021%20July%209%5D%3B101%281%29%3A012350.%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttps%3A%5C%2F%5C%2Flink.aps.org%5C%2Fdoi%5C%2F10.1103%5C%2FPhysRevA.101.012350%26%23039%3B%26gt%3Bhttps%3A%5C%2F%5C%2Flink.aps.org%5C%2Fdoi%5C%2F10.1103%5C%2FPhysRevA.101.012350%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DAMADTE66%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Phase-space-simulation%20method%20for%20quantum%20computation%20with%20magic%20states%20on%20qubits%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Robert%22%2C%22lastName%22%3A%22Raussendorf%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juani%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Emily%22%2C%22lastName%22%3A%22Tyhurst%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Cihan%22%2C%22lastName%22%3A%22Okay%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Michael%22%2C%22lastName%22%3A%22Zurel%22%7D%5D%2C%22abstractNote%22%3A%22We%20propose%20a%20method%20for%20classical%20simulation%20of%20finite-dimensional%20quantum%20systems%2C%20based%20on%20sampling%20from%20a%20quasiprobability%20distribution%2C%20i.e.%2C%20a%20generalized%20Wigner%20function.%20Our%20construction%20applies%20to%20all%20finite%20dimensions%2C%20with%20the%20most%20interesting%20case%20being%20that%20of%20qubits.%20For%20multiple%20qubits%2C%20we%20find%20that%20quantum%20computation%20by%20Clifford%20gates%20and%20Pauli%20measurements%20on%20magic%20states%20can%20be%20efficiently%20classically%20simulated%20if%20the%20quasiprobability%20distribution%20of%20the%20magic%20states%20is%20non-negative.%20This%20provides%20the%20so%20far%20missing%20qubit%20counterpart%20of%20the%20corresponding%20result%20%5BV.%20Veitch%20et%20al.%2C%20New%20J.%20Phys.%2014%2C%20113011%20%282012%29%5D%20applying%20only%20to%20odd%20dimension.%20Our%20approach%20is%20more%20general%20than%20previous%20ones%20based%20on%20mixtures%20of%20stabilizer%20states.%20Namely%2C%20all%20mixtures%20of%20stabilizer%20states%20can%20be%20efficiently%20simulated%2C%20but%20for%20any%20number%20of%20qubits%20there%20also%20exist%20efficiently%20simulable%20states%20outside%20the%20stabilizer%20polytope.%20Further%2C%20our%20simulation%20method%20extends%20to%20negative%20quasiprobability%20distributions%2C%20where%20it%20provides%20probability%20estimation.%20The%20simulation%20cost%20is%20then%20proportional%20to%20a%20robustness%20measure%20squared.%20For%20all%20quantum%20states%2C%20this%20robustness%20is%20smaller%20than%20or%20equal%20to%20robustness%20of%20magic.%22%2C%22date%22%3A%22January%2031%2C%202020%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%2210.1103%5C%2FPhysRevA.101.012350%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22https%3A%5C%2F%5C%2Flink.aps.org%5C%2Fdoi%5C%2F10.1103%5C%2FPhysRevA.101.012350%22%2C%22collections%22%3A%5B%22Y84PES7X%22%5D%2C%22dateModified%22%3A%222021-07-09T18%3A46%3A24Z%22%2C%22tags%22%3A%5B%7B%22tag%22%3A%22Quantum%20Physics%22%7D%5D%7D%7D%2C%7B%22key%22%3A%22GRZXG7UL%22%2C%22library%22%3A%7B%22id%22%3A6397418%7D%2C%22meta%22%3A%7B%22creatorSummary%22%3A%22Baez%20et%20al.%22%2C%22parsedDate%22%3A%222020-10-20%22%2C%22numChildren%22%3A1%7D%2C%22bib%22%3A%22%26lt%3Bdiv%20class%3D%26quot%3Bcsl-bib-body%26quot%3B%20style%3D%26quot%3Bline-height%3A%201.35%3B%20%26quot%3B%26gt%3B%5Cn%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-entry%26quot%3B%20style%3D%26quot%3Bclear%3A%20left%3B%20%26quot%3B%26gt%3B%5Cn%20%20%20%20%26lt%3Bdiv%20class%3D%26quot%3Bcsl-left-margin%26quot%3B%20style%3D%26quot%3Bfloat%3A%20left%3B%20padding-right%3A%200.5em%3B%20text-align%3A%20right%3B%20width%3A%201em%3B%26quot%3B%26gt%3B1.%26lt%3B%5C%2Fdiv%26gt%3B%26lt%3Bdiv%20class%3D%26quot%3Bcsl-right-inline%26quot%3B%20style%3D%26quot%3Bmargin%3A%200%20.4em%200%201.5em%3B%26quot%3B%26gt%3BBaez%20ML%2C%20Goihl%20M%2C%20Haferkamp%20J%2C%20Bermejo-Vega%20J%2C%20Gluza%20M%2C%20Eisert%20J.%20Dynamical%20structure%20factors%20of%20dynamical%20quantum%20simulators.%20Proceedings%20of%20the%20National%20Academy%20of%20Sciences%20%5BInternet%5D.%202020%20Oct%2020%20%5Bcited%202025%20Oct%2017%5D%3B117%2842%29%3A26123%26%23x2013%3B34.%20Available%20from%3A%20%26lt%3Ba%20class%3D%26%23039%3Bzp-ItemURL%26%23039%3B%20href%3D%26%23039%3Bhttps%3A%5C%2F%5C%2Fwww.pnas.org%5C%2Fdoi%5C%2F10.1073%5C%2Fpnas.2006103117%26%23039%3B%26gt%3Bhttps%3A%5C%2F%5C%2Fwww.pnas.org%5C%2Fdoi%5C%2F10.1073%5C%2Fpnas.2006103117%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3Ba%20title%3D%26%23039%3BCite%20in%20RIS%20Format%26%23039%3B%20class%3D%26%23039%3Bzp-CiteRIS%26%23039%3B%20data-zp-cite%3D%26%23039%3Bapi_user_id%3D6397418%26amp%3Bitem_key%3DGRZXG7UL%26%23039%3B%20href%3D%26%23039%3Bjavascript%3Avoid%280%29%3B%26%23039%3B%26gt%3BCite%26lt%3B%5C%2Fa%26gt%3B%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%20%20%26lt%3B%5C%2Fdiv%26gt%3B%5Cn%26lt%3B%5C%2Fdiv%26gt%3B%22%2C%22data%22%3A%7B%22itemType%22%3A%22journalArticle%22%2C%22title%22%3A%22Dynamical%20structure%20factors%20of%20dynamical%20quantum%20simulators%22%2C%22creators%22%3A%5B%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Maria%20Laura%22%2C%22lastName%22%3A%22Baez%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Marcel%22%2C%22lastName%22%3A%22Goihl%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Jonas%22%2C%22lastName%22%3A%22Haferkamp%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Juani%22%2C%22lastName%22%3A%22Bermejo-Vega%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Marek%22%2C%22lastName%22%3A%22Gluza%22%7D%2C%7B%22creatorType%22%3A%22author%22%2C%22firstName%22%3A%22Jens%22%2C%22lastName%22%3A%22Eisert%22%7D%5D%2C%22abstractNote%22%3A%22The%20dynamical%20structure%20factor%20is%20one%20of%20the%20experimental%20quantities%20crucial%20in%20scrutinizing%20the%20validity%20of%20the%20microscopic%20description%20of%20strongly%20correlated%20systems.%20However%2C%20despite%20its%20long-standing%20importance%2C%20it%20is%20exceedingly%20difficult%20in%20generic%20cases%20to%20numerically%20calculate%20it%2C%20ensuring%20that%20the%20necessary%20approximations%20involved%20yield%20a%20correct%20result.%20Acknowledging%20this%20practical%20difficulty%2C%20we%20discuss%20in%20what%20way%20results%20on%20the%20hardness%20of%20classically%20tracking%20time%20evolution%20under%20local%20Hamiltonians%20are%20precisely%20inherited%20by%20dynamical%20structure%20factors%20and%2C%20hence%2C%20offer%20in%20the%20same%20way%20the%20potential%20computational%20capabilities%20that%20dynamical%20quantum%20simulators%20do%3A%20We%20argue%20that%20practically%20accessible%20variants%20of%20the%20dynamical%20structure%20factors%20are%20bounded-error%20quantum%20polynomial%20time%20%28BQP%29-hard%20for%20general%20local%20Hamiltonians.%20Complementing%20these%20conceptual%20insights%2C%20we%20improve%20upon%20a%20novel%2C%20readily%20available%20measurement%20setup%20allowing%20for%20the%20determination%20of%20the%20dynamical%20structure%20factor%20in%20different%20architectures%2C%20including%20arrays%20of%20ultra-cold%20atoms%2C%20trapped%20ions%2C%20Rydberg%20atoms%2C%20and%20superconducting%20qubits.%20Our%20results%20suggest%20that%20quantum%20simulations%20employing%20near-term%20noisy%20intermediate-scale%20quantum%20devices%20should%20allow%20for%20the%20observation%20of%20features%20of%20dynamical%20structure%20factors%20of%20correlated%20quantum%20matter%20in%20the%20presence%20of%20experimental%20imperfections%2C%20for%20larger%20system%20sizes%20than%20what%20is%20achievable%20by%20classical%20simulation.%22%2C%22date%22%3A%222020-10-20%22%2C%22language%22%3A%22%22%2C%22DOI%22%3A%2210.1073%5C%2Fpnas.2006103117%22%2C%22ISSN%22%3A%22%22%2C%22url%22%3A%22https%3A%5C%2F%5C%2Fwww.pnas.org%5C%2Fdoi%5C%2F10.1073%5C%2Fpnas.2006103117%22%2C%22collections%22%3A%5B%22IYETWNLW%22%5D%2C%22dateModified%22%3A%222025-10-17T10%3A17%3A35Z%22%2C%22tags%22%3A%5B%5D%7D%7D%5D%7D
    1.
    Hangleiter D, Bermejo-Vega J, Schwarz M, Eisert J. Anti-concentration theorems for schemes showing a quantum computational speedup. Cite
    1.
    Bermejo-Vega J, Hangleiter D, Schwarz M, Raussendorf R, Eisert J. Architectures for quantum simulation showing a quantum speedup. Cite
    1.
    Bermejo-Vega J. Classical simulations of non-abelian quantum Fourier transforms. Master Thesis. [phd]. Technical University of Munich; 2011. Cite
    1.
    Bermejo-Vega J, Nest MV den. Classical simulations of Abelian-group normalizer circuits with intermediate measurements. arXiv:12103637 [quant-ph] [Internet]. 2013 Oct [cited 2021 Aug 13]; Available from: http://arxiv.org/abs/1210.3637 Cite
    1.
    Bermejo-Vega J, Nest MV den. Classical simulations of Abelian-group normalizer circuits with intermediate measurements. arXiv:12103637 [quant-ph] [Internet]. 2013 Oct 22 [cited 2021 Aug 13]; Available from: http://arxiv.org/abs/1210.3637 Cite
    1.
    Bermejo-Vega J, Van Den Nest M. Classical simulations of Abelian-group normalizer circuits with intermediate measurements. Quantum Info Comput. 2014;14(3–4). Cite
    1.
    Bermejo-Vega J, Lin CYY, Nest MV den. The computational power of normalizer circuits over black-box groups. arXiv:14094800 [quant-ph] [Internet]. 2014 Sept [cited 2021 Aug 13]; Available from: http://arxiv.org/abs/1409.4800 Cite
    1.
    Bermejo-Vega J, Lin CYY, Nest MV den. The computational power of normalizer circuits over black-box groups. arXiv:14094800 [quant-ph] [Internet]. 2014 Sept 16 [cited 2021 Aug 13]; Available from: http://arxiv.org/abs/1409.4800 Cite
    1.
    Bermejo-Vega J, Lin CYY, Nest MV den. Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems. arXiv:14093208 [math-ph, physics:quant-ph] [Internet]. 2015 Jan [cited 2021 Aug 13]; Available from: http://arxiv.org/abs/1409.3208 Cite
    1.
    Bermejo-Vega J, Lin CYY, Nest MV den. Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems. arXiv:14093208 [math-ph, physics:quant-ph] [Internet]. 2015 Jan 20 [cited 2021 Aug 13]; Available from: http://arxiv.org/abs/1409.3208 Cite
    1.
    Bermejo-Vega J, Lin CYY, Nest MV den. Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems [Internet]. arXiv; 2015 [cited 2025 Oct 17]. Available from: http://arxiv.org/abs/1409.3208 Cite
    1.
    Bermejo-Vega J, Zatloukal KC. Abelian Hypergroups and Quantum Computation. arXiv:150905806 [math-ph, physics:quant-ph] [Internet]. 2015 Sept [cited 2021 Aug 13]; Available from: http://arxiv.org/abs/1509.05806 Cite
    1.
    Bermejo-Vega J, Zatloukal KC. Abelian Hypergroups and Quantum Computation. arXiv:150905806 [math-ph, physics:quant-ph] [Internet]. 2015 Sept 18 [cited 2021 Aug 13]; Available from: http://arxiv.org/abs/1509.05806 Cite
    1.
    Bermejo-Vega J. Normalizer circuits and quantum computation. CoRR [Internet]. 2016;abs/1611.09274. Available from: http://arxiv.org/abs/1611.09274 Cite
    1.
    Bermejo-Vega J, Lin CYY, Van Den Nest M. Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems. Quantum Info Comput. 2016 Apr;16(5–6):361–422. Cite
    1.
    Stephen DT, Nautrup HP, Bermejo-Vega J, Eisert J, Raussendorf R. Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter. 2019 May 20 [cited 2025 Oct 17]; Available from: https://quantum-journal.org/papers/q-2019-05-20-142/ Cite
    1.
    Raussendorf R, Bermejo-Vega J, Tyhurst E, Okay C, Zurel M. Phase-space-simulation method for quantum computation with magic states on qubits. Physical Review A [Internet]. 2020 Jan [cited 2021 July 9];101(1):012350. Available from: https://link.aps.org/doi/10.1103/PhysRevA.101.012350 Cite
    1.
    Raussendorf R, Bermejo-Vega J, Tyhurst E, Okay C, Zurel M. Phase-space-simulation method for quantum computation with magic states on qubits. Physical Review A [Internet]. 2020 Jan [cited 2021 July 9];101(1):012350. Available from: https://link.aps.org/doi/10.1103/PhysRevA.101.012350 Cite
    1.
    Raussendorf R, Bermejo-Vega J, Tyhurst E, Okay C, Zurel M. Phase-space-simulation method for quantum computation with magic states on qubits. Phys Rev A [Internet]. 2020 Jan 31 [cited 2021 July 9];101(1):012350. Available from: https://link.aps.org/doi/10.1103/PhysRevA.101.012350 Cite
    1.
    Baez ML, Goihl M, Haferkamp J, Bermejo-Vega J, Gluza M, Eisert J. Dynamical structure factors of dynamical quantum simulators. Proceedings of the National Academy of Sciences [Internet]. 2020 Oct 20 [cited 2025 Oct 17];117(42):26123–34. Available from: https://www.pnas.org/doi/10.1073/pnas.2006103117 Cite

    Posts de Jara Juani Bermejo Vega