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Our algorithms prepare quantum states encoding higher-order finite differences and apply Fourier sampling together with amplitude estimation to obtain estimates of Gowers norms. As a central application, we study algebraic property testing problems of distinguishing whether a bounded function\n \n \n $$ f: \\mathbb {F}_p^n \\rightarrow \\mathbb {C} $$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n :<\/mml:mo>\n \n F<\/mml:mi>\n p<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msubsup>\n \u2192<\/mml:mo>\n C<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a low-degree phase polynomial or is far from any such structure. We show that whenever an inverse theorem for the\n \n \n $$ U^{d+1} $$<\/jats:tex-math>\n \n \n U<\/mml:mi>\n \n d<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -norm is available, our quantum framework yields a corresponding structure-testing algorithm whose query and measurement complexity depends explicitly on the quantitative bounds of that inverse theorem. In particular, using the recent quasipolynomial inverse theorem for the\n \n \n $$ U^4 $$<\/jats:tex-math>\n \n \n U<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -norm over\n \n \n $$ \\mathbb {F}_p^n $$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n p<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , we obtain quasipolynomial-time quantum algorithms for detecting cubic phase polynomials. For higher-order norms such as\n \n \n $$ U^5 $$<\/jats:tex-math>\n \n \n U<\/mml:mi>\n 5<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$ U^6 $$<\/jats:tex-math>\n \n \n U<\/mml:mi>\n 6<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n over\n \n \n $$ \\mathbb {F}_2^n $$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n 2<\/mml:mn>\n n<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the best known inverse theorems provide only tower-type quantitative bounds; accordingly, our detection algorithms remain correct but inherit complexity corresponding to tower-type quantitative bounds. We also present a quantum method for estimating the number of 3-term arithmetic progressions in Boolean functions via the\n \n \n $$ U^2 $$<\/jats:tex-math>\n \n \n U<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -norm. Although not query-optimal compared to Grover-style counting, this approach is sensitive to additive structure and naturally aligned with tools from higher-order Fourier analysis. Finally, we observe that Gowers norms are invariant under certain classes of shift-type noise, implying that our algorithms retain robustness under natural quantum noise models. This suggests that Gowers norm-based quantum procedures may serve as stable primitives for quantum property testing, learning theory, and the analysis of pseudorandomness in the NISQ regime.\n <\/jats:p>","DOI":"10.1007\/s11128-026-05165-6","type":"journal-article","created":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T08:28:36Z","timestamp":1776068916000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Quantum algorithms for Gowers norm estimation, polynomial testing, and arithmetic progression counting over finite abelian groups"],"prefix":"10.1007","volume":"25","author":[{"given":"En-Jui","family":"Kuo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,13]]},"reference":[{"key":"5165_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9458-7","volume-title":"Linear Representations of Finite Groups","author":"J-P Serre","year":"1977","unstructured":"Serre, J.-P.: Linear Representations of Finite Groups. 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