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arXiv:2508.01231 · Quantum Information Processing

Quantum algorithms for Gowers norm estimation, polynomial testing, and arithmetic progression counting over finite abelian groups

En-Jui Kuo

Published Apr 13, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract We propose a family of quantum algorithms for estimating Gowers uniformity norms $$ U^k $$ U k over finite abelian groups, extending earlier quantum methods for the $$ U^2 $$ U 2 -norm to arbitrary prime fields and higher-order uniformity norms. 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 $$ f: \mathbb {F}_p^n \rightarrow \mathbb {C} $$ f : F p n → C is a low-degree phase polynomial or is far from any such structure. We show that whenever an inverse theorem for the $$ U^{d+1} $$ U d + 1 -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 $$ U^4 $$ U 4 -norm over $$ \mathbb {F}_p^n $$ F p n , we obtain quasipolynomial-time quantum algorithms for detecting cubic phase polynomials. For higher-order norms such as $$ U^5 $$ U 5 and $$ U^6 $$ U 6 over $$ \mathbb {F}_2^n $$ F 2 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 $$ U^2 $$ U 2 -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.

DOI
10.1007/s11128-026-05165-6

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Abstract We propose a family of quantum algorithms for estimating Gowers uniformity norms UkU^kUk U k over finite abelian groups, extending earlier quantum methods for the U2U^2U2 U 2 -norm to arbitrary prime fields and higher-order uniformity norms. 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 f:Fpn→Cf: \mathbb {F}_p^n \rightarrow \mathbb {C}f:Fpn​→C f : F p n → C is a low-degree phase polynomial or is far from any such structure. We show that whenever an inverse theorem for the Ud+1U^{d+1}Ud+1 U d + 1 -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 U4U^4U4 U 4 -norm over Fpn\mathbb {F}_p^nFpn​ F p n , we obtain quasipolynomial-time quantum algorithms for detecting cubic phase polynomials. For higher-order norms such as U5U^5U5 U 5 and U6U^6U6 U 6 over F2n\mathbb {F}_2^nF2n​ F 2 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 U2U^2U2 U 2 -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.
We give a complete classification of the norm properties of quantum uniformity measures, introduced by Bu, Gu, and Jaffe to analytically characterize the Clifford hierarchy. Resolving an open question they posed, we show that quantum uniformity measures define norms precisely for k=2,3,4k=2,3,4k=2,3,4. In particular, we establish the triangle inequality for k=4k=4k=4, whereas for every k≥5k\geq5k≥5 it fails in some dimension. The higher-order result is proved using the structure theory of norming systems of linear equations over finite fields. We also study the geometry of quantum uniformity measures, establishing explicit bounds on the modulus of convexity whenever they define norms. Finally, we give a complete classification of the norm properties of matrix-valued Gowers uniformity functionals, showing that only k=2k=2k=2 defines a norm.
Attempts to separate the power of classical and quantum models of computation have a long history. The ultimate goal are exponential separations for computational (promise) problems, however, these do not come a dime a dozen: while there were some early successes in the form of hidden subgroup problems for abelian groups–which generalizes Shor’s factoring algorithm perhaps most faithfully–only for a handful of non-abelian groups efficient quantum algorithms were found. Recently, problems have gotten increased attention that seek to identify hidden sub-structures of other combinatorial and algebraic objects besides groups, notably the quantum algorithms by Childs, Schulman, and Vazirani for shifted subset problems for spheres over a finite field. In this paper we provide new examples for exponential separations by considering hidden shift problems that are defined for several classes of highly non-linear Boolean functions. These so-called bent functions arise in cryptography, where their property of having perfectly flat Fourier spectra on the Boolean hypercube gives them resilience against certain types of attack. We present quantum algorithms that solve the hidden shift problems for several well-known classes of bent functions in polynomial time with a constant number of queries, while the classical query complexity is shown to be exponential. Our approach uses a technique that exploits the duality between bent functions and their Fourier transforms. We also give a quantum algorithm that solves the hidden shift problem for quadratic forms that might be of independent interest. This algorithm can also be used to find approximating polynomials and shifts for functions with large Gowers U3 norm. 1 1
In this thesis, Quantum computing, norms, and polynomials, we investigate the interplay between quantum mechanics, complexity theory, and functional analysis, three central areas of physics, computer science, and mathematics, respectively. The unifying theme throughout the thesis is the dynamic exchange between quantum computing and functional analysis: we explore new applications of functional inequalities in quantum computing, and, in the process, establish novel results in functional analysis itself. In the first part, we study quantum query algorithms through their correspondence with completely bounded polynomials, as established in earlier work. We begin by revisiting this correspondence, extending it, and presenting it in a new form. Building on this foundation, we draw an analogy between quantum query algorithms and the Grothendieck inequality. Finally, we conclude this part by employing completely bounded polynomials to solve a special case of one of the main open problems in quantum query complexity, the Aaronson–Ambainis conjecture. In the second part, we turn to quantum learning theory, which seeks to determine how much information must be extracted from a quantum system to fully characterize it. We begin by applying existing versions of the Bohnenblust–Hille inequalities and deriving new ones to obtain results in the learning of low-degree quantum objects. We conclude by presenting some of the first results in the emerging area of Hamiltonian testing and learning. We also include a third part, as a bonus, where we gather three new proofs, that we find elegant and concise, of known results related to the analysis of Boolean functions.
We propose a mathematical framework that we call quantum, higher-order Fourier analysis. This generalizes the classical theory of higher-order Fourier analysis, which led to many recent advances in number theory and combinatorics. We define a family of "quantum measures" on linear transformations on a Hilbert space, that reduce in the case of diagonal matrices to the uniformity norms introduced by Timothy Gowers. We show that our quantum measures and our related theory of quantum higher-order Fourier analysis characterize the Clifford hierarchy, an important notion of complexity in quantum computation. In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the [Formula: see text] level of the Clifford hierarchy.
In this dissertation, we investigate three different problems in the field of Quantum computation. First, we discuss the quantum complexity of evaluating the Tutte polynomial of a planar graph. Furthermore, we devise a new quantum algorithm for approximating the phase of a unitary matrix. Finally, we provide quantum tools that can be utilized to extract the structure of black-box modules and algebras. While quantum phase estimation (QPE) is at the core of many quantum algorithms known to date, its physical implementation (algorithms based on quantum Fourier transform (QFT) ) is highly constrained by the requirement of high-precision controlled phase shift operators, which remain difficult to realize. In the second part of this dissertation, we introduce an alternative approach to approximately implement QPE with arbitrary constantprecision controlled phase shift operators. The new quantum algorithm bridges the gap between QPE algorithms based on QFT and Kitaev’s original approach. For approximating the eigenphase precise to the nth bit, Kitaev’s original approach does not require any controlled phase shift operator. In contrast, QPE algorithms based on QFT or approximate QFT require controlled phase shift operators with precision of at least Pi/2n. The new approach fills the gap and requires only arbitrary constant-precision controlled phase shift operators. From a physical implementation viewpoint, the new algorithm outperforms Kitaev’s approach. iii The other problem we investigate relates to approximating the Tutte polynomial. We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e 2πi/5 ,e−2πi/5 ) is DQC1-complete and at points (q k , 1 + 1−q−k (q 1/2−q−1/2) 2 ) for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones-Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid ˜b such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of ˜b are alternating links. The final part of the dissertation focuses on finding the structure of a black-box module or algebra. Suppose we are given black-box access to a finite module M or algebra over a finite ring R, and a list of generators for M and R. We show how to find a linear basis and structure constants for M in quantum poly(log |M|) time. This generalizes a recent quantum algorithm of Arvind et al. which finds a basis representation for rings. We then show that iv our algorithm is a useful primitive allowing quantum computers to determine the structure of a finite associative algebra as a direct sum of simple algebras. Moreover, it solves a wide variety of problems regarding finite modules and rings. Although our quantum algorithm is based on Abelian Fourier transforms, it solves problems regarding the multiplicative structure of modules and algebras, which need not be commutative. Examples include finding the intersection and quotient of two modules, finding the additive and multiplicative identities in a module, computing the order of an module, solving linear equations over modules, deciding whether an ideal is maximal, finding annihilators, and testing the injectivity and surjectivity of ring homomorphisms. These problems appear to be exponentially hard classically.
We present several new examples of speed-ups obtainable by quantum algorithms in the context of property testing. First, motivated by sampling algorithms, we consider probability distributions given in the form of an oracle f:[n]→[m]f:[n]\to[m]f:[n]→[m]. Here the probability $\PP_f(j)$ of an outcome j∈[m]j\in[m]j∈[m] is the fraction of its domain that fff maps to jjj. We give quantum algorithms for testing whether two such distributions are identical or εεε-far in L1L_1L1​-norm. Recently, Bravyi, Hassidim, and Harrow \cite{BHH10} showed that if $\PP_f$ and $\PP_g$ are both unknown (i.e., given by oracles fff and ggg), then this testing can be done in roughly m\sqrt{m}m​ quantum queries to the functions. We consider the case where the second distribution is known, and show that testing can be done with roughly m1/3m^{1/3}m1/3 quantum queries, which we prove to be essentially optimal. In contrast, it is known that classical testing algorithms need about m2/3m^{2/3}m2/3 queries in the unknown-unknown case and about m\sqrt{m}m​ queries in the known-unknown case. Based on this result, we also reduce the query complexity of graph isomorphism testers with quantum oracle access. While those examples provide polynomial quantum speed-ups, our third example gives a much larger improvement (constant quantum queries vs polynomial classical queries) for the problem of testing periodicity, based on Shor's algorithm and a modification of a classical lower bound by Lachish and Newman \cite{lachish&newman:periodicity}. This provides an alternative to a recent constant-vs-polynomial speed-up due to Aaronson \cite{aaronson:bqpph}.