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arXiv:2608.23929

Quantum Algorithms and Hardness for Point-Count Approximation over Finite Fields

Yota Maeda, Hiroshi Yano

quant-phcs.CCmath.AGmath.NTSubmitted Aug 25, 2026• Announced Aug 26, 2026• 0 citations
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Abstract

We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $ρ$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-ρ} \] with success probability $1-δ$. Provided that $ρ$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/δ))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.

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31 pages

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We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial [f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], ] let UUU be its augmented support matrix whose columns are (1,uj)(1,u_j)(1,uj​) with rank ρρρ and N(f):=#{x∈(Fq×)n∣f(x)=0}N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}N(f):=#{x∈(Fq×​)n∣f(x)=0} be its torus point count. Our first main result is a quantum algorithm that outputs N^(f)\widehat{N}(f)N(f) satisfying [ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-ρ} ] with success probability 1−δ1-δ1−δ. Provided that ρρρ and ∥U∥∞\|U\|_\infty∥U∥∞​ are bounded, the algorithm runs in both classical bit and quantum gate complexity poly(n,s,log⁡q,1/ε,log⁡(1/δ))\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/δ))poly(n,s,logq,1/ε,log(1/δ)). It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in log⁡q\log qlogq. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes #\##P-hard under randomized polynomial-time Turing reductions when the support matrix UUU varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.
What is the power of polynomial-time quantum computation with access to an NP oracle? In this work, we focus on two fundamental tasks from the study of Boolean satisfiability (SAT) problems: search-to-decision reductions, and approximate counting. We first show that, in strong contrast to the classical setting where a poly-time Turing machine requires Θ(n)Θ(n)Θ(n) queries to an NP oracle to compute a witness to a given SAT formula, quantumly Θ(log⁡n)Θ(\log n)Θ(logn) queries suffice. We then show this is tight in the black-box model - any quantum algorithm with "NP-like" query access to a formula requires Ω(log⁡n)Ω(\log n)Ω(logn) queries to extract a solution with constant probability. Moving to approximate counting of SAT solutions, by exploiting a quantum link between search-to-decision reductions and approximate counting, we show that existing classical approximate counting algorithms are likely optimal. First, we give a lower bound in the "NP-like" black-box query setting: Approximate counting requires Ω(log⁡n)Ω(\log n)Ω(logn) queries, even on a quantum computer. We then give a "white-box" lower bound (i.e. where the input formula is not hidden in the oracle) - if there exists a randomized poly-time classical or quantum algorithm for approximate counting making o(logn)o(log n)o(logn) NP queries, then BPPNP[o(n)]\text{BPP}^{\text{NP}[o(n)]}BPPNP[o(n)] contains a PNP\text{P}^{\text{NP}}PNP-complete problem if the algorithm is classical and FBQPNP[o(n)]\text{FBQP}^{\text{NP}[o(n)]}FBQPNP[o(n)] contains an FPNP\text{FP}^{\text{NP}}FPNP-complete problem if the algorithm is quantum.
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