Fanout Complexity of Symmetric Boolean Functions in $\mathsf{QAC}^0$

• URL: http://arxiv.org/abs/2609.05153v1
• Date: Fri, 04 Sep 2026 13:55:20 GMT
• Title: Fanout Complexity of Symmetric Boolean Functions in $\mathsf{QAC}^0$
• Abstract summary: Computing $mathttPARITY_n$ is equivalent to implementing $mathttFANOUT_(f)$ under $mathsfQAC0$ reductions.<n>We show that the answer is exactly the transition radius $(f)$: computing $f$ and implementing $mathttFANOUT_(f)$ are equivalent under $mathsfQAC0$ reductions.
• Score: 7.883691246794115
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Whether $\mathsf{QAC}^0$ can compute $\mathtt{PARITY}_n$ remains open. Computing $\mathtt{PARITY}_n$ is equivalent to implementing $\mathtt{FANOUT}_n$ under $\mathsf{QAC}^0$ reductions. This raises a more general question: for an arbitrary symmetric Boolean function $f:\{0,1\}^n\to\{0,1\}$, what fanout size is necessary and sufficient for computing $f$ in $\mathsf{QAC}^0$? We show that the answer is exactly the transition radius $ρ(f)$: computing $f$ and implementing $\mathtt{FANOUT}_{ρ(f)}$ are equivalent under $\mathsf{QAC}^0$ reductions. In particular, if $ρ(f)\ge n^δ$ for some constant $δ>0$, then computing $f$ is $\mathsf{QAC}^0_{\mathrm{f}}$-complete. Combined with Paturi's theorem, our characterization implies that if $\mathtt{PARITY}_n \notin \mathsf{QAC}^0$, then any Boolean function in $\mathsf{QAC}^0$ of approximate degree $n^{1/2+Ω(1)}$ must be nonsymmetric.

Related papers

• QAC0 Can Prepare Every Logarithmic-Qubit State [1.4018975578160688]
  $mathsfQAC0$ is the class of constant-depth $mathrmpoly(n)$-ancilla circuits.<n>We show that every $O(log n)$-qubit state can be exactly and cleanly prepared by a $mathrmpoly(n)$-ancilla $mathsfQAC0$ circuit.
  arXiv  Detail & Related papers  (2026-09-15T16:35:52Z)
• Hardness Amplification for (Sparse) LPN [5.1105538244022135]
  We prove new hardness amplification results for Learning Parity with Noise ($mathsfLPN$) and its sparse variants.<n>An algorithm that solves $mathsfLPN$ on a'small fraction of instances' can be converted into an algorithm that solves $mathsfLPN$ on 'almost all instances'<n>We extend the same amplification approach to $mathsfLPN$ over $mathbbF_q$ and to Sparse-$mathsfLPN$, where each query vector $vec a$ has
  arXiv  Detail & Related papers  (2026-05-11T06:34:37Z)
• $\mathsf{QAC}^0$ Contains $\mathsf{TC}^0$ (with Many Copies of the Input) [0.9023122463034333]
  $mathsfQAC0$ is the class of constant-depth quantum circuits constructed from arbitrary single-qubit gates and generalized Toffoli gates.<n>We show that $mathsfQAC0$ circuits are significantly more powerful than their classical counterparts.
  arXiv  Detail & Related papers  (2026-01-06T18:40:44Z)
• On the Capacity Region of Individual Key Rates in Vector Linear Secure Aggregation [55.126702858312456]
  We show that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature.<n>Our results uncover the novel fact that it is not necessary for every user to hold a key, thereby strictly enlarging the best-known achievable region in the literature.
  arXiv  Detail & Related papers  (2026-01-06T18:34:07Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• Quantum Sabotage Complexity [0.7812210699650152]
  We show $mathsfQ(f_mathsfsab)$, the quantum query complexity of $f_mathsfsab$. We show that when $f$ is the Indexing function, $mathsfQ(f_mathsfsab)=Theta(sqrtmathsfsab)$, ruling out the possibility that $mathsfQ(f_mathsfsab)=Theta(sqrtmathsf
  arXiv  Detail & Related papers  (2024-08-22T17:57:58Z)
• Efficient Continual Finite-Sum Minimization [52.5238287567572]
  We propose a key twist into the finite-sum minimization, dubbed as continual finite-sum minimization. Our approach significantly improves upon the $mathcalO(n/epsilon)$ FOs that $mathrmStochasticGradientDescent$ requires. We also prove that there is no natural first-order method with $mathcalOleft(n/epsilonalpharight)$ complexity gradient for $alpha 1/4$, establishing that the first-order complexity of our method is nearly tight.
  arXiv  Detail & Related papers  (2024-06-07T08:26:31Z)
• Noisy Computing of the $\mathsf{OR}$ and $\mathsf{MAX}$ Functions [22.847963422230155]
  We consider the problem of computing a function of $n$ variables using noisy queries. We show that an expected number of queries of [ (1 pm o(1)) fracnlog frac1deltaD_mathsfKL(p | 1-p) ] is both sufficient and necessary to compute both functions.
  arXiv  Detail & Related papers  (2023-09-07T19:37:52Z)
• Fast $(1+\varepsilon)$-Approximation Algorithms for Binary Matrix Factorization [54.29685789885059]
  We introduce efficient $(1+varepsilon)$-approximation algorithms for the binary matrix factorization (BMF) problem. The goal is to approximate $mathbfA$ as a product of low-rank factors. Our techniques generalize to other common variants of the BMF problem.
  arXiv  Detail & Related papers  (2023-06-02T18:55:27Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Threshold Phenomena in Learning Halfspaces with Massart Noise [56.01192577666607]
  We study the problem of PAC learning halfspaces on $mathbbRd$ with Massart noise under Gaussian marginals. Our results qualitatively characterize the complexity of learning halfspaces in the Massart model.
  arXiv  Detail & Related papers  (2021-08-19T16:16:48Z)
• The planted matching problem: Sharp threshold and infinite-order phase transition [25.41713098167692]
  We study the problem of reconstructing a perfect matching $M*$ hidden in a randomly weighted $ntimes n$ bipartite graph. We show that if $sqrtd B(mathcalP,mathcalQ) ge 1+epsilon$ for an arbitrarily small constant $epsilon>0$, the reconstruction error for any estimator is shown to be bounded away from $0$.
  arXiv  Detail & Related papers  (2021-03-17T00:59:33Z)
• Linear Bandits on Uniformly Convex Sets [88.3673525964507]
  Linear bandit algorithms yield $tildemathcalO(nsqrtT)$ pseudo-regret bounds on compact convex action sets. Two types of structural assumptions lead to better pseudo-regret bounds.
  arXiv  Detail & Related papers  (2021-03-10T07:33:03Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.