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Optimal Depth-Three Circuits for Inner Product

Authors: Mohit Gurumukhani, Daniel Kleber, Ramamohan Paturi, Christopher Rosin, and Navid Talebanfard

Published in: LIPIcs, Volume 383, 41st Computational Complexity Conference (CCC 2026)


Abstract
We show that Inner Product in 2n variables, IP_n(x, y) = x₁y₁ ⊕ … ⊕ x_ny_n, can be computed by depth-3 bottom fan-in 2 circuits of size poly(n)⋅ (9/5)ⁿ, matching the lower bound of Göös, Guan, and Mosnoi (Inform. Comput.'24). Our construction is obtained via the following steps. 1) We provide a general template for constructing optimal depth-3 circuits with bottom fan-in k for an arbitrary function f. We do this in two steps. First, we partition f^{-1}(1) into orbits of its automorphism group. Second, for each orbit, we construct one k-CNF that (a) accepts the largest number of inputs from that orbit and (b) rejects all inputs rejected by f. 2) We instantiate the template for IP_n and k = 2. Guided by the intuition (which we call modularity principle) that optimal 2-CNFs can be constructed by taking the conjunction of variable-disjoint copies of smaller 2-CNFs, we use computer search to identify a small set of building block 2-CNFs over at most 4 variables. 3) We again use computer search to discover appropriate combinations (disjoint conjunctions) of building blocks to arrive at optimal 2-CNFs and analyze them using techniques from analytic combinatorics. We believe that the approach outlined in this paper can be applied to a wide range of functions to determine their depth-3 complexity.

Cite as

Mohit Gurumukhani, Daniel Kleber, Ramamohan Paturi, Christopher Rosin, and Navid Talebanfard. Optimal Depth-Three Circuits for Inner Product. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 27:1-27:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)


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@InProceedings{gurumukhani_et_al:LIPIcs.CCC.2026.27,
  author =	{Gurumukhani, Mohit and Kleber, Daniel and Paturi, Ramamohan and Rosin, Christopher and Talebanfard, Navid},
  title =	{{Optimal Depth-Three Circuits for Inner Product}},
  booktitle =	{41st Computational Complexity Conference (CCC 2026)},
  pages =	{27:1--27:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-437-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{383},
  editor =	{Moshkovitz, Dana},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2026.27},
  URN =		{urn:nbn:de:0030-drops-270693},
  doi =		{10.4230/LIPIcs.CCC.2026.27},
  annote =	{Keywords: Depth 3 circuits, Circuit lower bounds, Inner product, Analytic combinatorics}
}
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