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Optimal Depth-Three Circuits for Inner Product

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

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LIPIcs.CCC.2026.27.pdf
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ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • Depth 3 circuits
  • Circuit lower bounds
  • Inner product
  • Analytic combinatorics

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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 Get BibTex

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) https://doi.org/10.4230/LIPIcs.CCC.2026.27

Author Details

Mohit Gurumukhani
  • Cornell University, Ithaca, NY, USA
Daniel Kleber
  • Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA, USA
Ramamohan Paturi
  • Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA, USA
Christopher Rosin
  • Constructive Codes, USA
Navid Talebanfard
  • University of Sheffield, UK

Funding

  • Gurumukhani, Mohit: Supported by a Sloan Research Fellowship, NSF CAREER Award 2045576, and NSF Award CCF-2514586.
  • Kleber, Daniel: Partially supported by NSF grant 2212136.
  • Paturi, Ramamohan: Partially supported by NSF grant 2212136.

Supplementary Materials

References

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