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A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity

Pavel Dvořák, Bruno Loff, Suhail Sherif

TL;DR

This paper develops a program to study Π_1 statements, especially PHP and KW-type lower bounds, through semidefinite relaxations of homogeneous quadratic feasibility (HQFP) problems. It introduces the quantum pigeonhole principle (QPHP) as a strengthened relaxation and constructs two SDP-based models of computation, γ_2 communication and quantum-lab protocols, to derive explicit lower and upper bounds for KW-type relations. A key no-go theorem links semidefinite relaxations to SOS-degree lower bounds, arguing that without high-degree ingredients, these relaxations would collapse to solving all KW games, which motivates seeking constructive protocols. The work also yields concrete results: a discrepancy-based lower bound for γ_2, a constant-depth γ_2 protocol for equality, a 3-round quantum-lab protocol for general functions, and a demonstration that the two models can lead to powerful, yet non-pathological, computational frameworks. Together, these results illuminate the trade-offs between SDP relaxations, proof complexity, and distributed computation, while offering explicit protocols in two quantum-inspired models.

Abstract

We study semidefinite relaxations of $Π_1$ combinatorial statements. By relaxing the pigeonhole principle, we obtain a new "quantum" pigeonhole principle which is a stronger statement. By relaxing statements of the form "the communication complexity of $f$ is $> k$", we obtain new communication models, which we call "$γ_2$ communication" and "quantum-lab protocols". We prove, via an argument from proof complexity, that any natural model obtained by such a relaxation must solve all Karchmer--Wigderson games efficiently. However, the argument is not constructive, so we work to explicitly construct such protocols in these two models.

A Quantum Pigeonhole Principle and Two Semidefinite Relaxations of Communication Complexity

TL;DR

This paper develops a program to study Π_1 statements, especially PHP and KW-type lower bounds, through semidefinite relaxations of homogeneous quadratic feasibility (HQFP) problems. It introduces the quantum pigeonhole principle (QPHP) as a strengthened relaxation and constructs two SDP-based models of computation, γ_2 communication and quantum-lab protocols, to derive explicit lower and upper bounds for KW-type relations. A key no-go theorem links semidefinite relaxations to SOS-degree lower bounds, arguing that without high-degree ingredients, these relaxations would collapse to solving all KW games, which motivates seeking constructive protocols. The work also yields concrete results: a discrepancy-based lower bound for γ_2, a constant-depth γ_2 protocol for equality, a 3-round quantum-lab protocol for general functions, and a demonstration that the two models can lead to powerful, yet non-pathological, computational frameworks. Together, these results illuminate the trade-offs between SDP relaxations, proof complexity, and distributed computation, while offering explicit protocols in two quantum-inspired models.

Abstract

We study semidefinite relaxations of combinatorial statements. By relaxing the pigeonhole principle, we obtain a new "quantum" pigeonhole principle which is a stronger statement. By relaxing statements of the form "the communication complexity of is ", we obtain new communication models, which we call " communication" and "quantum-lab protocols". We prove, via an argument from proof complexity, that any natural model obtained by such a relaxation must solve all Karchmer--Wigderson games efficiently. However, the argument is not constructive, so we work to explicitly construct such protocols in these two models.
Paper Structure (19 sections, 17 theorems, 53 equations)

This paper contains 19 sections, 17 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Quantum Pigeonhole Principle
  4. The Semidefinite Feasibility Problem
  5. A Non-tight Proof Using the AM-GM Inequality
  6. A Tight Proof Using a Geometric Argument
  7. An Explicit Proof via Duality
  8. gamma-2 Communication
  9. Definition of the Model
  10. A Lower-bound Using Discrepancy
  11. Upper Bound for Equality
  12. Quantum Lab Protocols
  13. Definition of the Model
  14. A 2-round Lower Bound for Equality
  15. Model Collapse -- All Functions Are Easy
  16. ...and 4 more sections

Key Result

Theorem 2.4

Let $\cK \subseteq \mathbb{R}^n$ be a closed convex cone. Let $\cA: \mathbb{R}^n \to \mathbb{R}^{m}$ be a linear map, and $b \in \mathbb{R}^m$. Suppose that $\ker(\cA) + \cK$ is a closed set (Ben-Israel's criterion). Then exactly one of the following two things are true:

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Ben-Israel benisrael_linear_1969
  • Lemma 2.5: Berman--Ben-Israel criterion
  • Theorem 3.1: QPHP
  • Theorem 3.2: Weak Quantitative QPHP
  • proof
  • Theorem 3.3: Quantitative QPHP
  • proof
  • ...and 30 more