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Distributed Quantum Advantage for Local Problems

Alkida Balliu, Sebastian Brandt, Xavier Coiteux-Roy, Francesco d'Amore, Massimo Equi, François Le Gall, Henrik Lievonen, Augusto Modanese, Dennis Olivetti, Marc-Olivier Renou, Jukka Suomela, Lucas Tendick, Isadora Veeren

TL;DR

The round elimination technique is used to prove that the iterated GHZ problem requires Ω(Δ) rounds for classical algorithms, and it is demonstrated that round elimination cannot be used to prove lower bounds for quantum-LOCAL.

Abstract

We present the first local problem that shows a super-constant separation between the classical randomized LOCAL model of distributed computing and its quantum counterpart. By prior work, such a separation was known only for an artificial graph problem with an inherently global definition [Le Gall et al. 2019]. We present a problem that we call iterated GHZ, which is defined using only local constraints. Formally, it is a family of locally checkable labeling problems [Naor and Stockmeyer 1995]; in particular, solutions can be verified with a constant-round distributed algorithm. We show that in graphs of maximum degree $Δ$, any classical (deterministic or randomized) LOCAL model algorithm will require $Ω(Δ)$ rounds to solve the iterated GHZ problem, while the problem can be solved in $1$ round in quantum-LOCAL. We use the round elimination technique to prove that the iterated GHZ problem requires $Ω(Δ)$ rounds for classical algorithms. This is the first work that shows that round elimination is indeed able to separate the two models, and this also demonstrates that round elimination cannot be used to prove lower bounds for quantum-LOCAL. To apply round elimination, we introduce a new technique that allows us to discover appropriate problem relaxations in a mechanical way; it turns out that this new technique extends beyond the scope of the iterated GHZ problem and can be used to e.g. reproduce prior results on maximal matchings [FOCS 2019, PODC 2020] in a systematic manner.

Distributed Quantum Advantage for Local Problems

TL;DR

The round elimination technique is used to prove that the iterated GHZ problem requires Ω(Δ) rounds for classical algorithms, and it is demonstrated that round elimination cannot be used to prove lower bounds for quantum-LOCAL.

Abstract

We present the first local problem that shows a super-constant separation between the classical randomized LOCAL model of distributed computing and its quantum counterpart. By prior work, such a separation was known only for an artificial graph problem with an inherently global definition [Le Gall et al. 2019]. We present a problem that we call iterated GHZ, which is defined using only local constraints. Formally, it is a family of locally checkable labeling problems [Naor and Stockmeyer 1995]; in particular, solutions can be verified with a constant-round distributed algorithm. We show that in graphs of maximum degree , any classical (deterministic or randomized) LOCAL model algorithm will require rounds to solve the iterated GHZ problem, while the problem can be solved in round in quantum-LOCAL. We use the round elimination technique to prove that the iterated GHZ problem requires rounds for classical algorithms. This is the first work that shows that round elimination is indeed able to separate the two models, and this also demonstrates that round elimination cannot be used to prove lower bounds for quantum-LOCAL. To apply round elimination, we introduce a new technique that allows us to discover appropriate problem relaxations in a mechanical way; it turns out that this new technique extends beyond the scope of the iterated GHZ problem and can be used to e.g. reproduce prior results on maximal matchings [FOCS 2019, PODC 2020] in a systematic manner.

Paper Structure

This paper contains 71 sections, 30 theorems, 19 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Setting: LOCAL vs. quantum-LOCAL
  3. Prior work vs. our main contribution
  4. Key techniques and new ideas
  5. CHSH game.
  6. Iterated CHSH.
  7. Iterated CHSH is a local problem.
  8. Iterated CHSH in super-quantum-LOCAL.
  9. Iterated CHSH in classical LOCAL.
  10. From super-quantum to quantum: replacing CHSH with GHZ.
  11. Iterated GHZ.
  12. Proving the classical lower bound.
  13. Round elimination: basic idea.
  14. Round elimination: formalism.
  15. Round elimination: basic strategy.
  16. ...and 56 more sections

Key Result

theorem 1

For any integer $\Delta > 0$, the iterated GHZ problem can be solved in $O(1)$ rounds in the quantum LOCAL model, if a $\Delta$-edge coloring is provided.

Figures (9)

  • Figure 1: The strength relation of the first color.
  • Figure 2: The strength relation of each present color $j$ different from the first and the special color.
  • Figure 3: The strength relation of the special color $j=\Delta-i-1$.
  • Figure 4: The strength relation of gone colors.
  • Figure 5: The strength relation of the first color.
  • ...and 4 more figures

Theorems & Definitions (64)

  • definition 1: LCL problem in the black-white formalism
  • theorem 1
  • proof
  • theorem 2
  • definition 2: Picking a configuration
  • proof
  • definition 3: Relaxation of a problem
  • definition 4: Right-closed
  • definition 5: A configuration generated by a condensed one
  • theorem 6: hideandseek, rephrased
  • ...and 54 more