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arXiv:quant-ph/0509047 · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

A Quantum Protocol to Win the Graph Colouring Game on All Hadamard Graphs

David Avis

Published May 1, 2006• 36 citations
arXivHTMLPDF

Abstract

This paper deals with graph colouring games, an example of pseudo-telepathy, in which two provers can convince a verifier that a graph $G$ is $c$-colourable where $c$ is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph $G_N$ with $N=c=16$. Their protocol applies only to Hadamard graphs where $N$ is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of $G_{12}$ having 1609 vertices, with $c=12$. Combined with a result of Frankl and Rodl, our result shows that all sufficiently large Hadamard graphs yield pseudo-telepathy games.

DOI
10.1093/ietfec/e89-a.5.1378

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This paper deals with graph colouring games, an example of pseudo-telepathy, in which two provers can convince a verifier that a graph GGG is ccc-colourable where ccc is less than the chromatic number of the graph. They win the game if they convince the verifier. It is known that the players cannot win if they share only classical information, but they can win in some cases by sharing entanglement. The smallest known graph where the players win in the quantum setting, but not in the classical setting, was found by Galliard, Tapp and Wolf and has 32,768 vertices. It is a connected component of the Hadamard graph GNG_NGN​ with N=c=16N=c=16N=c=16. Their protocol applies only to Hadamard graphs where NNN is a power of 2. We propose a protocol that applies to all Hadamard graphs. Combined with a result of Frankl, this shows that the players can win on any induced subgraph of G12G_{12}G12​ having 1609 vertices, with c=12c=12c=12. Combined with a result of Frankl and Rodl, our result shows that all sufficiently large Hadamard graphs yield pseudo-telepathy games.
The largest known gap between quantum and classical chromatic number of graphs, obtained via quantum protocols for colouring Hadamard graphs based on the Deutsch–Jozsa algorithm and the quantum Fourier transform, is exponential. We put forth a quantum pseudo-telepathy version of Khot’s ddd-to-1 Games Conjecture and prove that, conditional on its validity, the gap is unbounded: There exist graphs whose quantum chromatic number is 3 and whose classical chromatic number is arbitrarily large. Furthermore, we show that the existence of a certain form of pseudo-telepathic XOR games would imply the conjecture and, thus, the unboundedness of the quantum chromatic gap. As two technical steps of our proof that might be of independent interest, we establish a quantum adjunction theorem for Pultr functors between categories of relational structures, and we prove that the Dinur–Khot–Kindler–Minzer–Safra reduction, recently used for proving the 2-to-2 Games Theorem, is quantum complete.
We confirm a conjecture of the first author [SODA'26] on a strong form of pseudo-telepathy for Khot's ddd-to-111 games: for every d≥4d\geq 4d≥4, every ε>0ε>0ε>0, and every k∈Nk\in\mathbb Nk∈N, there is a ddd-to-111 label-cover game of classical value less than εεε that admits a perfect quantum strategy using finite-dimensional shared entanglement with measurement contexts of radius at least kkk. As a consequence, there exist graphs whose quantum chromatic number χqχ_qχq​ is 333 and whose classical chromatic number χχχ is arbitrarily large. This is the first improvement on the quantum chromatic gap since the work of Brassard, Cleve, and Tapp [Phys. Rev. Lett.'99], which produced graphs for which χχχ grows exponentially in χqχ_qχq​. To prove the conjecture, we proceed as follows. We start with a ``stretched'' version of the Mermin--Peres magic square, obtained by spacing out its equations and linking them with additional long equation wires. This operation increases the radius of the initial measurement contexts of a perfect quantum strategy for the game, while preserving classical unsatisfiability. We then apply parallel repetition to make the classical value small, and we turn the system into a pseudo-telepathic multi-layered Kneser hypergraph via a PCP construction of Dinur, Guruswami, Khot, and Regev [STOC'03]. Finally, we feed the resulting hypergraph into the reduction of Fei, Minzer, and Wang from hypergraph colouring to 444-to-111 games, we establish quantum completeness of this reduction, and we show that it preserves context radius.
In this work we have introduced two party pseudo telepathy games with respective winning conditions. One cannot win these games deterministically by any kind of classical protocols. Interestingly we find out that in quantum world, these winning conditions can be achieved if the players share an entangled state. We also introduced a game which is impossible to win if the players are form classical world, (both probabilistically and deterministically) yet there exists a perfect quantum strategy by following which, one can attain the winning condition of the game.
The Mermin-Peres magic square game is a cooperative two-player nonlocal game in which shared quantum entanglement allows the players to win with certainty, while players limited to classical operations cannot do so, a phenomenon dubbed "quantum pseudo-telepathy". The game has a referee separately ask each player to color a subset of a 3x3 grid. The referee checks that their colorings satisfy certain parity constraints that can't all be simultaneously realized. We define a generalization of these games to be played on an arbitrary arrangement of intersecting sets of elements. We characterize exactly which of these games exhibit quantum pseudo-telepathy, and give quantum winning strategies for those that do. In doing so, we show that it suffices for the players to share three Bell pairs of entanglement even for games on arbitrarily large arrangements. Moreover, it suffices for Alice and Bob to use measurements from the three-qubit Pauli group. The proof technique uses a novel connection of Mermin-style games to graph planarity.
Pseudo-telepathy provides an intuitive way of looking at Bell's inequalities, in which it is often obvious that feats achievable by use of quantum entanglement would be classically impossible. A~two-player pseudo-telepathy game proceeds as follows: Alice and Bob are individually asked a question and they must provide an answer. They are \emph{not} allowed any form of communication once the questions are asked, but they may have agreed on a common strategy prior to the execution of the game. We~say that they \emph{win} the game if the questions and answers fulfil a specific relation. A~game exhibits \emph{pseudo-telepathy} if there is a quantum strategy that makes Alice and Bob win the game for all possible questions, provided they share prior entanglement, whereas it would be impossible to win this game systematically in a classical setting. In~this paper, we show that any two-player pseudo-telepathy game requires the quantum players to share an entangled quantum system of dimension at least~\mbox{3×33 \times 33×3}. This is optimal for two-player games, but the most efficient pseudo-telepathy game possible, in terms of total dimension, involves \emph{three} players who share a quantum system of dimension~\mbox{2×2×22 \times 2 \times 22×2×2}.