[Paper Review] Algorithms, Bounds, and Strategies for Entangled XOR Games

Adam Bene Watts, Aram W. Harrow+2 more|arXiv (Cornell University)|Jan 2, 2018

Complexity and Algorithms in Graphs28 references3 citations

TL;DR

This paper presents a polynomial-time algorithm to determine whether the commuting-operator value of symmetric XOR games is exactly 1, resolving a long-standing decidability question. It introduces novel criteria for perfect quantum strategy realization, constructs explicit entanglement-based strategies, and establishes tight bounds on the NPA (ncSoS) hierarchy’s convergence, demonstrating doubly exponential slowdown in worst-case scenarios and proving the existence of random XOR games with an unsatisfiable phase.

ABSTRACT

We study the complexity of computing the commuting-operator value $ω^*$ of entangled XOR games with any number of players. We introduce necessary and sufficient criteria for an XOR game to have $ω^* = 1$, and use these criteria to derive the following results: 1. An algorithm for symmetric games that decides in polynomial time whether $ω^* = 1$ or $ω^* < 1$, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascués-Pironio-Acín (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known. 2. A family of games with three players and with $ω^* < 1$, where it takes doubly exponential time for the ncSoS algorithm to witness this (in contrast with our algorithm which runs in polynomial time). 3. A family of games achieving a bias difference $2(ω^* - ω)$ arbitrarily close to the maximum possible value of $1$ (and as a consequence, achieving an unbounded bias ratio), answering an open question of Briët and Vidick. 4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games: that is, we show that there exists a constant $C_k^{ ext{unsat}}$ depending only on the number $k$ of players, such that a random $k$-XOR game over an alphabet of size $n$ has $ω^* < 1$ with high probability when the number of clauses is above $C_k^{ ext{unsat}} n$. 5. A lower bound of $Ω(n \log(n)/\log\log(n))$ on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the $n$-th level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.

Motivation & Objective

• To determine whether the quantum value ω* of symmetric XOR games is exactly 1, a problem previously not known to be decidable.
• To construct explicit tensor-product strategies achieving ω* = 1 when possible, providing a constructive solution to the perfect strategy problem.
• To analyze the convergence behavior of the NPA (ncSoS) hierarchy for XOR games, particularly in worst-case scenarios.
• To establish the existence of a phase transition in random non-symmetric XOR games, where ω* < 1 with high probability when the number of clauses exceeds a threshold.
• To answer an open question on bias ratio in XOR games by constructing families with arbitrarily large differences between quantum and classical values.

Proposed method

• Introduces necessary and sufficient criteria for ω* = 1 in XOR games based on parity-permuted refutations (PREFs) and MERP (maximal entanglement, relative phase) strategies.
• Develops an algorithm for symmetric games that checks for the existence of a valid PREF refutation in polynomial time.
• Uses combinatorial tools and shuffle gadgets to construct and verify refutations, enabling efficient detection of unsatisfiability.
• Applies probabilistic analysis to bound the expected number of successful refutation paths in random games, leveraging Catalan number estimates and query mapping constraints.
• Employs Markov’s inequality and overcounting techniques to derive lower bounds on the number of NPA hierarchy levels required for refutation.
• Leverages duality between MERP strategies and PREF refutations to establish structural insights into quantum advantage in XOR games.

Experimental results

Research questions

• RQ1 Is it decidable whether the commuting-operator value ω* of a symmetric XOR game is exactly 1?
• RQ2 Can a polynomial-time algorithm be constructed to determine whether ω* = 1 for symmetric XOR games?
• RQ3 What is the worst-case convergence rate of the NPA (ncSoS) hierarchy for XOR games with ω* < 1?
• RQ4 Do random non-symmetric XOR games exhibit a phase transition where ω* < 1 with high probability above a certain clause density?
• RQ5 Can XOR games be constructed such that the bias difference 2(ω* − ω) approaches the maximum possible value of 1, leading to an unbounded bias ratio?

Key findings

• An algorithm is presented that decides in polynomial time whether ω* = 1 or ω* < 1 for symmetric XOR games, resolving a previously open decidability question.
• The algorithm constructs a simple tensor-product strategy achieving ω* = 1 when the value is 1, providing a constructive quantum advantage.
• A family of three-player XOR games is constructed where the NPA hierarchy requires doubly exponential time to detect ω* < 1, in contrast to the polynomial-time algorithm.
• The paper proves the existence of an unsatisfiable phase in random k-XOR games: for k ≥ 3, ω* < 1 with high probability when the number of clauses exceeds C_k^unsat * n.
• A family of games is constructed with bias difference 2(ω* − ω) arbitrarily close to 1, answering an open question by Briët and Vidick on unbounded bias ratios.
• A lower bound of Ω(n log n / log log n) is established on the number of NPA hierarchy levels required to detect unsatisfiability in most random 3-XOR games.