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arXiv:1801.00821 · ITCS

Algorithms, Bounds, and Strategies for Entangled XOR Games

Adam Bene Watts, Aram W. Harrow, Gurtej Kanwar, Anand Natarajan

Published Jan 2, 2018• 3 citations
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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 $C_k^{\text{unsat}}$ 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 $C_k^{\text{unsat}} n$. 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.

DOI
10.48550/arxiv.1801.00821

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Entangled games are a quantum analog of constraint satisfaction problems and have had important applications to quantum complexity theory, quantum cryptography, and the foundations of quantum mechanics. Given a game, the basic computational problem is to compute its entangled value: the supremum success probability attainable by a quantum strategy. We study the complexity of computing the (commuting-operator) entangled value omega^* of entangled XOR games with any number of players. Based on a duality theory for systems of operator equations, we introduce necessary and sufficient criteria for an XOR game to have omega^* = 1, and use these criteria to derive the following results: 1) An algorithm for symmetric games that decides in polynomial time whether omega^* = 1 or omega^* < 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 omega^* < 1, where it takes doubly exponential time for the ncSoS algorithm to witness this. By contrast, our algorithm runs in polynomial time. 3) Existence of an unsatisfiable phase for random (non-symmetric) XOR games. We show that there exists a constant C_k^{unsat} depending only on the number k of players, such that a random k-XOR game over an alphabet of size n has omega^* < 1 with high probability when the number of clauses is above C_k^{unsat} n. 4) A lower bound of Omega(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 (3n)^{th} level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.
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We study multipartite entanglement in the context of XOR games. In particular, we study the ratio of the entangled and classical \emph{biases}, which measure the maximum advantage of a quantum or classical strategy over a uniformly random strategy. For the case of two-player XOR games, Tsirelson proved that this ratio is upper bounded by the celebrated Grothendieck constant. In contrast, \PG proved the existence of entangled states that give quantum players an unbounded advantage over classical players in a three-player XOR game. We show that the multipartite entangled states that are most often seen in today's literature can only lead to a bias that is a constant factor larger than the classical bias. These states include GHZ states, any state local-unitarily equivalent to combinations of GHZ and maximally entangled states shared between different subsets of the players (e.g., stabilizer states), as well as generalizations of GHZ states of the form ∑iαi∣i⟩⋯∣i⟩\sum_i \alpha_i \ket{i}\cdots\ket{i}∑i​αi​∣i⟩⋯∣i⟩ for arbitrary amplitudes αi\alpha_iαi​. Our results have the following surprising consequence: \emph{classical} three-player XOR games do not follow an XOR parallel repetition theorem, even a very weak one. Besides this, we discuss implications of our results for communication complexity and hardness of approximation. Our proofs are based on novel applications of extensions of Grothendieck's inequality, due to Blei and Tonge, and Carne, generalizing Tsirelson's use of Grothendieck's inequality to bound the bias of two-player XOR games.
We bound separations between the entangled and classical values for several classes of nonlocal t-player games. Our motivating question is whether there is a family of t-player XOR games for which the entangled bias is 1 but for which the classical bias goes down to 0, for fixed t. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-m games and show that their classical value is always at least 1/m + (m-1)/m t^{1-t}. Secondly, for free XOR games, in which the input distribution is of product form, we show beta(G) >= beta^*(G)^{2^t} where beta(G) and beta^*(G) are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is 1-epsilon then the classical value is at least 1-O(sqrt{epsilon log k}) where k is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.
We characterize the amount of entanglement that is sufficient to play any XOR game near-optimally. We show that for any XOR game G and \eps>0 there is an \eps-optimal strategy for G using \lceil \eps^{-1} \rceil ebits of entanglement, irrespective of the number of questions in the game. By considering the family of XOR games CHSH(n) introduced by Slofstra (Jour. Math. Phys. 2011), we show that this bound is nearly tight: for any \eps>0 there is an n=\Theta(\eps^{-1/5}) such that \Omega(\eps^{-1/5}) ebits are required for any strategy achieving bias that is at least a multiplicative factor (1-\eps) from optimal in CHSH(n).