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arXiv:1603.05349

Parallel repetition via fortification: analytic view and the quantum case

Mohammad Bavarian, Thomas Vidick, Henry Yuen

Published Mar 17, 2016• 6 citations
arXivHTMLPDF

Abstract

In a recent work, Moshkovitz [FOCS '14] presented a transformation on two-player games called "fortification", and gave an elementary proof of an (exponential decay) parallel repetition theorem for fortified two-player projection games. In this paper, we give an analytic reformulation of Moshkovitz's fortification framework, which was originally cast in combinatorial terms. This reformulation allows us to expand the scope of the fortification method to new settings. First, we show any game (not just projection games) can be fortified, and give a simple proof of parallel repetition for general fortified games. Then, we prove parallel repetition and fortification theorems for games with players sharing quantum entanglement, as well as games with more than two players. This gives a new gap amplification method for general games in the quantum and multiplayer settings, which has recently received much interest. An important component of our work is a variant of the fortification transformation, called "ordered fortification", that preserves the entangled value of a game. The original fortification of Moshkovitz does not in general preserve the entangled value of a game, and this was a barrier to extending the fortification framework to the quantum setting.

DOI
10.48550/arxiv.1603.05349

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In a recent work, Moshkovitz [FOCS '14] presented a transformation on two-player games called "fortification", and gave an elementary proof of an (exponential decay) parallel repetition theorem for fortified two-player projection games. In this paper, we give an analytic reformulation of Moshkovitz's fortification framework, which was originally cast in combinatorial terms. This reformulation allows us to expand the scope of the fortification method to new settings. First, we show any game (not just projection games) can be fortified, and give a simple proof of parallel repetition for general fortified games. Then, we prove parallel repetition and fortification theorems for games with players sharing quantum entanglement, as well as games with more than two players. This gives a new gap amplification method for general games in the quantum and multiplayer settings, which has recently received much interest. An important component of our work is a variant of the fortification transformation, called "ordered fortification", that preserves the entangled value of a game. The original fortification of Moshkovitz does not in general preserve the entangled value of a game, and this was a barrier to extending the fortification framework to the quantum setting.
In a recent work, Moshkovitz [FOCS'14] presented a transformation n two-player games called "fortification", and gave an elementary proof of an (exponential decay) parallel repetition theorem for fortified two-player projection games. In this paper, we give an analytic reformulation of Moshkovitz's fortification framework, which was originally cast in combinatorial terms. This reformulation allows us to expand the scope of the fortification method to new settings. First, we show any game (not just projection games) can be fortified, and give a simple proof of parallel repetition for general fortified games. Then, we prove parallel repetition and fortification theorems for games with players sharing quantum entanglement, as well as games with more than two players. This gives a new gap amplification method for general games in the quantum and multiplayer settings, which has recently received much interest. An important component of our work is a variant of the fortification transformation, called "ordered fortification", that preserves the entangled value of a game. The original fortification of Moshkovitz does not in general preserve the entangled value of a game, and this was a barrier to extending the fortification framework to the quantum setting.
The Parallel Repetition Theorem upper-bounds the value of a repeated (tensored) two prover game G⊗k in terms of the value of the game G and the number of repetitions k. Contrary to what one might have guessed, the value of G⊗k is not val(G), but rather a more complicated expression depending on properties of G. Indeed, there are numerous works aiming to understand the value of repeated games, both for general games G and for special cases. A case of particular interest is that of projection games, where the answer of the first prover determines at most one acceptable answer for the second prover. In this work we give a simple transformation, which we call “fortification”, that can be applied to any projection game. We show that for fortified games G, the value of the k-repeated game is approximately val(G). This results in nearly a quadratic improvement in the size of projection PCP with the lowest error known today. Unlike previous proofs of the parallel repetition theorem that relied on information theory or linear algebra, our proof is purely combinatorial and quite short. We then discuss the problem of derandomizing parallel repetition, and the limitations of the fortification idea in this setting. We point out a connection between the problem of derandomizing parallel repetition and the problem of composition. This connection could shed light on the so-called Projection Games Conjecture, which asks for projection PCP with minimal error. ∗dmoshkov@csail.mit.edu. Department of Electrical Engineering and Computer Science, MIT. This material is based upon work supported by the National Science Foundation under Grant Number 1218547.
The Parallel Repetition Theorem upper-bounds the value of a repeated (tensored) two prover game in terms of the value of the base game and the number of repetitions. In this work we give a simple transformation on games -- "fortification" -- and show that for fortified games, the value of the repeated game decreases perfectly exponentially with the number of repetitions, up to an arbitrarily small additive error. Our proof is combinatorial and short. As corollaries, we obtain: (1) Starting from a PCP Theorem with soundness error bounded away from 1, we get a PCP with arbitrarily small constant soundness error. In particular, starting with the combinatorial PCP of Dinur, we get a combinatorial PCP with low error. The latter can be used for hardness of approximation as in the work of Håstad. (2) Starting from the work of the author and Raz, we get a projection PCP theorem with the smallest soundness error known today. The theorem yields nearly a quadratic improvement in the size compared to previous work. We then discuss the problem of derandomizing parallel repetition, and the limitations of the fortification idea in this setting. We point out a connection between the problem of derandomizing parallel repetition and the problem of composition. This connection could shed light on the so-called Projection Games Conjecture, which asks for projection PCP with minimal error.
Two major open problems regarding the parallel repetition of games are whether an analogue of Raz's parallel-repetition theorem holds for (a) games with more than two players, and (b) games with quantum players using entanglement. We make progress on both problems: we introduce a class of games we call anchored, and prove exponential-decay parallel repetition theorems for anchored games in the multiplayer and entangled-player settings. We introduce a simple transformation on games called anchoring and show that this transformation turns any game into an anchored game. Together, our parallel repetition theorem and our anchoring transformation provide a simple and efficient hardness-amplification technique in both the classical multiplayer and quantum settings.
We introduce a simple transformation on two-player nonlocal games, called “anchoring,” and prove an exponential-decay parallel repetition theorem for all anchored games in the setting of quantum entangled players. This transformation is inspired in part by the Feige--Kilian transformation [ SIAM J. Comput., 30 (2000), pp. 324--346], and has the property that if the quantum value of the original game GGG is vvv, then the quantum value of the anchored game G⊥G_{{\perp}}G⊥​ is 1−(1−α)2⋅(1−v)1 - (1 - \alpha)^2 \cdot (1 - v)1−(1−α)2⋅(1−v), where α\alphaα is a parameter of the transformation. In particular the anchored game has quantum value 1 if and only if the original game GGG has quantum value 1. This provides the first gap amplification technique for general two-player nonlocal games that achieves exponential decay of the quantum value.
We obtain quantitative estimates on the decay of the multiplayer optimal value under parallel repetition. In comparison to a previous work of the author in 2025 (arXiv: 2508.09380) which sought to generalize dependency-breaking and anchoring variables from two-player Quantum games, being able to establish quantitative estimates on the decay of the optimal value of a multiplayer game under parallel repetition is of interest to establish under different assumptions. Specifically, independently of the dependency-breaking and anchoring variables that have previously been employed to remove correlations from entangled information shared between Alice and Bob (hence removing dependencies), monotonic concave functions can be used in place of such variables to obtain rates of decay on the optimal value. The game-theoretic setting with two players was first analyzed with monotonic concave functions by Lanzenberger and Maurer. For qi,xi>0q_i , x_i > 0qi​,xi​>0 ∀1≤i≤N\forall 1 \leq i \leq N∀1≤i≤N where $N > 0 $ is the total number of players we adddress an open question raised in their work regarding potential generalizations of two-player monotonic concave functions, through amplification functions of the form ΨMult≡Ψ=N−∏1≤i≤Nexp[−qixi]Ψ_{\textit{Mult}} \equiv Ψ= N - \underset{1 \leq i \leq N}{\prod} \mathrm{exp} \big[ - q_i x_i \big]ΨMult​≡Ψ=N−1≤i≤N∏​exp[−qi​xi​], which in the multiplayer game-theoretic setting have more intricate combinatorial structures.