A Parallel Repetition Theorem for Entangled Projection Games

Dinur, Irit; Steurer, David; Vidick, Thomas

doi:10.1109/ccc.2014.28

Cited by 14 publications

(11 citation statements)

References 41 publications

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“…Matrix approach can, as in Theorem 1.11, also treat tensor product of different games. If we consider a ''quantum version'' of two-prover one-round games, the situation becomes a bit different: Both of information theoretic [13,28] and matrix analysis [18] approaches showed parallel repetition theorems for different classes of games.…”

Section: Tamakimentioning

confidence: 99%

Parallel Repetition of Two-Prover One-Round Games: An Exposition

Tamaki

2015

IIS

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A two-prover one-round game is a fundamental combinatorial optimization problem arising from such areas as interactive proof systems, hardness of approximation, cryptography and quantum mechanics. The parallel repetition theorem states that for any two-prover one-round game with value smaller than 1, k-fold parallel repetition reduces the value of the game exponentially in k. The theorem was originally proved by Raz (SICOMP 1998) and later simplified and improved by Holenstein (Theory of Computing 2009) and Rao (SICOMP 2011). All the known proofs are based on information theoretic arguments. Very recently, Dinur and Steurer (STOC 2014) obtained a new proof of the parallel repetition theorem based on a matrix analysis argument. In this paper, we describe a special case of Dinur and Steurer's proof. We also describe an application of the parallel repetition theorem to inapproximability results of two-prover one-round games. Our presentation is almost self-contained in the sense that we only assume the PCP theorem. To do so, we also present proofs for the necessary results related to algebraic graph theory and hardness of approximation.

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Section: Tamakimentioning

confidence: 99%

Parallel Repetition of Two-Prover One-Round Games: An Exposition

Tamaki

2015

IIS

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“…In a different line of work, Dinur, Steurer and Vidick show that projection games (with an arbitrary input distribution) also have an exponential decay in entangled value under parallel repetition: if G be a 2-player projection game with classical inputs and outputs, and val * (G) = 1−ǫ, then val * (G ⊗n ) ≤ (1− ǫ 12 ) Ω(n) [DSV14]. This result is not comparable with our work, nor with the work of [CS14b,JPY14].…”

Section: Related Workmentioning

confidence: 99%

“…This result is known as the Parallel Repetition Theorem, and is central in the study of hardness of approximation, probabilistically checkable proofs, and hardness amplification in classical theoretical computer science.Recently, quantum analogues of the Parallel Repetition Theorem have been studied, and for certain types of games, it has been shown that the entangled game value also goes down exponentially with the number of repetitions. In particular, parallel repetition theorems have been shown for 2-player free games (see [CS14a,CS14b,JPY14]) and projection games (see [DSV14]). Free games are where the input distribution to the players is a product distribution (i.e.…”

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confidence: 99%

Parallel repetition for entangled k-player games via fast quantum search

Chung¹,

Wu²,

Yuen³

2015

Preprint

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We present two parallel repetition theorems for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our first theorem shows that for a k-player free game G with entangled value val * (G) = 1 − ǫ, the n-fold repetition of G has entangled value val * (G ⊗n ) at most (1 − ǫ 3/2 ) Ω(n/sk 4 ) , where s is the answer length of any player. In contrast, the best known parallel repetition theorem for the classical value of two-player free games is val(G ⊗n ) ≤ (1 − ǫ 2 ) Ω(n/s) , due to Barak, et al. (RANDOM 2009). This suggests the possibility of a separation between the behavior of entangled and classical free games under parallel repetition.Our second theorem handles the broader class of free games G where the players can output (possibly entangled) quantum states. For such games, the repeated entangled value is upper bounded by (1 − ǫ 2 ) Ω(n/sk 2 ) . We also show that the dependence of the exponent on k is necessary: we exhibit a k-player free game G and n ≥ 1 such that val * (G ⊗n ) ≥ val * (G) n/k .Our analysis exploits the novel connection between communication protocols and quantum parallel repetition, first explored by Chailloux and Scarpa (ICALP 2014). We demonstrate that better communication protocols yield better parallel repetition theorems: in particular, our first theorem crucially uses a quantum search protocol by Aaronson and Ambainis, which gives a quadratic Grover speed-up for distributed search problems. Finally, our results apply to a broader class of games than were previously considered before; in particular, we obtain the first parallel repetition theorem for entangled games involving more than two players, and for games involving quantum outputs.val * (G) of game G, which is the maximum success probability over all possible entangled strategies for the players.Recently, there has been significant interest in the parallel repetition of entangled games [KV11, CS14a, CS14b, JPY14, DSV14]. More formally, the n-fold parallel repetition of a game G is a game G ⊗n where the referee will sample n independent pairs of questions (x 1 , y 1 ), . . . , (x n , y n ) from the distribution µ. Alice receives (x 1 , . . . , x n ) and Bob receives (y 1 , . . . , y n ). They produce outputs (a 1 , . . . , a n ) and (b 1 , . . . , b n ), respectively, and they win only if V (x i , y i , a i , b i ) = 1 for all i. We call each i a "coordinate" of G ⊗n or "repetition" of G.Suppose we have a game G where val * (G) = 1 − ǫ. Intuitively, one should expect that val * (G ⊗n ) should behave as (1−ǫ) n . Indeed, this would be the case if the game G were played n times sequentially. However, there are counterexamples of games G and n > 1 where val * (G ⊗n ) = val * (G) (see Section 7). Despite such counterexamples, it has been shown that the classical value val(G ⊗n ) (i.e. where the players are restricted to using classical strategies) of a repeated game G ⊗n goes down exponentially with n, for large enough n [Raz98, Hol07]. This result is known as the Parallel...

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“…Note here that the non trivial part is that Bob's second measurement is applied on the post-measurement state resulting from his first measurement. Because we are in the quantum setting, this first measurement will generally perturb the state shared by Alice and Bob, which makes it non trivial to relate the success probability of this strategy for G coup with the entangled value ω * (G) of the original game G. A similar construction of squared games was introduced in [DS14,DSV15] to study projective classical and entangled games. There, the input x is not revealed to the players but they receive respectively y and y and output b and b .…”

Section: Consecutive Measurementsmentioning

confidence: 99%

Relativistic (or 2-Prover 1-Round) Zero-Knowledge Protocol for $$\mathsf {NP}$$ Secure Against Quantum Adversaries

Chailloux¹,

Leverrier²

2017

Lecture Notes in Computer Science

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Abstract. In this paper, we show that the zero-knowledge construction for Hamiltonian Cycle remains secure against quantum adversaries in the relativistic setting. Our main technical contribution is a tool for studying the action of consecutive measurements on a quantum state which in turn gives upper bounds on the value of some entangled games. This allows us to prove the security of our protocol against quantum adversaries. We also prove security bounds for the (single-round) relativistic string commitment and bit commitment in parallel against quantum adversaries. As an additional consequence of our result, we answer an open question from [Unr12] and show tight bounds on the quantum knowledge error of some Σ-protocols.

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A Parallel Repetition Theorem for Entangled Projection Games

Cited by 14 publications

References 41 publications

Parallel Repetition of Two-Prover One-Round Games: An Exposition

Parallel Repetition of Two-Prover One-Round Games: An Exposition

Parallel repetition for entangled k-player games via fast quantum search

Relativistic (or 2-Prover 1-Round) Zero-Knowledge Protocol for $$\mathsf {NP}$$ Secure Against Quantum Adversaries

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