In this note, we show that $O(\log (1/ε))$ rounds of parallel repetition with a threshold suffice to reduce completeness and soundness error to $ε$ for single-prover quantum interactive proof systems. This improves on a previous $O(\log (1/ε) \log \log (1/ε))$ bound from Hornby (2018), while also simplifying its proof. A key element in our proof is a concentration bound from Impagliazzo and Kabanets (2010).
We consider one-round games between a classical verifier and two provers. One of the main questions in this area is the \emph{parallel repetition question}: If the game is played $\ell$ times in parallel, does the maximum winning probability decay exponentially in $\ell$? In the classical setting, this question was answered in the affirmative by Raz. More recently the question arose whether the decay is of the form $(1-Θ(\eps))^\ell$ where $1-\eps$ is the value of the game and $\ell$ is the number of repetitions. This question is known as the \emph{strong parallel repetition question} and was …
We prove that parallel repetition of the (3-player) GHZ game reduces the value of the game polynomially fast to 0. That is, the value of the GHZ game repeated in parallel $t$ times is at most $t^{-Ω(1)}$. Previously, only a bound of $\approx \frac{1}{α(t)}$, where $α$ is the inverse Ackermann function, was known. The GHZ game was recently identified by Dinur, Harsha, Venkat and Yuen as a multi-player game where all existing techniques for proving strong bounds on the value of the parallel repetition of the game fail. Indeed, to prove our result we use a completely new proof technique. Dinur, H…
The behavior of games repeated in parallel, when played with quantumly entangled players, has received much attention in recent years. Quantum analogues of Raz's classical parallel repetition theorem have been proved for many special classes of games. However, for general entangled games no parallel repetition theorem was known. We prove that the entangled value of a two-player game $G$ repeated $n$ times in parallel is at most $c_G n^{-1/4} \log n$ for a constant $c_G$ depending on $G$, provided that the entangled value of $G$ is less than 1. In particular, this gives the first proof that the…
We use a recently discovered constrained de Finetti reduction (aka "Post-Selection Lemma") to study the parallel repetition of multi-player non-local games under no-signalling strategies. Since the technique allows us to reduce general strategies to independent plays, we obtain parallel repetition (corresponding to winning all rounds) in the same way as exponential concentration of the probability to win a fraction larger than the value of the game. Our proof technique leads us naturally to a relaxation of no-signalling (NS) strategies, which we dub sub-no-signalling (SNOS). While for two play…
In this thesis, we answer several questions about the behaviour of prover-verifier interactions under parallel repetition when quantum information is allowed, and the verifier acts independently in them. We first consider the case in which a value is associated with each of the possible outcomes of an interaction. We prove that it is not possible for the prover to improve on the optimum average value per repetition by repeating the protocol multiple times in parallel. We look then at games in which the outcomes are classified into two types, winning outcomes and losing outcomes. We ask what is…
We show a parallel repetition theorem for the entangled value $ω^*(G)$ of any two-player one-round game $G$ where the questions $(x,y) \in \mathcal{X}\times\mathcal{Y}$ to Alice and Bob are drawn from a product distribution on $\mathcal{X}\times\mathcal{Y}$. We show that for the $k$-fold product $G^k$ of the game $G$ (which represents the game $G$ played in parallel $k$ times independently), $ ω^*(G^k) =\left(1-(1-ω^*(G))^3\right)^{Ω\left(\frac{k}{\log(|\mathcal{A}| \cdot |\mathcal{B}|)}\right)} $, where $\mathcal{A}$ and $\mathcal{B}$ represent the sets from which the answers of Alice and Bob…
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 $\mathrm{val}^*(G) = 1 - ε$, the $n$-fold repetition of $G$ has entangled value $\mathrm{val}^*(G^{\otimes n})$ at most $(1 - ε^{3/2})^{Ω(n/sk^4)}$, 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 $\mathrm{val}(G^{\otimes n}) \leq (1 - ε^2)^{Ω(n/s)…
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 …
We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game $G$ of entangled value 1-eps < 1, the value of the $k$-fold repetition of G goes to zero as O((1-eps^c)^k), for some universal constant c\geq 1. Previously parallel repetition with an exponential decay in $k$ was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework recently introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proo…
In this work we show a barrier towards proving a randomness-efficient parallel repetition, a promising avenue for achieving many tight inapproximability results. Feige and Kilian (STOC'95) proved an impossibility result for randomness-efficient parallel repetition for two prover games with small degree, i.e., when each prover has only few possibilities for the question of the other prover. In recent years, there have been indications that randomness-efficient parallel repetition (also called derandomized parallel repetition) might be possible for games with large degree, circumventing the impo…
We present a strong parallel repetition theorem for the entangled value of multi-player, one-round free games (games where the inputs come from a product distribution). Our result is the first parallel repetition theorem for entangled games involving more than two players. Furthermore, our theorem applies to games where the players are allowed to output (possibly entangled) quantum states as answers. More specifically, let $G$ be a $k$-player free game, with entangled value $\mathrm{val}^*(G) = 1 - ε$. We show that the entangled value of the $n$-fold repetition of $G$, $\mathrm{val}^*(G^{\otim…
We propose an analytical framework for studying parallel repetition, a basic product operation for one-round two-player games. In this framework, we consider a relaxation of the value of a game, $\mathrm{val}_+$, and prove that for projection games, it is both multiplicative (under parallel repetition) and a good approximation for the true value. These two properties imply a parallel repetition bound as $$ \mathrm{val}(G^{\otimes k}) \approx \mathrm{val}_+(G^{\otimes k}) = \mathrm{val}_+(G)^{k} \approx \mathrm{val}(G)^{k}. $$ Using this framework, we can also give a short proof for the NP-hard…
We give a new proof of the fact that the parallel repetition of the (3-player) GHZ game reduces the value of the game to zero polynomially quickly. That is, we show that the value of the $n$-fold GHZ game is at most $n^{-Ω(1)}$. This was first established by Holmgren and Raz [HR20]. We present a new proof of this theorem that we believe to be simpler and more direct. Unlike most previous works on parallel repetition, our proof makes no use of information theory, and relies on the use of Fourier analysis. The GHZ game [GHZ89] has played a foundational role in the understanding of quantum inform…
We show that the value of the $n$-fold repeated GHZ game is at most $2^{-Ω(n)}$, improving upon the polynomial bound established by Holmgren and Raz. Our result is established via a reduction to approximate subgroup type questions from additive combinatorics.
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 (SICOMP 2000), and has the property that if the quantum value of the original game $G$ is $v$ then the quantum value of the anchored game $G_\bot$ is $1 - (1 - α)^2 \cdot (1 - v)$ where $α$ is a parameter of the transformation. In particular the anchored game has quantum value $1$ if and only if the original g…
We investigate the value of parallel repetition of one-round games with any number of players $k\ge 2$. It has been an open question whether an analogue of Raz's Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whet…
For distinguishing quantum states sampled from a fixed ensemble, the gap in bipartite and single-party distinguishability can be interpreted as a nonlocality of the ensemble. In this paper, we consider bipartite state discrimination in a composite system consisting of $N$ subsystems, where each subsystem is shared between two parties and the state of each subsystem is randomly sampled from a particular ensemble comprising the Bell states. We show that the success probability of perfectly identifying the state converges to $1$ as $N\rightarrow\infty$ if the entropy of the probability distributi…
We prove that for every 3-player game with binary questions and answers and value $<1$, the value of the $n$-fold parallel repetition of the game decays polynomially fast to 0. That is, for every such game, there exists a constant $c>0$, such that the value of the $n$-fold parallel repetition of the game is at most $n^{-c}$. Along the way to proving this theorem, we prove two additional parallel repetition theorems for multiplayer games, that may be of independent interest: Playerwise Connected Games (with any number of players and any Alphabet size): We identify a large class of multiplayer g…
We prove that a sufficiently strong parallel repetition theorem for a special case of multiplayer (multiprover) games implies super-linear lower bounds for multi-tape Turing machines with advice. To the best of our knowledge, this is the first connection between parallel repetition and lower bounds for time complexity and the first major potential implication of a parallel repetition theorem with more than two players. Along the way to proving this result, we define and initiate a study of block rigidity, a weakening of Valiant's notion of rigidity. While rigidity was originally defined for ma…
Consider a game where a refereed a referee chooses (x,y) according to a publicly known distribution P_XY, sends x to Alice, and y to Bob. Without communicating with each other, Alice responds with a value "a" and Bob responds with a value "b". Alice and Bob jointly win if a publicly known predicate Q(x,y,a,b) holds. Let such a game be given and assume that the maximum probability that Alice and Bob can win is v<1. Raz (SIAM J. Comput. 27, 1998) shows that if the game is repeated n times in parallel, then the probability that Alice and Bob win all games simultaneously is at most v'^(n/log(s)), …
Strong Parallel Repetition for Unique Games on Small Set Expanders The strong parallel repetition problem for unique games is to efficiently reduce the 1-delta vs. 1-C*delta gap problem of Boolean unique games (where C>1 is a sufficiently large constant) to the 1-epsilon vs. epsilon gap problem of unique games over large alphabet. Due to its importance to the Unique Games Conjecture, this problem garnered a great deal of interest from the research community. There are positive results for certain easy unique games (e.g., unique games on expanders), and an impossibility result for hard unique g…
We prove that for every 3-player (3-prover) game $\mathcal G$ with value less than one, whose query distribution has the support $\mathcal S = \{(1,0,0), (0,1,0), (0,0,1)\}$ of hamming weight one vectors, the value of the $n$-fold parallel repetition $\mathcal G^{\otimes n}$ decays polynomially fast to zero; that is, there is a constant $c = c(\mathcal G)>0$ such that the value of the game $\mathcal G^{\otimes n}$ is at most $n^{-c}$. Following the recent work of Girish, Holmgren, Mittal, Raz and Zhan (STOC 2022), our result is the missing piece that implies a similar bound for a much more gen…
A PCP is a proof system for NP in which the proof can be checked by a probabilistic verifier. The verifier is only allowed to read a very small portion of the proof, and in return is allowed to err with some bounded probability. The probability that the verifier accepts a false proof is called the soundness error, and is an important parameter of a PCP system that one seeks to minimize. Constructing PCPs with sub-constant soundness error and, at the same time, a minimal number of queries into the proof (namely two) is especially important due to applications for inapproximability. In this work…
What is the least surface area of a symmetric body $B$ whose $\mathbb{Z}^n$ translations tile $\mathbb{R}^n$? Since any such body must have volume $1$, the isoperimetric inequality implies that its surface area must be at least $Ω(\sqrt{n})$. Remarkably, Kindler et al.\ showed that for general bodies $B$ this is tight, i.e.\ that there is a tiling body of $\mathbb{R}^n$ whose surface area is $O(\sqrt{n})$. In theoretical computer science, the tiling problem is intimately to the study of parallel repetition theorems (which are an important component in PCPs), and more specifically in the questi…
We prove that QIP(2), the class of problems having two-message quantum interactive proof systems, is a subset of PSPACE. This relationship is obtained by means of an efficient parallel algorithm, based on the multiplicative weights update method, for approximately solving a certain class of semidefinite programs.
In a recent breakthrough, Mahadev constructed an interactive protocol that enables a purely classical party to delegate any quantum computation to an untrusted quantum prover. In this work, we show that this same task can in fact be performed non-interactively and in zero-knowledge. Our protocols result from a sequence of significant improvements to the original four-message protocol of Mahadev. We begin by making the first message instance-independent and moving it to an offline setup phase. We then establish a parallel repetition theorem for the resulting three-message protocol, with an asym…
This paper presents stronger methods of achieving perfect completeness in quantum interactive proofs. First, it is proved that any problem in QMA has a two-message quantum interactive proof system of perfect completeness with constant soundness error, where the verifier has only to send a constant number of halves of EPR pairs. This in particular implies that the class QMA is necessarily included by the class QIP_1(2) of problems having two-message quantum interactive proofs of perfect completeness, which gives the first nontrivial upper bound for QMA in terms of quantum interactive proofs. It…
In two-prover one-round interactive proof systems, no-signaling provers are those who are allowed to use arbitrary strategies, not limited to local operations, as long as their strategies cannot be used for communication between them. Study of multi-prover interactive proof systems with no-signaling provers is motivated by study of those with provers sharing quantum states. The relation between them is that no-signaling strategies include all the strategies realizable by provers sharing arbitrary entangled quantum states, and more. This paper shows that two-prover one-round interactive proof s…
A recent result of Moshkovitz \cite{Moshkovitz14} presented an ingenious method to provide a completely elementary proof of the Parallel Repetition Theorem for certain projection games via a construction called fortification. However, the construction used in \cite{Moshkovitz14} to fortify arbitrary label cover instances using an arbitrary extractor is insufficient to prove parallel repetition. In this paper, we provide a fix by using a stronger graph that we call fortifiers. Fortifiers are graphs that have both $\ell_1$ and $\ell_2$ guarantees on induced distributions from large subsets. We t…
We consider the problem of a particular kind of quantum correlation that arises in some two-party games. In these games, one player is presented with a question they must answer, yielding an outcome of either 'win' or 'lose'. Molina and Watrous (arXiv:1104.1140) studied such a game that exhibited a perfect form of hedging, where the risk of losing a first game can completely offset the corresponding risk for a second game. This is a non-classical quantum phenomenon, and establishes the impossibility of performing strong error-reduction for quantum interactive proof systems by parallel repetiti…
