The class QMA(k), introduced by , consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Does QMA(k)=QMA(2) for k>=2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on the existence of very short PCPs. Second, we show that assuming a weak version of the Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k)=QMA(2) for all k>=2. Third, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.
In 2003, Kobayashi, Matsumoto, and Yamakami @cite_20 defined a generalization of @math called @math . Here there are @math Merlins, who send Arthur @math quantum proofs @math respectively that are guaranteed to be unentangled with each other. (Thus @math .) Notice that in the classical case, this generalization is completely uninteresting: we have @math for all @math , since we can always simulate @math Merlins by a single Merlin who sends Arthur a concatenation of the @math proofs. In the quantum case, however, a single Merlin could cheat by the @math proofs, and we know of no general way to detect such entanglement.
@cite_20
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This paper introduces quantum multiple-Merlin''-Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multi-proof systems are obviously equivalent to classical single-proof systems (i.e., usual Merlin-Arthur proof systems), it is unclear whether or not quantum multi-proof systems collapse to quantum single-proof systems (i.e., usual quantum Merlin-Arthur proof systems). This paper presents a necessary and sufficient condition under which the number of quantum proofs is reducible to two. It is also proved that, in the case of perfect soundness, using multiple quantum proofs does not increase the power of quantum Merlin-Arthur proof systems. It is well known that probabilistic boolean decision trees cannot be much more powerful than deterministic ones (N. Nisan, SIAM J. Comput.20, No. 6 (1991), 999?1007). Motivated by a question if randomization can significantly speed up a nondeterministic computation via a boolean decision tree, we address structural properties of Arthur?Merlin games in this model and prove some lower bounds. We consider two cases of interest, the first when the length of communication between the players is limited and the second, if it is not. While in the first case we can carry over the relations between the corresponding Turing complexity classes, in the second case we observe in contrast with Turing complexity that a one-round Merlin?Arthur protocol is as powerful as a general interactive proof system and, in particular, can simulate a one-round Arthur?Merlin protocol. Moreover, we show that sometimes a Merlin?Arthur protocol can be more efficient than an Arthur?Merlin protocol and than a Merlin?Arthur protocol with limited communication. This is the case for a boolean function whose set of zeroes is a code with high minimum distance and a natural uniformity condition. Such functions provide an example when the Merlin?Arthur complexity is 1 with one-sided error ??(23, 1), but at the same time the nondeterministic decision tree complexity is ?(n). The latter should be contrasted with another fact we prove. Namely, if a function has Merlin?Arthur complexity 1 with one-sided error probability ??(0, 23, then its nondeterministic complexity is bounded by a constant. Other results of the paper include connections with the block sensitivity and related combinatorial properties of a boolean function. We show that any language in nondeterministic time @math , where the number of iterated exponentials is an arbitrary function @math , can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness @math and soundness @math , where the number of iterated exponentials is @math and @math is a universal constant. The result was previously known for @math and @math ; we obtain it for any time-constructible function @math . The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC'17). As a separate consequence of this technique we obtain a different proof of Slofstra's recent result (unpublished) on the uncomputability of the entangled value of multiprover games. Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP* contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson's problem on the relation between the commuting operator and tensor product models for quantum correlations. In quantum zero knowledge, the assumption was made that the verifier is only using unitary operations. Under this assumption, many nice properties have been shown about quantum zero knowledge, including the fact that Honest-Verifier Quantum Statistical Zero Knowledge ( @math ) is equal to Cheating-Verifier Quantum Statistical Zero Knowledge ( @math ) (see Wat02,Wat06 ). In this paper, we study what happens when we allow an honest verifier to flip some coins in addition to using unitary operations. Flipping a coin is a non-unitary operation but doesn 't seem at first to enhance the cheating possibilities of the verifier since a classical honest verifier can flip coins. In this setting, we show an unexpected result: any classical Interactive Proof has an Honest-Verifier Quantum Statistical Zero Knowledge proof with coins. Note that in the classical case, honest verifier @math is no more powerful than @math and hence it is not believed to contain even @math . On the other hand, in the case of cheating verifiers, we show that Quantum Statistical Zero Knowledge where the verifier applies any non-unitary operation is equal to Quantum Zero-Knowledge where the verifier uses only unitaries. One can think of our results in two complementary ways. If we would like to use the honest verifier model as a means to study the general model by taking advantage of their equivalence, then it is imperative to use the unitary definition without coins, since with the general one this equivalence is most probably not true. On the other hand, if we would like to use quantum zero knowledge protocols in a cryptographic scenario where the honest-but-curious model is sufficient, then adding the unitary constraint severely decreases the power of quantum zero knowledge protocols. We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is @math -complete to decide whether the classical value of a non-local game is 1 or @math . Furthermore, as long as @math is small enough, this result does not depend on the gap @math . In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap @math decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of @math (proofs with two provers, one round, completeness probability @math , soundness probability @math , and commuting-operator strategies) can increase without bound as the gap @math gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class @math - @math of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap @math gets arbitrarily small. While we do not know any computable time upper bound on the class @math , a result of the first author and Vidick shows that for @math and @math , the class @math , with constant communication from the provers, is contained in @math . We give a lower bound of @math (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis. Suppose that one of n people knows a rumor. At the first stage, he passes the rumor to someone chosen at random; at each stage, each person already informed (“knower”) communicates the rumor to a person chosen at random and independently of all other past and present choices. Denote by @math the random number of stages before everybody is informed. How large is @math typically? Frieze and Grimmet, who introduced this problem, proved that, in probability, @math . In this paper we show that, in fact, @math in probability. Our proof demonstrates that the number @math of persons informed after t stages obeys very closely, with high probability, a deterministic equation @math , @math . A case when each knower passes the rumor to several members at every stage is also discussed. We show that the greedy spanner algorithm constructs a @math -spanner of weight @math for a point set in metrics of doubling dimension @math , resolving an open problem posed by Gottlieb. Our result generalizes the result by Narasimhan and Smid who showed that a point set in @math -dimension Euclidean space has a @math -spanner of weight at most @math . Our proof only uses the packing property of doubling metrics and thus implies a much simpler proof for the same result in Euclidean space. This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given @math binary matrix @math , the GM-MDS conjecture, proposed by , states that if @math satisfies the so-called MDS condition, then for any field @math of size @math , there exists an @math MDS code whose generator matrix @math , with entries in @math , fits the matrix @math (i.e., @math is the support matrix of @math ). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by and , that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if @math satisfies the MDS condition, then the determinant of a transform matrix @math , such that @math fits @math , is not identically zero, where @math is a Vandermonde matrix with distinct parameters. In this work, we first reformulate the TM-MDS conjecture in terms of the Wronskian determinant, and then present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ( @math ) of @math is upper bounded by @math . For this class of special cases of @math where the only additional constraint is on @math , only cases with @math were previously proven theoretically, and the previously used proof techniques are not applicable to cases with @math . A family of sets is said to be if any two sets in the family have nonempty intersection. In 1973, Erd o s raised the problem of determining the maximum possible size of a union of @math different intersecting families of @math -element subsets of an @math -element set, for each triple of integers @math . We make progress on this problem, proving that for any fixed integer @math and for any @math , if @math is an @math -element set, and @math , where each @math is an intersecting family of @math -element subsets of @math , then @math , with equality only if @math for some @math with @math . This is best possible up to the size of the @math term, and improves a 1987 result of Frankl and F "uredi, who obtained the same conclusion under the stronger hypothesis @math , in the case @math . Our proof utilises an isoperimetric, influence-based method recently developed by Keller and the authors. Consider two parties holding samples from correlated distributions @math and @math , respectively, where these samples are within distance @math of each other in some metric space. The parties wish to agree on a close-to-uniformly distributed secret key @math by sending a single message over an insecure channel controlled by an all-powerful adversary who may read and modify anything sent over the channel. We consider both the keyless case, where the parties share no additional secret information, and the keyed case, where the parties share a long-term secret @math that they can use to generate a sequence of session keys @math using multiple pairs @math . The former has applications to, e.g., biometric authentication, while the latter arises in, e.g., the bounded-storage model with errors. We show solutions that improve upon previous work in several respects. The best prior solution for the keyless case with no errors (i.e., @math ) requires the min-entropy of @math to exceed @math , where @math is the bit length of @math . Our solution applies whenever the min-entropy of @math exceeds the minimal threshold @math , and yields a longer key. We study arithmetic proof systems @math and @math operating with arithmetic circuits and arithmetic formulas, respectively, and that prove polynomial identities over a field @math . We establish a series of structural theorems about these proof systems, the main one stating that @math proofs can be balanced: if a polynomial identity of syntactic degree @math and depth @math has a @math proof of size @math , then it also has a @math proof of size @math in which every circuit has depth @math . As a corollary, we obtain a quasi-polynomial simulation of @math by @math . Using these results we obtain the following: consider the identities @math where @math , and @math are @math square matrices and @math is a triangular matrix with $z_... Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is @math for @math and @math when @math . In the case @math , the process is the concatenation of two BBM extremal processes, as expected. In the case @math , a new family of cluster point processes arises, that are similar, but distinctively different from the BBM process. Our proofs follow the strategy of Arguin, Bovier, and Kistler. In this paper we show that sufficient multi-partite quantum entanglement helps in fair and unbiased election of a leader in a distributed network of processors with only linear classical communication complexity. We show that a total of @math distinct multi-partite maximally entanglement sets (ebits) are capable of supporting such a protocol in the presence of nodes that may lie and thus be biased. Here, @math is the number of nodes in the network. We also demonstrate the difficulty of performing unbiased and fair election of a leader with linear classical communication complexity in the absence of quantum entanglement even if all nodes have perfect random bit generators. We show that the presence of a sufficient number @math of biased agents leads to a non-zero limiting probability of biased election of the leader, whereas, the presence of a smaller number @math of biased agents matters little. We define two new related complexity classes motivated by the our leader election problem and discuss a few open questions. In this paper we present several results on the expected complexity of a convex hull of @math points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of @math points, chosen uniformly and independently from a disk is @math , and @math for the case a convex polygon with @math sides. Those results are well known (see rs-udkhv-63,r-slcdn-70,ps-cgi-85 ), but we believe that the elementary proof given here are simpler and more intuitive. (ii) Let @math be a set of directions in the plane, we define a generalized notion of convexity induced by @math , which extends both rectilinear convexity and standard convexity. We prove that the expected complexity of the @math -convex hull of a set of @math points, chosen uniformly and independently from a disk, is @math , where @math is the largest angle between two consecutive vectors in @math . This result extends the known bounds for the cases of rectilinear and standard convexity. (iii) Let @math be an axis parallel hypercube in @math . We prove that the expected number of points on the boundary of the quadrant hull of a set @math of @math points, chosen uniformly and independently from @math is @math . Quadrant hull of a set of points is an extension of rectilinear convexity to higher dimensions. In particular, this number is larger than the number of maxima in @math , and is also larger than the number of points of @math that are vertices of the convex hull of @math . Those bounds are known bkst-anmsv-78 , but we believe the new proof is simpler. This text is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type @math , where the rank of @math stays bounded as the dimension goes to infinity and where the matrix @math is a non Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if @math has some eigenvalues out of the maximal circle of the single ring, then @math has some eigenvalues (called outliers) in the neighborhood of those of @math , which is not the case for the eigenvalues of @math in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of @math around the eigenvalues of @math and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such facts had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of @math ) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in the Hermitian case, but only in the case of non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a previous result by Tao for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new. We show an optimal data-dependent hashing scheme for the approximate near neighbor problem. For an @math -point data set in a @math -dimensional space our data structure achieves query time @math and space @math , where @math for the Euclidean space and approximation @math . For the Hamming space, we obtain an exponent of @math . Our result completes the direction set forth in [AINR14] who gave a proof-of-concept that data-dependent hashing can outperform classical Locality Sensitive Hashing (LSH). In contrast to [AINR14], the new bound is not only optimal, but in fact improves over the best (optimal) LSH data structures [IM98,AI06] for all approximation factors @math . From the technical perspective, we proceed by decomposing an arbitrary dataset into several subsets that are, in a certain sense, pseudo-random. We consider the scenario where Alice wants to send a secret (classical) @math -bit message to Bob using a classical key, and where only one-way transmission from Alice to Bob is possible. In this case, quantum communication cannot help to obtain perfect secrecy with key length smaller then @math . We study the question of whether there might still be fundamental differences between the case where quantum as opposed to classical communication is used. In this direction, we show that there exist ciphers with perfect security producing quantum ciphertext where, even if an adversary knows the plaintext and applies an optimal measurement on the ciphertext, his Shannon uncertainty about the key used is almost maximal. This is in contrast to the classical case where the adversary always learns @math bits of information on the key in a known plaintext attack. We also show that there is a limit to how different the classical and quantum cases can be: the most probable key, given matching plain- and ciphertexts, has the same probability in both the quantum and the classical cases. We suggest an application of our results in the case where only a short secret key is available and the message is much longer. This paper makes two contributions to the areas of anti-concentration and non-asymptotic random matrix theory. First, we study the counting problem in inverse Littlewood-Offord theory for general random variables: for random variables @math which are i.i.d. copies of a random variable @math (satisfying some mild hypotheses), how many integer vectors @math in a prescribed box have large @math ? Building on recent work of Ferber, Jain, Luh, and Samotij (who treated the case when @math is a Rademacher random variable), we provide significantly better bounds for this problem than those obtained using the inverse Littlewood-Offord theorems of Tao and Vu, and Nguyen and Vu. Next, we study the non-asymptotic behavior of the least singular value @math of a random @math square matrix @math with i.i.d. entries, which are only assumed to be centered with variance @math . As an application of our counting theorem, and utilizing and developing a recent work by the author, we show that for all @math , @math , where @math is an absolute constant, thereby providing a considerably simpler proof of an approximate version of an essentially optimal result due to Rebrova and Tikhomirov. In this work we show that for every @math , such that for all @math where the parameters of the model do not depend on @math . They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than @math . Our proof exploits in novel ways the local treelike structure of Erd o s-R 'enyi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex @math of the graph has a neighborhood @math of radius @math in which the induced sub-graph is a tree union at most @math edges and where for each simple path in @math the sum of the vertex degrees along the path is @math . Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for @math it applies for all external fields and @math , where @math is the critical point for decay of correlation for the Ising model on @math .