A strong direct product theorem for quantum query complexity

We show that quantum query complexity satisfies a strong direct product theorem. This means that computing $k$ copies of a function with less than $k$ times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in $k$. For a boolean function $f$ we also show an XOR lemma---computing the parity of $k$ copies of $f$ with less than $k$ times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, is always at least as large as the additive adversary method, which is known to characterize quantum query complexity.Comment: V2: 19 pages (various additions and improvements, in particular: improved parameters in the main theorems due to a finer analysis of the output condition, and addition of an XOR lemma and a threshold direct product theorem in the boolean case). V3: 19 pages (added grant information

Multipartite entanglement in XOR games

We study multipartite entanglement in the context of XOR games. In particular, we study the ratio of the entangled and classical 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, Pérez-García et al. 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 ∑iɑi|i〉...|i〉 for arbitrary amplitudes ɑi. Our results have the following surprising consequence: 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

Concentration Bounds from Parallel Repetition Theorems

This thesis contributes to the study of parallel repetition theorems and concentration bounds for nonlocal games and quantum interactive proofs. We make the following contributions: - A lemma that is useful for converting parallel repetition theorems (bounds on the probability of winning all instances of a nonlocal game which is being repeated in parallel) into concentration bounds (bounds on winning a certain fraction of the instances). - Exponentially-decaying concentration bounds for two-player games on the uniform distribution and k-player free games, against quantum strategies. - A proof that given a quantum interactive proof system with parameters α (the probability with which the verifier can be convinced to accept when they should accept) and β (the soundness error), as long as α > β, both the soundness error and completeness error can be reduced exponentially by repeating the protocol in parallel and requiring an (α + β)/2 fraction of the repetitions to be won. Our result requires a log-factor more repetitions than are necessary in the classical case

Multiplayer XOR games and quantum communication complexity with clique-wise entanglement

XOR games are a simple computational model with connections to many areas of complexity theory. Perhaps the earliest use of XOR games was in the study of quantum correlations. XOR games also have an interesting connection to Grothendieck's inequality, a fundamental theorem of analysis, which shows that two players sharing entanglement can achieve at most a constant factor advantage over players following classical strategies in an XOR game. Perez-Garcia et al. show that when the players share GHZ states, this advantage is bounded by a constant. We use a multilinear generalization of Grothendieck's inequality due to Blei and Tonge to simplify the proof of the second result and extend it to the case of so-called Schmidt states, answering an open problem of Perez-Garcia et al. Via a reduction given in that paper, this answers a 35-year-old problem in operator algebras due to Varopoulos, showing that the space of compact operators on a Hilbert space is a Q-algebra under Schur product. A further generalization of Grothendieck's inequality due to Carne lets us show that the gap between the entangled and classical value is at most a constant in any multiplayer XOR game in which the players are allowed to share combinations of GHZ states and EPR pairs of any dimension. As an application of our results, we show that the discrepancy method in communication complexity remains a lower bound in the multiparty model where the players have quantum communication and the kinds of entanglement discussed above. This answers an open question of Lee, Schechtman, and Shraibman

Improved Direct Product Theorems for Randomized Query Complexity

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The complexity of joint computation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 253-266).Joint computation is the ubiquitous scenario in which a computer is presented with not one, but many computational tasks to perform. A fundamental question arises: when can we cleverly combine computations, to perform them with greater efficiency or reliability than by tackling them separately? This thesis investigates the power and, especially, the limits of efficient joint computation, in several computational models: query algorithms, circuits, and Turing machines. We significantly improve and extend past results on limits to efficient joint computation for multiple independent tasks; identify barriers to progress towards better circuit lower bounds for multiple-output operators; and begin an original line of inquiry into the complexity of joint computation. In more detail, we make contributions in the following areas: Improved direct product theorems for randomized query complexity: The "direct product problem" seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T-query algorithm has success probability at most 1-[epsilon] in computing the Boolean function f on input distribution [mu], then for [alpha] 0, the worst-case success probability of any [alpha]R₂(f)k-query randomized algorithm for f k falls exponentially with k. The best previous statement of this type, due to Klauck, Spalek, and de Wolf, required a query bound of O(bs(f)k). Our proof technique involves defining and analyzing a collection of martingales associated with an algorithm attempting to solve f*k. Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest. Joint complexity in the Decision Tree Model: We study the diversity of possible behaviors of the joint computational complexity of a collection f1,... , fk of Boolean functions over a shared input. We focus on the deterministic decision tree model, with depth as the complexity measure; in this model, we prove a result to the effect that the "obvious" constraints on joint computational complexity are essentially the only ones. The proof uses an intriguing new type of cryptographic data structure called a "mystery bin," which we construct using a polynomial separation between deterministic and unambiguous query complexity shown by Savický. We also pose a conjecture in the communication model which, if proved, would extend our result to that model. Limitations of Lower-Bound Methods for the Wire Complexity of Boolean Operators: We study the circuit complexity of Boolean operators, i.e., collections of Boolean functions defined over a common input. Our focus is the well-studied model in which arbitrary Boolean functions are allowed as gates, and in which a circuit's complexity is measured by its depth and number of wires. We show sharp limitations of several existing lower-bound methods for this model. First, we study an information-theoretic lower-bound method due to Cherukhin, which gave the first improvement over the lower bounds provided by the well-known superconcentrator technique for constant depths. (The lower bounds are still barelysuperlinear, however) Cherukhin's method was formalized by Jukna as a general lower-bound criterion for Boolean operators, the "Strong Multiscale Entropy" (SME) property. It seemed plausible that this property could imply significantly better lower bounds by an improved analysis. However, we show that this is not the case, by exhibiting an explicit operator with the SME property that is computable in constant depths whose wire-complexity essentially matches the Cherukhin-Jukna lower bound (to within a constant multiplicative factor, for depths d = 2,3 and for even depths d >/= 6). Next, we show limitations of two simpler lower-bound criteria given by Jukna: the "entropy method" for general operators, and the "pairwise-distance method" for linear operators. We show that neither method gives super-linear lower bounds for depth 3. In the process, we obtain the first known polynomial separation between the depth-2 and depth-3 wire complexities for an explicit operator. We also continue the study (initiated by Jukna) of the complexity of "representing" a linear operator by bounded-depth circuits, a weaker notion than computing the operator. New limits to classical and quantum instance compression: Given an instance of a decision problem that is too difficult to solve outright, we may aim for the more limited goal of compressing that instance into a smaller, equivalent instance of the same or a different problem. As a representative problem, say we are given Boolean formulas [psi]1,... ,[psi]t, each of length n << t, and we want to determine if at least one [psi]j is satisfiable. Can we efficiently reduce this "OR-SAT" question to an equivalent problem instance (of SAT or another problem) of size poly(n), independent of t? We call any such reduction a "strong compression" reduction for OR-SAT. This would amount to a major gain from compressing [psi]1,. .. , [psi]t jointly, since we know of no way to reliably compress an individual SAT instance. Harnik and Naor (FOCS '06/SICOMP '10) and Bodlaender, Downey, Fellows, and Hermelin (ICALP '08/JCSS '09) showed that the infeasibility of strong compression for OR-SAT would also imply limits to instance compression schemes for a large number of other, natural problems; this is significant because instance compression is a central technique in the design of so-called fixed-parameter tractable algorithms. Bodlaender et al. also showed that the infeasibility of strong compression for the analogous "AND-SAT" problem would establish limits to instance compression for another family of problems. Fortnow and Santhanam (STOC '08) showed that deterministic (or 1-sided error randomized) strong compression for OR-SAT is not possible unless NP C coNP/poly; the case of AND-SAT remained mysterious. We give new and improved evidence against strong compression schemes for both OR-SAT and AND-SAT; our method applies to probabilistic compression schemes with 2-sided error. We also give versions of these results for an analogous task of quantum instance compression, in which a polynomial-time quantum reduction must output a quantum state that, in an appropriate sense, "preserves the answer" to the input instance. We give quantitatively similar evidence against strong compression for AND- and OR-SAT in this setting, albeit under less well-studied hypotheses about the relationship between NP and quantum complexity classes. To prove all of these results, we exploit the information bottleneck of an instance compression scheme, using a new method to "disguise" information being fed into a compressive mapping.by Andrew Donald Drucker.Ph.D