The Complexity of Quantum Disjointness

We introduce the communication problem QNDISJ, short for Quantum (Unique) Non-Disjointness, and study its complexity under different modes of communication complexity. The main motivation for the problem is that it is a candidate for the separation of the quantum communication complexity classes QMA and QCMA. The problem generalizes the Vector-in-Subspace and Non-Disjointness problems. We give tight bounds for the QMA, quantum, randomized communication complexities of the problem. We show polynomially related upper and lower bounds for the MA complexity. We also show an upper bound for QCMA protocols, and show that the bound is tight for a natural class of QCMA protocols for the problem. The latter lower bound is based on a geometric lemma, that states that every subset of the n-dimensional sphere of measure 2^-p must contain an ortho-normal set of points of size Omega(n/p). We also study a "small-spaces" version of the problem, and give upper and lower bounds for its randomized complexity that show that the QNDISJ problem is harder than Non-disjointness for randomized protocols. Interestingly, for quantum modes the complexity depends only on the dimension of the smaller space, whereas for classical modes the dimension of the larger space matters

Zero-Information Protocols and Unambiguity in Arthur–Merlin Communication

We study whether information complexity can be used to attack the long-standing open problem of proving lower bounds against Arthur–Merlin (AM) communication protocols. Our starting point is to show that—in contrast to plain randomized communication complexity—every boolean function admits an AM communication protocol where on each yes-input, the distribution of Merlin’s proof leaks no information about the input and moreover, this proof is unique for each outcome of Arthur’s randomness. We posit that these two properties of zero information leakage and unambiguity on yes-inputs are interesting in their own right and worthy of investigation as new avenues toward AM. Zero-information protocols (ZAM): Our basic ZAM protocol uses exponential communication for some functions, and this raises the question of whether more efficient protocols exist. We prove that all functions in the classical space-bounded complexity classes NL and ⊕ L have polynomial-communication ZAM protocols. We also prove that ZAM complexity is lower bounded by conondeterministic communication complexity. Unambiguous protocols (UAM): Our most technically substantial result is a Ω (n) lower bound on the UAM complexity of the NP-complete set-intersection function; the proof uses information complexity arguments in a new, indirect way and overcomes the “zero-information barrier” described above. We also prove that in general, UAM complexity is lower bounded by the classic discrepancy bound, and we give evidence that it is not generally lower bounded by the classic corruption bound

Fine-grained Complexity Meets IP = PSPACE

In this paper we study the fine-grained complexity of finding exact and approximate solutions to problems in P. Our main contribution is showing reductions from exact to approximate solution for a host of such problems. As one (notable) example, we show that the Closest-LCS-Pair problem (Given two sets of strings $A$ and $B$, compute exactly the maximum $\textsf{LCS}(a, b)$ with $(a, b) \in A \times B$) is equivalent to its approximation version (under near-linear time reductions, and with a constant approximation factor). More generally, we identify a class of problems, which we call BP-Pair-Class, comprising both exact and approximate solutions, and show that they are all equivalent under near-linear time reductions. Exploring this class and its properties, we also show: $\bullet$ Under the NC-SETH assumption (a significantly more relaxed assumption than SETH), solving any of the problems in this class requires essentially quadratic time. $\bullet$ Modest improvements on the running time of known algorithms (shaving log factors) would imply that NEXP is not in non-uniform $\textsf{NC}^1$. $\bullet$ Finally, we leverage our techniques to show new barriers for deterministic approximation algorithms for LCS. At the heart of these new results is a deep connection between interactive proof systems for bounded-space computations and the fine-grained complexity of exact and approximate solutions to problems in P. In particular, our results build on the proof techniques from the classical IP = PSPACE result

Zero-Information Protocols and Unambiguity in Arthur-Merlin Communication

We study whether information complexity can be used to attack the long-standing open problem of proving lower bounds against Arthur–Merlin (AM) communication protocols. Our starting point is to show that—in contrast to plain randomized communication complexity— every boolean function admits an AM communication protocol where on each yes-input, the distribution of Merlin’s proof leaks no information about the input and moreover, this proof is unique for each outcome of Arthur’s randomness. We posit that these two properties of zero information leakage and unambiguity on yes-inputs are interesting in their own right and worthy of investigation as new avenues toward AM. • Zero-information protocols (ZAM). Our basic ZAM protocol uses exponential com-munication for some functions, and this raises the question of whether more efficient protocols exist. We prove that all functions in the classical space-bounded complexity classes NL and ⊕L have polynomial-communication ZAM protocols. We also prove that ZAM complexity is lower bounded by conondeterministic communication complexity. • Unambiguous protocols (UAM). Our most technically substantial result is a Ω(n) lower bound on the UAM complexity of the NP-complete set-intersection function; the proof uses information complexity arguments in a new, indirect way and overcomes the “zero-information barrier” described above. We also prove that in general, UAM complexity is lower bounded by the classic discrepancy bound, and we give evidence that it is not generally lower bounded by the classic corruption bound

Zero-Information Protocols and Unambiguity in Arthur–Merlin Communication

Non UBCUnreviewedAuthor affiliation: University of TorontoPostdoctora