From communication complexity to an entanglement spread area law in the ground state of gapped local Hamiltonians

In this work, we make a connection between two seemingly different problems. The first problem involves characterizing the properties of entanglement in the ground state of gapped local Hamiltonians, which is a central topic in quantum many-body physics. The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory. We construct a communication protocol for testing (or measuring) the ground state and use its communication complexity to reveal a new structural property for the ground state entanglement. This property, known as the entanglement spread, roughly measures the ratio between the largest and the smallest Schmidt coefficients across a cut in the ground state. Our main result shows that gapped ground states possess limited entanglement spread across any cut, exhibiting an "area law" behavior. Our result quite generally applies to any interaction graph with an improved bound for the special case of lattices. This entanglement spread area law includes interaction graphs constructed in [Aharonov et al., FOCS'14] that violate a generalized area law for the entanglement entropy. Our construction also provides evidence for a conjecture in physics by Li and Haldane on the entanglement spectrum of lattice Hamiltonians [Li and Haldane, PRL'08]. On the technical side, we use recent advances in Hamiltonian simulation algorithms along with quantum phase estimation to give a new construction for an approximate ground space projector (AGSP) over arbitrary interaction graphs.Comment: 29 pages, 1 figur

Universal Communication, Universal Graphs, and Graph Labeling

We introduce a communication model called universal SMP, in which Alice and Bob receive a function f belonging to a family ?, and inputs x and y. Alice and Bob use shared randomness to send a message to a third party who cannot see f, x, y, or the shared randomness, and must decide f(x,y). Our main application of universal SMP is to relate communication complexity to graph labeling, where the goal is to give a short label to each vertex in a graph, so that adjacency or other functions of two vertices x and y can be determined from the labels ?(x), ?(y). We give a universal SMP protocol using O(k^2) bits of communication for deciding whether two vertices have distance at most k in distributive lattices (generalizing the k-Hamming Distance problem in communication complexity), and explain how this implies a O(k^2 log n) labeling scheme for deciding dist(x,y) ? k on distributive lattices with size n; in contrast, we show that a universal SMP protocol for determining dist(x,y) ? 2 in modular lattices (a superset of distributive lattices) has super-constant ?(n^{1/4}) communication cost. On the other hand, we demonstrate that many graph families known to have efficient adjacency labeling schemes, such as trees, low-arboricity graphs, and planar graphs, admit constant-cost communication protocols for adjacency. Trees also have an O(k) protocol for deciding dist(x,y) ? k and planar graphs have an O(1) protocol for dist(x,y) ? 2, which implies a new O(log n) labeling scheme for the same problem on planar graphs

Approximate F_2-Sketching of Valuation Functions

We study the problem of constructing a linear sketch of minimum dimension that allows approximation of a given real-valued function f : F_2^n - > R with small expected squared error. We develop a general theory of linear sketching for such functions through which we analyze their dimension for most commonly studied types of valuation functions: additive, budget-additive, coverage, alpha-Lipschitz submodular and matroid rank functions. This gives a characterization of how many bits of information have to be stored about the input x so that one can compute f under additive updates to its coordinates. Our results are tight in most cases and we also give extensions to the distributional version of the problem where the input x in F_2^n is generated uniformly at random. Using known connections with dynamic streaming algorithms, both upper and lower bounds on dimension obtained in our work extend to the space complexity of algorithms evaluating f(x) under long sequences of additive updates to the input x presented as a stream. Similar results hold for simultaneous communication in a distributed setting

Graphs and Circuits: Some Further Remarks

We consider the power of single level circuits in the context of graph complexity. We first prove that the single level conjecture fails for fanin-$2$ circuits over the basis ${oplus,land,1}$. This shows that the (surpisingly tight) phenomenon, established by Mirwald and Schnorr (1992) for quadratic functions, has no analogon for graphs. We then show that the single level conjecture fails for unbounded fanin circuits over ${lor,land,1}$. This partially answers the question of Pudl\u27ak, R"odl and Savick\u27y (1986). We also prove that $Sigma_2 eq Pi_2$ in a restricted version of the hierarhy of communication complexity classes introduced by Babai, Frankl and Simon (1986). Further, we show that even depth-$2$ circuits are surprisingly powerful: every bipartite $n imes n$ graph of maximum degree $Delta$ can be represented by a monotone CNF with $O(Deltalog n)$ clauses. We also discuss a relation between graphs and $ACC$-circuits

08381 Abstracts Collection -- Computational Complexity of Discrete Problems

From the 14th of September to the 19th of September, the Dagstuhl Seminar 08381 ``Computational Complexity of Discrete Problems\u27\u27 was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work as well as open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this report. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Open Diophantine Problems

We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.Comment: 58 pages. to appear in the Moscow Mathematical Journal vo. 4 N.1 (2004) dedicated to Pierre Cartie