A polynomial lower bound for testing monotonicity

We show that every algorithm for testing n-variate Boolean functions for monotonicity has query complexity Ω(n1/4). All previous lower bounds for this problem were designed for nonadaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only Ω(logn). Combined with the query complexity of the non-adaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity. By contrast, we show that there is an exponential gap between the query complexity of adaptive and non-adaptive algorithms for testing regular linear threshold functions (LTFs) for monotonicity. Chen, De, Servedio, and Tan (STOC 2015) recently showed that non-adaptive algorithms require almost Ω(n1/2) queries for this task. We introduce a new adaptive monotonicity testing algorithm which has query complexity O(logn) when the input is a regular LTF

Motion of magnetotactic bacteria swarms in an external field

Magnetotactic bacteria moving on circular orbits form hydrodynamically bound states. When close to a surface and with the tilting of the field in a direction close to the perpendicular to this surface these swarms move perpendicularly to the tilting angle. We describe quantitatively this motion by a continuum model with couple stress arising from the torques produced by the rotary motors of the amphitrichous magnetotactic bacteria. The model not only correctly describes the change of direction of swarm motion while inverting the tangential field but also predicts reasonable value of the torque produced by the rotary motors

Gyromagnetic effects in dynamics of magnetic microparticles

We derive equations of motion for paramagnetic and ferromagnetic particles fully accounting for gyromagnetic effects. Considering the Einstein-de Haas effect for an ellipsoidal paramagnetic particle we find that starting from a quiescent non-magnetized state, after the field is switched on a rotation along the short axis is established. This is confirmed by the stability analysis of the fixed points of the corresponding ordinary differential equations. In the case of a ferromagnetic particle we integrate the equations of motion in the dissipationless case by finding the integrals of motion. We also reformulate the equations in a Hamiltonian framework in this case and find a period of small nutation oscillations.Comment: 24 pages, 15 figure

Quantum Algorithms for Finding Constant-sized Sub-hypergraphs

We develop a general framework to construct quantum algorithms that detect if a $3$-uniform hypergraph given as input contains a sub-hypergraph isomorphic to a prespecified constant-sized hypergraph. This framework is based on the concept of nested quantum walks recently proposed by Jeffery, Kothari and Magniez [SODA'13], and extends the methodology designed by Lee, Magniez and Santha [SODA'13] for similar problems over graphs. As applications, we obtain a quantum algorithm for finding a $4$-clique in a $3$-uniform hypergraph on $n$ vertices with query complexity $O(n^{1.883})$, and a quantum algorithm for determining if a ternary operator over a set of size $n$ is associative with query complexity $O(n^{2.113})$.Comment: 18 pages; v2: changed title, added more backgrounds to the introduction, added another applicatio

On the polynomial parity argument complexity of the combinatorial nullstellensatz

The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems. Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combinatorial Nullstellensatz and to the Chevalley-Warning Theorem over the two elements field GF(2). The input of these problems contain PPA-circuits which are arithmetic circuits with special symmetric properties that assure that the polynomials computed by them have always an even number of zeros. In the proof of the result we relate the multilinear degree of the polynomials to the parity of the maximal parse subcircuits that compute monomials with maximal multilinear degree, and we show that the maximal parse subcircuits of a PPA-circuit can be paired in polynomial time

Quantum Communication Complexity of Distribution Testing

The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $\epsilon$-far apart in the $l_1$-distance. In the present paper we show that the quantum communication complexity of this problem is $\tilde{O}(n/(t\epsilon^2))$ qubits when the distributions have low $l_2$-norm, which gives a quadratic improvement over the classical communication complexity obtained by Andoni, Malkin and Nosatzki. We also obtain a matching lower bound by using the pattern matrix method. Let us stress that the samples received by each of the parties are classical, and it is only communication between them that is quantum. Our results thus give one setting where quantum protocols overcome classical protocols for a testing problem with purely classical samples.Comment: 11 page

Efficient Distributed Quantum Computing

We provide algorithms for efficiently addressing quantum memory in parallel. These imply that the standard circuit model can be simulated with low overhead by the more realistic model of a distributed quantum computer. As a result, the circuit model can be used by algorithm designers without worrying whether the underlying architecture supports the connectivity of the circuit. In addition, we apply our results to existing memory intensive quantum algorithms. We present a parallel quantum search algorithm and improve the time-space trade-off for the Element Distinctness and Collision problems.Comment: Some material rearranged and references adde

From SICs and MUBs to Eddington

This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg group, while the latter are believed to be orbits under the Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are understandable in terms of elliptic curves, but a general statement escapes us. The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.Comment: 12 pages; from the Festschrift for Tony Sudber

Practical implementation of a quantum backtracking algorithm

In previous work, Montanaro presented a method to obtain quantum speedups for backtracking algorithms, a general meta-algorithm to solve constraint satisfaction problems (CSPs). In this work, we derive a space efficient implementation of this method. Assume that we want to solve a CSP with $m$ constraints on $n$ variables and that the union of the domains in which these variables take their value is of cardinality $d$. Then, we show that the implementation of Montanaro's backtracking algorithm can be done by using $O(n \log d)$ data qubits. We detail an implementation of the predicate associated to the CSP with an additional register of $O(\log m)$ qubits. We explicit our implementation for graph coloring and SAT problems, and present simulation results. Finally, we discuss the impact of the usage of static and dynamic variable ordering heuristics in the quantum setting.Comment: 18 pages, 10 figure

El federalismo cooperativo como factor catalizador de un Gobierno Abierto

Este artículo tiene como propósito articular un marco de análisis que justifica la importancia de la colaboración entre gobiernos, bajo el supuesto de que un sistema rígido de competencias tiende a fragmentar la solución de los asuntos públicos en las agendas de gobierno. Un federalismo cooperativo, en tanto variable productora de relaciones intergubernamentales, da forma a un contexto que favorece impulsar soluciones más integrales en torno a demandas inscritas en el plan de acción, elaborado como requisito de pertenencia a la Alianza Internacional por un Gobierno Abierto. La propuesta llevada a Brasilia por la representación de México en junio de 2012, con la idea de promover la apertura gubernamental en el plano subnacional y local, tendrá mayores posibilidades de éxito si previamente se establecen bases mínimas para un federalismo cooperativo que facilite la gestión de los asuntos públicos establecidos en el plan de acción