A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\log n)^{1/2}}$ for some constant $c>0$. This yields a nearly tight $n^{1/2 - o(1)}$ bound on the value of this program for any degree $d = o(\log n)$. Moreover we introduce a new framework that we call \emph{pseudo-calibration} to construct Sum of Squares lower bounds. This framework is inspired by taking a computational analog of Bayesian probability theory. It yields a general recipe for constructing good pseudo-distributions (i.e., dual certificates for the Sum-of-Squares semidefinite program), and sheds further light on the ways in which this hierarchy differs from others.Comment: 55 page

Computation of Kalman Decompositions of Periodic Systems

We consider the numerically reliable computation of reachability and observability Kalman decompositions of a periodic system with time-varying dimensions. These decompositons generalize the controllability/observability Kalman decompositions for standard state space systems and have immediate applications in the structural analysis of periodic systems. We propose a structure exploiting numerical algorithm to compute the periodic controllability form by employing exclusively orthogonal similarity transformations. The new algorithm is computationally efficient and strongly backward stable, thus fulfils all requirements for a satisfactory algorithm for periodic systems

Energy-tunable sources of entangled photons: a viable concept for solid-state-based quantum relays

We propose a new method of generating triggered entangled photon pairs with wavelength on demand. The method uses a micro-structured semiconductor-piezoelectric device capable of dynamically reshaping the electronic properties of self-assembled quantum dots (QDs) via anisotropic strain-engineering. Theoretical models based on kp theory in combination with finite-element calculations show that the energy of the polarization-entangled photons emitted by QDs can be tuned in a range larger than 100 meV without affecting the degree of entanglement of the quantum source. These results pave the way towards the deterministic implementation of QD entanglement resources in all-electrically-controlled solid-state-based quantum relays

Markov two-components processes

We propose Markov two-components processes (M2CP) as a probabilistic model of asynchronous systems based on the trace semantics for concurrency. Considering an asynchronous system distributed over two sites, we introduce concepts and tools to manipulate random trajectories in an asynchronous framework: stopping times, an Asynchronous Strong Markov property, recurrent and transient states and irreducible components of asynchronous probabilistic processes. The asynchrony assumption implies that there is no global totally ordered clock ruling the system. Instead, time appears as partially ordered and random. We construct and characterize M2CP through a finite family of transition matrices. M2CP have a local independence property that guarantees that local components are independent in the probabilistic sense, conditionally to their synchronization constraints. A synchronization product of two Markov chains is introduced, as a natural example of M2CP.Comment: 34 page

Undecidability of the Spectral Gap (full version)

We show that the spectral gap problem is undecidable. Specifically, we construct families of translationally-invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lower-bounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.Comment: v1: 146 pages, 56 theorems etc., 15 figures. See shorter companion paper arXiv:1502.04135 (same title and authors) for a short version omitting technical details. v2: Small but important fix to wording of abstract. v3: Simplified and shortened some parts of the proof; minor fixes to other parts. Now only 127 pages, 55 theorems etc., 10 figures. v4: Minor updates to introductio

Fully-Automated Verification of Linear Systems Using Inner- and Outer-Approximations of Reachable Sets

Reachability analysis is a formal method to guarantee safety of dynamical systems under the influence of uncertainties. A major bottleneck of all reachability algorithms is the requirement to adequately tune certain algorithm parameters such as the time step size, which requires expert knowledge. In this work, we solve this issue with a fully-automated reachability algorithm that tunes all algorithm parameters internally such that the reachable set enclosure satisfies a user-defined accuracy in terms of distance to the exact reachable set. Knowing the distance to the exact reachable set, an inner-approximation of the reachable set can be efficiently extracted from the outer-approximation using the Minkowski difference. Finally, we propose a novel verification algorithm that automatically refines the accuracy of the outer- and inner-approximation until specifications given by time-varying safe and unsafe sets can either be verified or falsified. The numerical evaluation demonstrates that our verification algorithm successfully verifies or falsifies benchmarks from different domains without any requirement for manual tuning.Comment: 16 page