Bifurcation diagram for saddle/source bimodal linear dynamical systems

We continue the study of the structural stability and the bifurcations of planar bimodal linear dynamical systems (BLDS) (that is, systems consisting of two linear dynamics acting on each side of a straight line, assuming continuity along the separating line). Here, we enlarge the study of the bifurcation diagram of saddle/spiral BLDS to saddle/source BLDS and in particular we study the position of the homoclinic bifurcation with regard to the new improper node bifurcationPostprint (published version

Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory

A hybrid stochastic-deterministic approach for computing the second-order perturbative contribution $E^{(2)}$ within multireference perturbation theory (MRPT) is presented. The idea at the heart of our hybrid scheme --- based on a reformulation of $E^{(2)}$ as a sum of elementary contributions associated with each determinant of the MR wave function --- is to split $E^{(2)}$ into a stochastic and a deterministic part. During the simulation, the stochastic part is gradually reduced by dynamically increasing the deterministic part until one reaches the desired accuracy. In sharp contrast with a purely stochastic MC scheme where the error decreases indefinitely as $t^{-1/2}$ (where $t$ is the computational time), the statistical error in our hybrid algorithm displays a polynomial decay $\sim t^{-n}$ with $n=3-4$ in the examples considered here. If desired, the calculation can be carried on until the stochastic part entirely vanishes. In that case, the exact result is obtained with no error bar and no noticeable computational overhead compared to the fully-deterministic calculation. The method is illustrated on the F$_2$ and Cr$_2$ molecules. Even for the largest case corresponding to the Cr$_2$ molecule treated with the cc-pVQZ basis set, very accurate results are obtained for $E^{(2)}$ for an active space of (28e,176o) and a MR wave function including up to $2 \times 10^7$ determinants.Comment: 8 pages, 5 figure

Supersymmetric Path Integrals II: The Fermionic Integral and Pfaffian Line Bundles

The Pfaffian line bundle of the covariant derivative and the transgression of the spin lifting gerbe are two canonically given real line bundles on the loop space of an oriented Riemannian manifold. It has been shown by Prat-Waldron that these line bundles are naturally isomorphic as metric line bundles and that the isomorphism maps their canonical sections to each other. In this paper, we provide a vast generalization of his results, by showing that there are natural sections of the corresponding line bundles for any $N \in \N$, which are mapped to each other under this isomorphism (with the previously known being the one for $N=0$). These canonical sections are important to define the fermionic part of the supersymmetric path integral on the loop space.Comment: The contents of this article have been integrated into arXiv:1709.10027, which now contains a simplified presentation of the result

Verifying the quantumness of bipartite correlations

Entanglement is at the heart of most quantum information tasks, and therefore considerable effort has been made to find methods of deciding the entanglement content of a given bipartite quantum state. Here, we prove a fundamental limitation to deciding if an unknown state is entangled or not: we show that any quantum measurement which can answer this question necessarily gives enough information to identify the state completely. Therefore, only prior information regarding the state can make entanglement detection less expensive than full state tomography in terms of the demanded quantum resources. We also extend our treatment to other classes of correlated states by considering the problem of deciding if a state is NPT, discordant, or fully classically correlated. Remarkably, only the question related to quantum discord can be answered without resorting to full state tomography

Universality of single quantum gates

We supply a rigorous proof that an open dense set of all possible 2-qubit gates G has the property that if the quantum circuit model is restricted to only permit swap of qubits lines and the application of G to pairs of lines, then the model is still computationally universal.Comment: 6 pages, 1 figure; added references to previous proof