Generalized quantum PageRank algorithm with arbitrary phase rotations

CAM/FEDER Project [S2018/TCS-4342]; Spanish MINECO/FEDER Project [PGC2018-099169- B-I00FIS2018]; MCIN; European Union NextGenerationEU [PRTR-C17.I1]; Ministry of Economic Affairs Quantum ENIA project; U.S. Army Research Office [W911NF-14-1-0103]; QUITEMAD grant; Universidad Complutense de Madrid-Banco Santander [CT58/21-CT59/21]The quantization of the PageRank algorithm is a promising tool for a future quantum internet. Here we present a modification of the quantum PageRank, introducing arbitrary phase rotations (APR) in the underlying Szegedy's quantum walk. We define three different APR schemes with only one phase as a degree of freedom. We have analyzed the behavior of these algorithms in a small generic graph, observing that a decrease of the phase reduces the standard deviation of the instantaneous PageRank, so the nodes of the network can be distinguished better. However, the algorithm takes more time to converge, so the phase cannot be decreased arbitrarily. With these results we choose a concrete value for the phase to later apply the algorithm to complex scale-free graphs. In these networks, the original quantum PageRank is able to break the degeneracy of the residual nodes and detect secondary hubs that the classical algorithm suppresses. Nevertheless, not all of the detected secondary hubs are real according to the PageRank's definition. Some APR schemes can overcome this problem, restoring the degeneration of the residual nodes and highlighting the truly secondary hubs of the networks. Finally, we have studied the stability of the new algorithms. The original quantum algorithm was known to be more stable than the classical. We have found that one of our algorithms, whose PageRank distribution resembles the classical one, has a stability similar to the original quantum algorithm.Depto. de Física TeóricaFac. de Ciencias FísicasTRUEMinisterio de Economia y Competitividad (MINECO)/FEDERMinisterio de Ciencia e Innovación (MICINN)/AEIMinisterio de Economía y Competitividad (MINECO)Comunidad de Madrid/FEDERU.S. Army Research Office W911NF-14-1-0103Universidad Complutense de Madrid/Banco de Santanderpu

Analytic formulations of the density matrix renormalization group

We present two new analytic formulations of the density matrix renormalization group (DMRG) method. In these formulations we combine the block renormalization group (BRG) procedure with the variational and Fokker-Planck methods. The BRG method is used to reduce the lattice size while the latter are used to construct approximate target states to compute the block density matrix. We apply our DMRG methods to the Ising model in a transverse field (ITF model) and compute several of its critical properties, which are then compared with the old BRG results.Work partially supported in part by CICYT under contracts AEN93-0776 (M.A.M.-D.) and PB92- 1092, European Community Grant ERBCHRXCT920069 (G.S.).Peer Reviewe

Dualities in spin ladders

We introduce a set of discrete modular transformations Tℓ, Uℓ and Sℓ in order to study the relationships between the different phases of the Heisenberg ladders obtained with all possible exchange coupling constants. For the two-legged ladder we show that the resonating valence bond (RVB) phase is invariant under the Sℓ transformation, while the Haldane phase is invariant under Uℓ. These two phases are related by Tℓ. Moreover, there is a 'mixed' phase, that is invariant under Tℓ, and which under Uℓ becomes the RVB phase, while under Sℓ becomes the Haldane phase. For odd ladders there exists only the Tℓ transformation which, for strong coupling, maps the effective antiferromagnetic spin 1/2 chain onto the spin 3/2 chain. Our work is based on a combination of approximate methods such as bosonization, perturbation theory and the nonlinear sigma model, adapted to the different regimes of coupling constants. 1998 IOP Publishing LtdWork partially supported by the grants NSF PHY94-07194 and the DGES (GS) and by CICYT under contract AEN93-0776 (MAM-D).Peer Reviewe

Density matrix renormalization group approach to an asymptotically free model with bound states

We apply the density matrix renormalization group (DMRG) method to the two-dimensional delta function potential which is a simple quantum mechanical model with asymptotic freedom and formation of bound states. The system block and the environment block of the DMRG contain the low energy and high energy degrees of freedom, respectively. The ground state energy and the lowest excited states are obtained with very high accuracy. We compare the DMRG method with the similarity renormalization group method and propose its generalization to field theoretical models in high energy physics. © 1999 The American Physical Society.This work was supported by the DGES Spanish Grant No. PB97-1190.Peer Reviewe

Bosonization on a lattice: The emergence of the higher harmonics

6 págs.; 2 figs.A general and transparent procedure to bosonize fermions placed on a lattice is presented. Harmonics higher than kF in the one-particle Green function are shown to appear due to the compact character of real electron bands. Quantitative estimations of the role of higher harmonics are made possible by this bosonization technique. © 1995 The American Physical Society.J.F. was supported by DEYCIT Project No. PB93-1248.Peer Reviewe

Density-matrix Chern insulators: finite-temperature generalization of topological insulators

Thermal noise can destroy topological insulators (TI). However, we demonstrate how TIs can be made stable in dissipative systems. To that aim, we introduce the notion of band Liouvillian as the dissipative counterpart of band Hamiltonian, and show a method to evaluate the topological order of its steady state. This is based on a generalization of the Chern number valid for general mixed states (referred to as density-matrix Chern value), which witnesses topological order in a system coupled to external noise. Additionally, we study its relation with the electrical conductivity at finite temperature, which is not a topological property. Nonetheless, the density-matrix Chern value represents the part of the conductivity which is topological due to the presence of quantum mixed edge states at finite temperature. To make our formalism concrete, we apply these concepts to the two-dimensional Haldane model in the presence of thermal dissipation, but our results hold for arbitrary dimensions and density matrices

Localization and oscillations of Majorana fermions in a two-dimensional electron gas coupled with d-wave superconductors

We study the localization and oscillation properties of the Majorana fermions that arise in a two-dimensional electron gas (2DEG) with spin-orbit coupling (SOC) and a Zeeman field coupled with a d-wave superconductor. Despite the angular dependence of the d-wave pairing, localization and oscillation properties are found to be similar to the ones seen in conventional s-wave superconductors. In addition, we study a microscopic lattice version of the previous system that can be characterized by a topological invariant. We derive its real space representation that involves nearest and next-to-nearest-neighbors pairing. Finally, we show that the emerging chiral Majorana fermions are indeed robust against static disorder. This analysis has potential applications to quantum simulations and experiments in high-T_(c) superconductors

The localization of phonons in ion traps with controlled quantum disorder

We show that the vibrations of a chain of trapped ions offer an interesting route to explore the physics of disordered quantum systems. By preparing the internal state of the ions in a quantum superposition, we show how the local vibrational energy becomes a stochastic variable, its statistical properties inherited from the underlying quantum parallelism of the internal state. We describe a minimally perturbing measurement of the resonance fluorescence, which allows us to study effects such as Anderson localization without the need for ground-state cooling or individual addressing and thus paves the way for high-temperature ion experiments