Universality and optimality of programmable quantum processors

We analyze and compare the optimality of approximate and probabilistic universal programmable quantum processors. We define several characteristics how to quantify the optimality and we study in detail performance of three types of programmable quantum processors based on (1) the C-NOT gate, (2) the SWAP operation, and (3) the model of the quantum information distributor - the QID processor. We show under which conditions the measurement assisted QID processor is optimal. We also investigate optimality of the so-called U-processors and we also compare the optimal approximative implementation of U(1) qubit rotations with the known probabilistic implementation as introduced by Vidal, Masanes and Cirac [ {\em Phys. Rev. Lett.} {\bf 88}, 047905 (2002)].Comment: 9 page

Supersymmetric variational energies of 3d confined potentials

Within the approach of Supersymmetric Quantum Mechanics associated with the variational method a recipe to construct the superpotential of three dimensional confined potentials in general is proposed. To illustrate the construction, the energies of the Harmonic Oscillator and the Hulth\'en potential, both confined in three dimensions are evaluated. Comparison with the corresponding results of other approximative and exact numerical results is presented.Comment: 9 pages, Late

Quantum robustness and phase transitions of the 3D Toric Code in a field

We study the robustness of 3D intrinsic topogical order under external perturbations by investigating the paradigmatic microscopic model, the 3D toric code in an external magnetic field. Exact dualities as well as variational calculations reveal a ground-state phase diagram with first and second-order quantum phase transitions. The variational approach can be applied without further approximations only for certain field directions. In the general field case, an approximative scheme based on an expansion of the variational energy in orders of the variational parameters is developed. For the breakdown of the 3D intrinsic topological order, it is found that the (im-)mobility of the quasiparticle excitations is crucial in contrast to their fractional statistics

Ground states of the generalized Falicov-Kimball model in one and two dimensions

A combination of small-cluster exact-diagonalization calculations and a well-controlled approximative method is used to study the ground-state phase diagram of the spin-one-half Falicov-Kimball model extended by the spin-dependent on-site interaction between localized ($f$) and itinerant ($d$) electrons. Both the magnetic and charge ordering are analysed as functions of the spin-dependent on-site interaction ($J$) and the total number of itinerant ($N_d$) and localized ($N_f$) electrons at selected $U$ (the spin-independent interaction between the $f$ and $d$ electrons). It is shown that the spin-dependent interaction (for $N_f=L$, where $L$ is the number of lattice sites) stabilizes the ferromagnetic (F) and ferrimagnetic (FI) state, while the stability region of the antiferromagnetic (AF) phase is gradually reduced. The precisely opposite effect on the stability of F, FI and AF phases has a reduction of $N_f$. Moreover, the strong coupling between the $f$ and $d$-electron subsystems is found for both $N_f=L$ as well as $N_f < L$.Comment: LaTex, 20 pages, 7 figure

Scalable approximate FRNN-OWA classification

Fuzzy Rough Nearest Neighbour classification with Ordered Weighted Averaging operators (FRNN-OWA) is an algorithm that classifies unseen instances according to their membership in the fuzzy upper and lower approximations of the decision classes. Previous research has shown that the use of OWA operators increases the robustness of this model. However, calculating membership in an approximation requires a nearest neighbour search. In practice, the query time complexity of exact nearest neighbour search algorithms in more than a handful of dimensions is near-linear, which limits the scalability of FRNN-OWA. Therefore, we propose approximate FRNN-OWA, a modified model that calculates upper and lower approximations of decision classes using the approximate nearest neighbours returned by Hierarchical Navigable Small Worlds (HNSW), a recent approximative nearest neighbour search algorithm with logarithmic query time complexity at constant near-100% accuracy. We demonstrate that approximate FRNN-OWA is sufficiently robust to match the classification accuracy of exact FRNN-OWA while scaling much more efficiently. We test four parameter configurations of HNSW, and evaluate their performance by measuring classification accuracy and construction and query times for samples of various sizes from three large datasets. We find that with two of the parameter configurations, approximate FRNN-OWA achieves near-identical accuracy to exact FRNN-OWA for most sample sizes within query times that are up to several orders of magnitude faster