Geometric Computing Based on Computerized Descriptive Geometric

Computer-aided Design （CAD）, video games and other computer graphic related technology evolves substantial processing to geometric elements. A novel geometric computing method is proposed with the integration of descriptive geometry, math and computer algorithm. Firstly, geometric elements in general position are transformed to a special position in new coordinate system. Then a 3D problem is projected to new coordinate planes. Finally, according to 2D/3D correspondence principle in descriptive geometry, the solution is constructed computerized drawing process with ruler and compasses. In order to make this method a regular operation, a two-level pattern is established. Basic Layer is a set algebraic packaged function including about ten Primary Geometric Functions （PGF） and one projection transformation. In Application Layer, a proper coordinate is established and a sequence of PGFs is sought for to get the final results. Examples illustrate the advantages of our method on dimension reduction, regulatory and visual computing and robustness.

Geometric basis:a geometric solving cell for geometric computing

Geometric computing is an important tool in design and manufacturing and in arts. Conventionally, geometric computing is taken by algebraic computing. The vivid intuition of objects in visualization is lost in numeric functions, which is however very useful to human cognition as well as emotion. In this paper, we proposed a concept and theory of geometric basis （GB） as the solving cell for geometric computing. Each GB represents a basic geometric operation. GB works as both expressing and solving cell just like the concept of basis in linear algebra by which every element of the vector space can be expressed. For 3D problems, with a procedure of a projections reduction, the problem can be reduced to plane and the reduction function can be designed as a GB. A sequence of GB can construct a higher layer GB. Then, by the traversal of tree, a sequence of GB is got and this sequence is just the construction process and also the solution of this geometric problem.

New progress in geometric computing for image and video processing

In recent years, geometry-based image and video processing methods have aroused significant interest. This paper considers progress from four aspects： geometric characteristics and shape, geometric transformations, embedded geometric structure, and differential geometry methods. Current research trends are also pointed out.

Realization of high-fidelity and robust geometric gates with time-optimal control technique in superconducting quantum circuit

One of the key features required to realize fault-tolerant quantum computation is the robustness of quantum gates against errors.Since geometric quantum gate is naturally insensitivity to noise,it appears to be a promising routine to achieve high-fidelity,robust quantum gates.The implementation of geometric quantum gate however faces some troubles such as its complex interaction among multiple energy levels.Moreover,traditional geometric schemes usually take more time than equivalent dynamical ones.Here,we experimentally demonstrate a geometric gate scheme with the time-optimal control(TOC)technique in a superconducting quantum circuit.With a transmon qubit and operations restricted to two computational levels,we implement a set of geometric gates which exhibit better robustness features against control errors than the dynamical counterparts.The measured fidelities of TOC X gate and X/2 gate are 99.81%and 99.79%respectively.Our work shows a promising routine toward scalable fault-tolerant quantum computation.

Computational Geometric Analysis for <i>C. elegans</i>Trajectories on Thermal and Salinity Gradient

Elegans are one of the best model organisms in neural researches, and tropism movement is a typical learning and memorizing activity. Based on one imaging technique called Fast Track-Capturing Microscope (FTCM), we investigated the movement regulation. Two movement patterns are extracted from various trajectories through analysis on turning angle. Then we applied this classification on trajectory regulation on the compound gradient field, and theoretical results corresponded with experiments well, which can initially verify the conclusion. Our breakthrough is performed computational geometric analysis on trajectories. Several independent features were combined to describe movement properties by principal composition analysis (PCA) and support vector machine (SVM). After normalizing all data sets, no-supervising machine learning was processed along with some training under certain supervision. The final classification results performed perfectly, which indicates the further application of such computational analysis in biology researches combining with machine learning.

Nonadiabatic geometric quantum computation with optimal control on superconducting circuits

Quantum gates,which are the essent ial building blocks of quantum computers,are very fragile.Thus,to realize robust quanturm gates with high fidelity is the ultimate goal of quantum manipulation.Here,we propose a nonadiabatic geometric quantum computation scheme on superconducting circuits to engineer arbitrary quantum gates,which share both the robust merit of geometric phases and the capacity to combine with optimal control technique to further enhance the gate robustness.Specif-ically,in our proposal,arbitrary geometric single-qubit gates can be realized on a transmon qubit,by a resonant microwave field driving,with both the amplitude and phase of the driving being time-dependent.Meanwhile,nontrivial two-qubit gometric gates can be implemented by two capacitively coupled transmon qubits,with one of the transmon qubits'frequency being modulated to obtain ef-fective resonant coupling between them.Therefore,our scheme provides a promising step towards fault-tolerant solid-state quantum computation.

Experimental realization of nonadiabatic geometric gates with a superconducting Xmon qubit

Geometric phases are only dependent on evolution paths but independent of evolution details so that they possess some intrinsic noise-resilience features. Based on different geometric phases, various quantum gates have been proposed, such as nonadiabatic geometric gates based on nonadiabatic Abelian geometric phases and nonadiabatic holonomic gates based on nonadiabatic nonAbelian geometric phases. Up to now, nonadiabatic holonomic one-qubit gates have been experimentally demonstrated with superconducting transmons, where the three lowest levels are all utilized in operation. However, the second excited state of transmons has a relatively short coherence time, which results in a decreased fidelity of quantum gates. Here, we experimentally realize Abelian-geometric-phase-based nonadiabatic geometric one-qubit gates with a superconducting Xmon qubit. The realization is performed on the two lowest levels of an Xmon qubit and thus avoids the influence from the short coherence time of the second excited state. The experimental result indicates that the average fidelities of single-qubit gates can be up to 99.6% and 99.7% characterized by quantum process tomography and randomized benchmarking.

Nonadiabatic geometric quantum computation protected by dynamical decoupling via the XXZ Hamiltonian

Nonadiabatic geometric quantum computation protected by dynamical decoupling combines the robustness of nonadiabatic geometric gates and the decoherence-resilience feature of dynamical decoupling. Solid-state systems provide an appealing candidate for the realization of nonadiabatic geometric quantum computation protected dynamical decoupling since the solid-state qubits are easily embedded in electronic circuits and scaled up to large registers. In this paper, we put forward a scheme of nonadiabatic geometric quantum computation protected by dynamical decoupling via the XXZ Hamiltonian, which not only combines the merits of nonadiabatic geometric gates and dynamical decoupling but also can be realized in a number of solid-state systems, such as superconducting circuits and quantum dots.

An Optimized Divide-and-Conquer Algorithm for the Closest-Pair Problem in the Planar Case

We present an engineered version of the divide-and-conquer algorithm for finding the closest pair of points, within a given set of points in the XY-plane. For this version of the algorithm we show that only two pairwise comparisons are required in the combine step, for each point that lies in the 25-wide vertical slab. The correctness of the algorithm is shown for all Minkowski distances with p ≥ 1. We also show empirically that, although the time complexity of the algorithm is still O（n lgn）, the reduction in the total number of comparisons leads to a significant reduction in the total execution time, for inputs with size sufficiently large.