Topological computing

Topological computing is the designing and building of hardware and software based on the processing of topologically modulated electromagnetic impulses that differ from each other in their spatio-temporal topology.
Science and theory
The topology of figures and the logical systems were first brought together by J.C.C. McKinsey and A. Tarski. They showed that the logical units could be associated with the topologically different geometrical figures. This idea can be implemented in electromagnetics and optics if the waves carry logical units spatially written in their geometrical properties.
There are different means to describe the geometry: the field-force lines and their graphical images and representations of field by sets of the field-level surfaces.
The field-force lines and their graphical images were initially introduced by M. Faraday for the electric (E, D) and magnetic (H, B) fields. J.C. Maxwell used the field-force lines for explaining the EM phenomena. Additionally, field geometry can be represented by sets of the field-level surfaces. Field geometry has a certain topological content, and small perturbations do not destroy this content except for some bifurcation cases.
Geometrical and topological representations of fields and geometrical relations between them were used by G.A. Deschamps and others to build a theory of the EM field which was based on the external differential forms introduced by Grassmann in the 19th Century. In this theory, the field intensity (E, H) forms are associated with surfaces, the field flux quantities (D, B) correspond to tubes, and 3-D images of charges (ρ) and currents (j) are depicted with boxes or spheres. The topological Maxwell’s equations define correspondence of these shapes.
At the end of the 1980s, the topology of field-force-line pictures was studied in detail, and a topological theory (topological electromagnetics) of guided waves and components was proposed.
This theory describes the electric and magnetic fields by their skeletons, and these topological schemes are composed of separatrices of field-force maps and field-equilibrium manifolds. These skeletons (schemes) are coupled to each other through the geometro-topological analogs of Maxwell's equations. Topological schemes, being stable towards the boundary field variations, are found by simplified analytical or numerical algorithms, and many approximate models of integrated microwave components are built.
Then these models are substituted to any variational algorithm to obtain their more accurate parameters. This approach was verified by comparing its results with the theoretical and experimental data, and more detailed information on it is available in and references therein.
Other aspects of topology applications in physics and electromagnetism can be found in.
Topologically modulated signals
There are different ways how the ideas of topology may be implemented in signaling, but all of them deal with the correspondence of the signal topology to a logical unit. In transmission lines and optical waveguides the EM impulses can be generated with the topologically different spatio-temporal content, which is managed by a control signal and formed by reflections of elementary waves and their interference inside waveguides. These are the topologically modulated signals. They can excel in increased noise immunity due to their topological nature and space-time orthogonality of the signal fields.
In open space, where there are no reflections from walls, other effects are used to generate a carrier of topology-related information. As a general theory, topology, can be applied to any characteristics defined in real or phase space. One of them is the angular momentum density of EM waves. The momentum density is defined as the Poynting vector oriented along the energy-transfer direction.
As an example, the plane TEM wave has only one component of the Poynting vector oriented normally from the magnitude/phase plane. In waveguides, the transversal components of the Poynting vector are imaginary, which means zero average energy flow through the ideal walls of waveguides. These momentary Poynting vector distributions in these waveguides can have more complicated structures and radio telecommunications. Generation and nonlinear transformations of these helical waves are presented in.
Discretely-modulated propagation-invariant spatial wave packets can be used for free-space optical communications, as well. The transverse structure of light intensity of these packets is modulated discretely, and this structure is invariant while a beam is propagating through open space. These signals are considered as the scalar topologically-modulated ones.
Transmission lines for topologically modulated signals
Topological signals are the spatially enriched impulses propagating through media of an extended band of spatial frequencies. Then, the open space, multimodal microwave, and optical waveguides are the natural supporters of these signals. Today, the coupled and multi-coupled strips, microstrip lines, and the microwave and optical multimodal waveguides have been known for these signals. and a large amount of work should be performed in the future to reach better cost-per-bit parameters in comparison to the presently known multiplexing technologies.
Elementary circuitry and devices based on the ideas of topology
Taking into account the spatial nature of topologically-modulated signals, the passive spatial filters, multiplexers, and de-multiplexors can be used for their processing. Circuitry particular design depends on the signal frequencies and the signal time-dependence. In micro- and millimeter-wave range, the circuits for switching and comparison of signals are based on the interference and diffraction effects although their size can be rather large.
The optical range allows miniature holograms, interferometers, prisms, etc. to compare the spatial signals, shape them, and perform some operations over these signals. Some special launchers,
The condensed matter physics is rich in different topological effects, and many of them can be used in future advanced electronics. For example, in ferromagnetic solids, due to quantum-mechanical interactions, the nontrivial topological 2-D and 3-D spin textures emerge. These structures are several-nanometers-large and rather stable. Then, binary digits can be assigned to these vortices according to their topology or direction of the vortices, and the spin textures hold out promises to be the elements of nonvolatile memory.
In electron waveguides, the non-degenerated wave functions are topologically different from each other. This feature can be used for design of advanced semiconductor devices based on the interference of electron wave functions. In 2005, a new effect was predicted by a theory using topology and quantum physics. It is shown that in some materials, due to interaction of electron plasma and inner material filling, electron gas concentrates only in the atomic-thin layer, and this effect is topologically-persistent, as it is determined by bulk interactions of electrons and material (i.e., shape deformation of a chunk of this material leaves this effect unchanged). This sort of conductors is now called the topological insulators, following C.L. Kane and his co-authors. Later, this prediction was confirmed experimentally by several research teams. An optical analog of the topological insulators is shown, for instance, in. A mechanism of switching between different topological states in 3-D BiTeI compound systems is theoretically demonstrated in. Topological insulators are anticipated to be important in fault-tolerant quantum computing.
Topological processors and computers
A computing unit performing operations with topologically modulated impulses is called a topological processor. Combined with other necessary conventional digital units, this processor makes up the topological computer. The original ideas on this matter can be found in, and the authors focused on particular circuits for microwave and digital electronics simulated and tested in measurements.
In this processor, the elementary predicates are coded by unipolar signals along two wires, and the main modules of the designed processor perform predicate logic operations. The Boolean ones are executed as well in this architecture. The proposed, designed, and FPGA-verified processor can be further enhanced for performing the advanced artificial intelligence operations, including proofs of mathematical theorems, etc. Today, the multi-core systems and the heterogeneous processing are common, and the predicate logic processors can be embedded into these multi-processor systems as the co-units (e.g. for enhancing the artificial intelligence operations).
A theory of a particular case of topological processors has been proposed by Ryabov and Serov (2007); it encompasses parallel operations with elementary image voxels. The main feature of the C++ algorithm developed for the 3-D-image processing is the voxels, and a 3-D image composed of them is processed according to the topology-supported strategies. The required memory and calculation time are reduced by two orders of magnitude compared to conventional algorithms. The proposed idea on topological computing can be realized in hardware as a dedicated micro-coprocessor.
The noise-immunity of topological representations is used in quantum computing, and the first quantum topological computer and its theory were proposed by A. Kitaev et al. in 2003. In this approach, the quantum information is handled in a non-local manner, and the carriers of this information are anyons, which are the collective excitations or effective particles. Quantum computation is performed with the exchange of the anyon position along a set of lines which have a certain topology in phase space. These topological peculiarities provide increased noise-immunity of operations. A review on the contemporary state of these ideas and the first experimentations are presented in.
Related results
1. Relations of topology, physics, and computations are considered by Baez and Stay in.
2. Logic of topology is a particular case of the spatial one.
3. Some information on digital topology, which is basic for topological computing, is in.
4. Spatial logic circuits which can be used in nanoelectronics are considered in.
5. Applications of topology in molecular biology and the role of topologically constrained DNAs in intracellular communications are reviewed in.

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