scholarly journals Quantum Computing in Four Spatial Dimensions

2019 ◽  
Author(s):  
Arturo Tozzi ◽  
Muhammad Zubair Ahmad ◽  
James F. Peters
Keyword(s):  
Set Theory ◽  
Quantum Hall Effect ◽  
Topological Charge ◽  
Electromagnetic Waves ◽  
Quantum Computer ◽  
Quantum Hall ◽  
Higher Dimensions ◽  
Spatial Dimension ◽  
Optical Computers ◽  
Spatial Dimensions

Relationships among near set theory, shape maps and recent accounts of the Quantum Hall effect pave the way to quantum computations performed in higher dimensions.  We illustrate the operational procedure to build a quantum computer able to detect, assess and quantify a fourth spatial dimension.  We show how, starting from two-dimensional shapes embedded in a 2D topological charge pump, it is feasible to achieve the corresponding four-dimensional shapes, which encompass a larger amount of information.  This novel, relatively straightforward architecture not only permits to increase the amount of available qbits in a fixed volume, but also converges towards a solution to the problem of optical computers, that are not allowed to tackle quantum entanglement through their canonical superposition of electromagnetic waves.

scholarly journals Topological charges and conservation laws involving an arbitrary function of time for dynamical PDEs

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200442
Author(s):  
Stephen C. Anco ◽  
Elena Recio
Keyword(s):  
Conservation Laws ◽  
Arbitrary Function ◽  
Topological Charge ◽  
Applied Mathematics ◽  
Initial Boundary ◽  
Spatial Dimension ◽  
Nonlinear Pdes ◽  
Local Conservation ◽  
Potential System ◽  
Spatial Dimensions

Dynamical PDEs that have a spatial divergence form possess conservation laws that involve an arbitrary function of time. In one spatial dimension, such conservation laws are shown to describe the presence of an x -independent source/sink; in two and more spatial dimensions, they are shown to produce a topological charge. Two applications are demonstrated. First, a topological charge gives rise to an associated spatial potential system, allowing non-local conservation laws and symmetries to be found for a given dynamical PDE. This type of potential system has a different form and different gauge freedom compared to potential systems that arise from ordinary conservation laws. Second, when a topological charge arises from a conservation law whose conserved density is non-trivial off of solutions to the dynamical PDE, then this relation yields a constraint on initial/boundary data for which the dynamical PDE will be well posed. Several examples of nonlinear PDEs from applied mathematics and integrable system theory are used to illustrate these results.

HOPPING TRANSPORT AND QUANTUM HALL EFFECT: ABSORPTION OF SURFACE ACOUSTIC WAVES

1994 ◽  
Vol 08 (07) ◽  
pp. 801-807 ◽  
Author(s):  
I.L. Aleiner ◽  
B.I. Shklovskii
Keyword(s):  
Hall Effect ◽  
Quantum Hall Effect ◽  
Acoustic Waves ◽  
Electromagnetic Waves ◽  
Surface Acoustic Waves ◽  
Quantum Hall ◽  
Localization Length ◽  
Hopping Transport ◽  
Drastic Increase ◽  
Peak Widths

New theory of the width of the peaks of the diagonal conductivity in the quantum Hall effect suggested recently by Polyakov and Shklovskii is extended to describe the peak widths of attenuation of the surface acoustic waves (SAW). In contrast to electromagnetic waves, wavelength of SAW can be smaller than the relevant localization length. This leads to a drastic increase of the peaks widths. The latter are shown to grow as (Ts(ω)/T1)0.4 where the effective temperature Ts is about 400ħω for GaAs.

scholarly journals Quantum Hall effect in higher dimensions

2002 ◽  
Vol 641 (3) ◽  
pp. 533-546 ◽  
Author(s):  
Dimitra Karabali ◽  
V.P. Nair
Keyword(s):  
Hall Effect ◽  
Quantum Hall Effect ◽  
Quantum Hall ◽  
Higher Dimensions
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