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Paper 1
Quantum Algorithm for the Longest Trail Problem
Kamil Khadiev, Ruslan Kapralov
- Year
- 2021
- Journal
- arXiv preprint
- DOI
- arXiv:2112.13847
- arXiv
- 2112.13847
We present the quantum algorithm for the Longest Trail Problem. The problem is to search the longest edge-simple path for a graph with $n$ vertexes and $m$ edges. Here edge-simple means no edge occurs in the path twice, but vertexes can occur several times. The running time of our algorithm is $O^*(1.728^m)$.
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Confined Klein-Gordon oscillators in Minkowski spacetime and a pseudo-Minkowski spacetime with a space-like dislocation: PDM KG-oscillators, isospectrality and invariance
Omar Mustafa
- Year
- 2021
- Journal
- arXiv preprint
- DOI
- arXiv:2111.10077
- arXiv
- 2111.10077
We revisit the a confined (in a Cornell-type Lorentz scalar potential) KG-oscillator in Minkowski spacetime with space-like dislocation background. We show that the effect of space-like dislocation is to shift the energy levels along the dislocation parameter axis, and consequently energy levels crossings are unavoidable. We report some KG-particles in a pseudo-Minkowski spacetime with space-like dislocation that admit isospectrality and invariance with the confined KG-oscillator in Minkowski spacetime with space-like dislocation. An alternative PDM setting for the KG-particles (relativistic particles in general) is introduced. We discuss the effects of space-like dislocation and PDM settings on the confined KG-oscillators in Minkowski spacetime with space-like dislocation. Three confined PDM KG-oscillators are discussed as illustrative examples, (i) a PDM KG-oscillator from a dimensionless scalar multiplier $g\left( r\right) =\, exp(2αr^2)\geq0,\, α\,\geq 0$, (ii) a PDM KG-oscillator from a power law type dimensionless scalar multiplier $g\left( r\right) =Ar^{σ}\geq0$, and (iii) a PDM KG-oscillator in a Cornnell-type confinement with a dimensionless scalar multiplier $g\left( r\right) =\exp \left( ξr\right)\geq0$
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