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Paper 1

A Quantum Complexity Lowerbound from Differential Geometry

Adam R. Brown

Year

2021

Journal

arXiv preprint

DOI

arXiv:2112.05724

arXiv

2112.05724

The Bishop-Gromov bound -- a cousin of the focusing lemmas that Hawking and Penrose used to prove their black hole singularity theorems -- is a differential geometry result that upperbounds the rate of growth of volume of geodesic balls in terms of the Ricci curvature. In this paper, I apply the Bishop-Gromov bound to Nielsen's complexity geometry to prove lowerbounds on the quantum complexity of a typical unitary. For a broad class of penalty schedules, the typical complexity is shown to be exponentially large in the number of qubits. This technique gives results that are tighter than all known lowerbounds in the literature, as well as establishing lowerbounds for a much broader class of complexity geometry metrics than has hitherto been bounded. For some metrics, I prove these lowerbounds are tight. This method realizes the original vision of Nielsen, which was to apply the tools of differential geometry to study quantum complexity.

Open paper

Paper 2

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 $g\left( r\right) =\exp \left( ξr\right)\geq0$

Open paper