Compare Papers

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

Affine Quantization of the Harmonic Oscillator on the Semi-bounded domain $(-b,\infty)$ for $b: 0 \rightarrow \infty$

Carlos R. Handy

Year

2021

Journal

arXiv preprint

DOI

arXiv:2111.10700

arXiv

2111.10700

The transformation of a classical system into its quantum counterpart is usually done through the well known procedure of canonical quantization. However, on non-Cartesian domains, or on bounded Cartesian domains, this procedure can be plagued with theoretical inconsistencies. An alternative approach is {\it affine quantization} (AQ) Fantoni and Klauder (arXiv:2109.13447,Phys. Rev. D {\bf 103}, 076013 (2021)), resulting in different conjugate variables that lead to a more consistent quantization formalism. To highlight these issues, we examine a deceptively simple, but important, problem: that of the harmonic oscillator potential on the semibounded domain: ${\cal D} = (-b,\infty)$. The AQ version of this corresponds to the (rescaled) system, ${\cal H} = \frac{1}{2}\Big(-\partial_x^2 + \frac{3}{4(x+b)^2} + x^2\Big)$. We solve this system numerically for $b > 0$. The case $b = 0$ corresponds to an {\it exactly solvable} potential, for which the eigenenergies can be determined exactly (through non-wavefunction dependent methods), confirming the results of Gouba ( arXiv:2005.08696 ,J. High Energy Phys., Gravitation Cosmol. {\bf 7}, 352-365 (2021)). We investigate the limit $b \rightarrow \infty$, confirming that the full harmonic oscillator problem is recovered. The adopted computational methods are in keeping with the underlying theoretical framework of AQ. Specifically, one method is an affine map invariant variational procedure, made possible through a moment problem quantization reformulation. The other method focuses on boundedness (i.e. $L^2$) as an explicit quantization criteria. Both methods lead to converging bounds to the discrete state energies; and thus confirming the accuracy of our results, particularly as applied to a singular potential problem.

Open paper