Very ample line bundles, contextuality and quantum computation

TL;DR

This paper explores the geometric relationship between line bundles and contextuality, demonstrating that contextuality is essential for quantum computational advantage in measurement-based quantum computation (MBQC).

Contribution

It introduces a geometric framework linking line bundles to contextuality and categorifies MBQC as a subfunctor of the spectral presheaf, highlighting the resource's necessity.

Findings

Contextuality is geometrically linked to line bundles in algebraic geometry.

Existence of certain rational maps relates to the computational power of MBQC.

MBQC's categorical formulation reveals the triviality of the functor despite computational complexity.

Abstract

I relate contextuality to line bundles. Line bundles are important in algebraic geometry, they determine through their global sections rational maps to projective spaces. I explain how such maps, if they exist, relate rationally the input and output of measurement based computation (MBQC) and show geometrically that, indeed, contextuality is a necessary resource for the computational advantage in MBQC. I also leverage the definition of MBQC to category theory and present it as a "subfunctor" of the spectral presheaf. In general, the MBQC functor is pointless whereas the computation is trivial.