Cohomological framework for contextual quantum computations

TL;DR

This paper introduces a cohomological framework for measurement-based quantum computation, emphasizing the role of symmetry and topological invariants in understanding quantum contextuality and computational output.

Contribution

It presents a novel cohomological approach that links topological invariants to quantum computation and contextuality, revealing fundamental algebraic structures.

Findings

Topological invariants encode computational output.

Invariants distinguish deterministic and probabilistic cases.

Invariants witness quantum contextuality.

Abstract

We describe a cohomological framework for measurement based quantum computation, in which symmetry plays a central role. Therein, the essential information about the computational output is contained in topological invariants, namely elements of two cohomology groups. One of those invariants applies to the deterministic case, and the other to the general probabilistic case. The same invariants also witness quantumness in the form of contextuality. In result, they give rise to fundamental algebraic structures underlying quantum computation.