Cohomology and the Algebraic Structure of Contextuality in Measurement Based Quantum Computation

Abstract

Okay, Roberts, Bartlett and Raussendorf recently introduced a new cohomological approach to contextuality in measurement based quantum computing. We give an abstract description of their obstruction and the algebraic structure it exploits, using the sheaf theoretic framework of Abramsky and Brandenburger. At this level of generality we contrast their approach to theČech cohomology obstruction of Abramsky, Mansfield and Barbosa and give a direct proof thatČech cohomology is at least as powerful.

Simplicial quantum contextuality

“…The 1-outcomes for the remaining 1-contexts can be obtained (inferred) from a i 's as a consequence of the formula in Eq. (19). A similar analysis works for ∂∆ n when n ≥ 3.…”

“…First we prove the statement for X = ∆ n , in which case it suffices to prove Eq. (19). In this case r is determined by an n-tuple…”

“…Building off of these efforts, here we introduce a new framework that subsumes both the topological and sheaf-theoretic approaches and is based on the theory of simplicial sets, which are combinatorial models of topological spaces generalizing simplicial complexes. For previous work where topology and sheaf-theory come in contact, see e.g., [16][17][18][19][20].…”

Combining contextuality and causality: a game semantics approach

“…In future work, we aim to employ our formalism to describe unconditional quantum advantage in shallow circuits, building on [ 31 , 32 ]. We will also investigate other applications to quantum advantage.…”

The Sheaf-Theoretic Structure of Definite Causality

“…The core framework itself has grown through the addition of extended operational semantics [10], All-vs-Nothing arguments [5], logical characterisation of no-signalling polytopes [6], contextual fraction [7], an extension to continuous variable models [15], and the development of a categorical structure on empirical models [4,8,48,49]. Deep connections have been discovered with cohomology [1,11,14,22,23], logic [3,13,51,52,55,58,59] and other frameworks, including database theory [2], Spekkens's framework for contextuality [70], effect algebras [69] and non-commutative geometry [33].…”