No quantum solutions to linear constraint systems from monomial measurement-based quantum computation in odd prime dimension

TL;DR

This paper investigates the relationship between contextuality in measurement-based quantum computation and quantum solutions to linear constraint systems, showing that certain MBQC setups do not admit quantum solutions in odd prime dimensions, thus clarifying the limits of quantum advantage.

Contribution

It establishes a connection between state-dependent contextuality in MBQC and the non-existence of quantum solutions to associated linear constraint systems in odd prime dimensions.

Findings

Quantum solutions do not exist for a large class of MBQC in odd prime dimensions.

The work sharpens the distinction between state-dependent and state-independent contextuality.

It generalizes previous results on the non-existence of quantum solutions to linear constraint systems.

Abstract

We combine the study of resources in measurement-based quantum computation (MBQC) with that of quantum solutions to linear constraint systems (LCS). Contextuality of the input state in MBQC has been identified as a key resource for quantum advantage, and in a stronger form, underlies algebraic relations between (measurement) operators which obey classically unsatisfiable (linear) constraints. Here, we compare these two perspectives on contextuality, and study to what extent they are related. More precisely, we associate a LCS to certain MBQC which exhibit strong forms of state-dependent contextuality, and ask if the measurement operators in such MBQC give rise to state-independent contextuality in the form of quantum solutions of its associated LCS. Our main result rules out such quantum solutions for a large class of MBQC. This both sharpens the distinction between state-dependent and state-independent forms of contextuality, and further generalises results on the non-existence of quantum solutions to LCS in finite odd (prime) dimension.