Symmetry constraints on temporal order in measurement-based quantum computation

TL;DR

This paper explores how symmetry constraints and measurement settings influence the possible temporal orders in measurement-based quantum computation, revealing a classification framework and invariance properties.

Contribution

It introduces a classification of temporal relations using matroids and links classical processing to resource states, highlighting symmetry transformations.

Findings

Temporal orders are constrained by quantum measurement randomness.

A matroid-based classification of measurement order relations is provided.

Symmetry transformations preserve temporal relations in the measurement setup.

Abstract

We discuss the interdependence of resource state, measurement setting and temporal order in measurement-based quantum computation. The possible temporal orders of measurement events are constrained by the principle that the randomness inherent in quantum measurement should not affect the outcome of the computation. We provide a classification for all temporal relations among measurement events compatible with a given initial stabilizer state and measurement setting, in terms of a matroid. Conversely, we show that classical processing relations necessary for turning the local measurement outcomes into computational output determine the resource state and measurement setting up to local equivalence. Further, we find a symmetry transformation related to local complementation that leaves the temporal relations invariant.