Hierarchies of resources for measurement-based quantum computation

TL;DR

This paper investigates the resource requirements of measurement-based quantum computation (MBQC), identifying hierarchies of resources and minimal qubit counts needed for computing Boolean functions within various MBQC subtheories.

Contribution

It refines the understanding of MBQC resources by classifying which Boolean functions are computable within certain resource constraints and quantifies minimal qubits needed in stabiliser MBQC.

Findings

Identifies Boolean functions computable in non-adaptive MBQC with Clifford hierarchy operations.

Computes minimal qubits for Boolean functions in stabiliser MBQC.

Proposes hierarchies of resources that characterize MBQC power beyond contextuality.

Abstract

For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where non-locality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality.