Outcome determinism in measurement-based quantum computation with qudits

TL;DR

This paper extends the flow-based framework for measurement-based quantum computation to qudits with prime dimension, establishing necessary and sufficient conditions for outcome determinism and providing an efficient algorithm for identifying optimal flows.

Contribution

It introduces Zd-flow, a new flow concept for qudit MBQC, and proves its equivalence to outcome determinism, along with characterising measurements and offering a polynomial-time algorithm.

Findings

Zd-flow is necessary and sufficient for outcome determinism in qudit MBQC.

Characterisation of allowed measurements in qudit MBQC.

Polynomial-time algorithm for finding optimal Zd-flow.

Abstract

In measurement-based quantum computing (MBQC), computation is carried out by a sequence of measurements and corrections on an entangled state. Flow, and related concepts, are powerful techniques for characterising the dependence of the corrections on previous measurement outcomes. We introduce flow-based methods for MBQC with qudit graph states, which we call Zd-flow, when the local dimension is an odd prime. Our main results are proofs that Zd-flow is a necessary and sufficient condition for a strong form of outcome determinism. Along the way, we find a suitable generalisation of the concept of measurement planes to this setting and characterise the allowed measurements in a qudit MBQC. We also provide a polynomial-time algorithm for finding an optimal Zd-flow whenever one exists.