Measurement-based quantum computing is a universal model where successive product measurements of an entangled resource state drive the computation. The non-deterministic nature of measurements necessitates adaptivity to ensure deterministic computation. Flow structures characterize cases where such adaptive correction procedures are feasible. Recently, flow has been defined for prime-dimensional qudit graph states, which is more cumbersome than for qubit graph states.
Here, we provide a simpler definition of qudit flow and explore various useful properties, drawing on results from the qubit case. We show how to focus qudit flow and argue that focused flow is canonical.
Our improved algebraic formulation captures focused flow and leads to an $O(n^3)$ flow-finding algorithm (where $n$ is the number of qudits), matching the best-known complexity for qubit flows and improving the previous $O(n^4)$ result for qudits. Moreover, we explore multiple flow-preserving transformations, opening pathways for optimization. These transformations include pivoting, removal and insertion of certain types of vertices, and reversibility of flow.
Lastly, we propose an algorithmic approach to generating large qudit computations with flow for testing or machine learning.
Blogger's Review: This paper significantly enhances the efficiency of flow-finding algorithms by simplifying the definition of qudit flow, advancing measurement-based quantum computing in the realm of multi-dimensional qubits and laying the groundwork for future optimization and applications.