Working with measurement-based computations on qudits

2026-06-29

AI Summary

This work addresses the lack of an efficient and concise flow structure for measurement-based quantum computation on high-dimensional qudits, which has hindered their application in fault tolerance and optimization. Focusing on qudit graph states, the paper proposes a streamlined definition of qudit flow, establishes the canonicity of focused flows, and introduces a flow-finding algorithm with O(n³) time complexity—significantly improving upon the previous O(n⁴) bound. Furthermore, it develops a suite of flow-preserving graph transformations, including pivoting and vertex addition/removal, to construct an algorithmic framework capable of generating large-scale flow-equipped qudit computations. This framework lays the groundwork for qudit circuit optimization and future applications in machine learning.

Abstract

Measurement-based quantum computing is a universal model of quantum computation in which successive product measurements of an entangled resource state drive the computation. The non-deterministic nature of measurements necessitates adaptivity to ensure an overall deterministic computation. Flow structures characterise cases in which such an adaptive correction procedure is possible. Recently, flow has been defined in a setting where the resource states are prime-dimensional qudit graph states rather than the usual qubit graph states. Yet, this qudit flow definition is more burdensome to work with than analogous definitions for qubits. Here, we give a simpler definition of qudit flow and consider various useful properties of this flow, drawing on results for the qubit case. In particular, we show how to focus qudit flow and argue that focused flow is canonical. We improve the previous algebraic formulation to capture focused flow and use it to obtain an $O(n^3)$ flow-finding algorithm (where $n$ is the number of qudits), matching the best known complexity for qubit flows and improving on the previous $O(n^4)$ result for qudits. Furthermore, we explore multiple flow-preserving transformations, thus opening a pathway to using flow for optimisation. 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.

Problem

• measurement-based quantum computing
• qudit
• flow
• graph states
• adaptivity

Innovation

• qudit flow
• measurement-based quantum computing
• focused flow
• flow-preserving transformations
• flow-finding algorithm