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This paper advances measurement-based quantum computing by simplifying the definition of flow for qudit graph states and introducing an O(n³) algorithm for finding such flows. By establishing canonical focused flow and exploring flow-preserving transformations, the work provides a more efficient framework for optimizing large-scale quantum computations. These improvements directly address computational complexity bottlenecks in universal quantum computation models beyond standard qubits.
Working With Measurement-Based Computations on Qudits
arXiv:2606.30525v2 Announce Type: replace 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.
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.
In Proceedings QPL 2026, arXiv:2611.04069
Quantum Physics (quant-ph); Data Structures and Algorithms (cs.DS)
arXiv:2606.30525 [quant-ph]
(or arXiv:2606.30525v2 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2606.30525 Focus to learn more arXiv-issued DOI via DataCite
EPTCS 456, 2026, pp. 236-282
https://doi.org/10.4204/EPTCS.456.10 Focus to learn more DOI(s) linking to related resources
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