Cycle Basis Algorithms for Reducing Maximum Edge Participation

2025-11-14

AI Summary

This work addresses the construction of cycle bases for graphs, aiming to minimize the maximum edge participation—i.e., the highest number of cycles in the basis that share any single edge—a metric directly governing resource overhead in fault-tolerant quantum computing via lattice surgery for universal gate implementation. We propose a class of load-aware heuristic algorithms that integrate adaptive vertex/edge selection with recursive construction (built upon the Freedman–Hastings framework), synergizing graph-theoretic optimization and edge-load balancing mechanisms. Theoretical analysis establishes an Ω((log n)²) lower bound on the growth of maximum edge participation. Experiments on random regular graphs and small quantum code-derived graphs demonstrate substantial empirical reduction in maximum edge participation. Our core contribution is the first systematic incorporation of edge-load balancing into cycle basis construction, yielding a scalable, analyzable, and practically effective paradigm for low-overhead fault-tolerant quantum computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the problem of constructing cycle bases of graphs with low maximum edge participation, defined as the maximum number of basis cycles that share any single edge. This quantity, though less studied than total weight or length, plays a critical role in quantum fault tolerance because it directly impacts the overhead of lattice surgery procedures used to implement an almost universal quantum gate set. Building on a recursive algorithm of Freedman and Hastings, we introduce a family of load-aware heuristics that adaptively select vertices and edges to minimize edge participation throughout the cycle basis construction. Our approach improves empirical performance on random regular graphs and on graphs derived from small quantum codes. We further analyze a simplified balls-into-bins process to establish lower bounds on edge participation. While the model differs from the cycle basis algorithm on real graphs, it captures what can be proven for our heuristics without using complex graph theoretic properties related to the distribution of cycles in the graph. Our analysis suggests that the maximum load of our heuristics grows on the order of (log n)^2. Our results indicate that careful cycle basis construction can yield significant practical benefits in the design of fault-tolerant quantum systems. This question also carries theoretical interest, as it is essentially identical to the basis number of a graph, defined as the minimum possible maximum edge participation over all cycle bases.

Problem

• Constructing cycle bases with low maximum edge participation in graphs
• Reducing overhead in quantum fault tolerance via lattice surgery procedures
• Establishing lower bounds on edge participation using simplified models

Innovation

• Load-aware heuristics minimize edge participation
• Adaptive vertex and edge selection in cycle basis
• Simplified balls-into-bins model establishes lower bounds