Cycle Basis Algorithms for Reducing Maximum Edge Participation

TL;DR

This paper introduces load-aware heuristics for constructing cycle bases with low maximum edge participation, which is crucial for quantum fault tolerance, and demonstrates their effectiveness through empirical results and theoretical analysis.

Contribution

It presents novel heuristics for cycle basis construction that minimize maximum edge participation, improving quantum fault-tolerant system design.

Findings

Heuristics reduce maximum edge participation in random regular graphs.

Empirical performance surpasses previous methods on quantum code graphs.

Theoretical analysis bounds the maximum load growth as (log n)^2.

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.