Cycle Basis Algorithms for Reducing Maximum Edge Participation

TL;DR

The paper studies cycle bases with minimal maximum edge participation, a metric linked to quantum fault-tolerance overhead via lattice surgery. Building on the Freedman–Hastings framework, it introduces load-aware heuristics that adaptively select vertices and edges to reduce edge congestion, with Version 3 typically delivering the best empirical performance on random regular graphs and quantum-code-derived graphs. A theoretical proxy via a balls-into-bins model yields a lower bound $L_{\max} = \Omega(\log^2 M)$ on edge load, suggesting inherent limits and guiding understanding of the heuristics' behavior. The work demonstrates meaningful practical gains for fault-tolerant quantum systems and contributes to the theoretical understanding of the basis number in graphs. Key implications include reduced ancilla overhead in LDPC-based quantum codes and improved scalability of cycle-basis constructions in relevant graph families.

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.

Figures (10)

• Figure 1.1: Illustration of Case 2A and Case 2B in the Freedman--Hastings cycle basis algorithm. Vertex $v$ has degree two, with neighbors $x$ and $y$. In Case 2A, we proceed recursively on the updated graph and after computing a cycle basis, reinsert v if any cycle basis contains $(x,y)$; in Case 2B, we add the triangle $[x,v,y]$ to the cycle basis and proceed recursively on the updated graph.
• Figure 3.1: Illustration of how Case 2 introduces additional reductions in a 3-regular graph following Case 3. Suppose edge $(u, v)$ is removed as part of a short cycle in Case 3. After removal, both $u$ and $v$ become degree-two vertices. By Case 2A, edges $(x, v)$ and $(v, y)$ are replaced by $(x, y)$, and the load on $(x, y)$ is set to the maximum of the loads on $(x, v)$ and $(v, y)$. This gives an advantage only if all 3 edges incident to $v$ has high load. A similar reduction applies to the pair $(w, z)$ resulting from the vertex $u$.
• Figure 3.2: Boxplot comparison of the maximum edge participation across five algorithmic variants of the Freedman--Hastings algorithm, evaluated on random $d$-regular graphs. The left panel corresponds to $d = 3$, and the right panel to $d = 8$. The boxplot aggregates the results from thousands of random graph instances (see Table \ref{['tab:trialcounts']}). Each variant is color-coded, and the black dashed curve in both panels shows the baseline $\log_2(n)$ scaling for reference. The boxplots summarize the distribution of the maximum edge participation over all trials: the central line indicates the median, the box bounds the interquartile range (IQR), and the whiskers extend to 1.5×IQR. Outliers beyond that range are represented by small circles.
• Figure 3.3: Comparison of the ratio between the actual median maximum edge participation and fitted functions $c\log_2(n)$ (left) and $c\log_2^2(n)$ (right), for Versions 0 and 3 of the algorithm. Each curve is normalized by its own fitted coefficient $c$, independently optimized per curve. Solid lines are used throughout, with marker shape distinguishing the degree: squares (degree 3) and triangles (degree 8) for Version 0 (blue), and circles (degree 3) and inverted triangles (degree 8) for Version 3 (red). A flat curve near 1 indicates better scaling agreement with the fitted asymptotic model.
• Figure 4.1: Median maximum edge participation across algorithmic variants for the graphs arising from

Theorems & Definitions (8)

• Theorem 5.1
• Lemma 5.2: Few bad buckets in one epoch
• Lemma 5.3: Few bad buckets over $L$ epochs
• Proof 1
• Proof 2
• Proof 3
• Theorem C .1: Pointwise comparison between P1 and P2
• Proof 4