Fig. 3: Cycle sparsification and cellulation.
From: Low-overhead fault-tolerant quantum computation by gauging logical operators
Left: cycle sparsification with two layers. Thin, black edges are copies of the original graph and grey edges connect them. A complete cycle basis consists of the highlighted cycles and all length-four cycles between layers that traverse exactly two black and two grey edges. The constant c in Definition 3 is 1 after this sparsification, whereas it was 2 for the original graph, and therefore, \({R}_{G}^{1}=2\). Right: cellulating a weight-six cycle (black) into triangles by adding additional edges (grey).
