Figure 3

The sweep rule in the rhombic dodecahedral lattice. The sweep direction \(\vec {\omega } = -(1,1,1)\) is indicated by the arrow. (a) The future \({{{\uparrow }\,}}(v)\) of v (the blue vertex), is the power set of the black vertices and the blue vertex. (b) The causal diamond of U, where U is the set of blue vertices. The red vertices are \(\sup U\) and \(\inf U\), and \(\lozenge \left( U\right)\) is the power set of the red, black and blue vertices. (c) In the rhombic dodecahedral lattice, there are two types of vertices: one type is degree four (red and black) and the other is degree eight (blue). The red (blue) shaded faces are the qubits that the rule may return, depending on the syndrome at the highlighted red (blue) vertex. The rule returns nothing at the black vertices because there are no syndromes whose restriction to a black vertex is in the future of that vertex.