Fig. 1: Tanner graph of the gauging measurement and a surface code example.
From: Low-overhead fault-tolerant quantum computation by gauging logical operators
Left: Tanner graph of the deformed code can be compactly represented by creating ordered subsets of qubits (circles) and checks (squares) of X, Z or mixed (unlabelled) type. Each edge connecting an X or Z check set \({\mathcal{P}}\) and qubit set \({\mathcal{Q}}\) is labelled by a binary matrix H with \(| {\mathcal{P}}|\) rows and \(| {\mathcal{Q}}|\) columns, where Hij = 1 if and only if the ith check in \({\mathcal{P}}\) acts non-trivially on the jth qubit in \({\mathcal{Q}}\). Edges from the mixed-type check sets are instead labelled with a symplectic matrix of the form [HX∣HZ], where HX indicates qubits acted on by X and HZ, those by Z. The original code may not be CSS, but we assume (without loss of generality) that L—the operator being measured—is X type and its qubit support is \({\mathcal{L}}\). Set \({\mathcal{C}}\) contains checks from the original code that do not have Z-type support on \({\mathcal{L}}\), whereas set \({\mathcal{S}}\) contains checks that do. Also, \({\mathcal{A}}\) is the set of Gauss’s law operators Av, \({\mathcal{B}}\) is the set of flux operators Bp, \({\mathcal{E}}\) is the set of edge qubits and we abuse the notation so that G also denotes the auxiliary graph’s incidence matrix. Matrix N specifies a cycle basis of G, and M indicates how original stabilizers are deformed by perfect matching in G. Right: applying the gauging measurement procedure to a product of X-type logicals on a pair of surface codes (dark grey edges) and choosing the graph G to be a ladder (light grey edges) results in a standard surface code lattice surgery procedure.
