Site and bond percolation thresholds in $K_{n,n}$-based lattices: Vulnerability of quantum annealers to random qubit and coupler failures on chimera topologies

Abstract

We estimate the critical thresholds of bond and site percolation on nonplanar, effectively two-dimensional graphs with chimera like topology. The building blocks of these graphs are complete and symmetric bipartite subgraphs of size $2n$, referred to as $K_{n,n}$ graphs. For the numerical simulations we use an efficient union-find based algorithm and employ a finite-size scaling analysis to obtain the critical properties for both bond and site percolation. We report the respective percolation thresholds for different sizes of the bipartite subgraph and verify that the associated universality class is that of standard two-dimensional percolation. For the canonical chimera graph used in the D-Wave Systems Inc.~quantum annealer ($n = 4$), we discuss device failure in terms of network vulnerability, i.e., we determine the critical fraction of qubits and couplers that can be absent due to random failures prior to losing large-scale connectivity throughout the device.