Abstract: A method for cluster-state quantum computing method includes transforming a Gaussian graph state into a non-Gaussian percolated graph state by probabilistically subtracting one photon from each of a plurality of modes forming the Gaussian graph state. The method also includes determining cat-basis qubits of the non-Gaussian percolated graph state for which one photon was successfully subtracted from a corresponding one of the modes, and identifying in the non-Gaussian percolated graph state a renormalized graph of logical qubits connected by percolation highways. The logical qubits and percolation highways are formed from the cat-basis qubits. The renormalized graph and the non-Gaussian percolated graph state are outputted to a one-way quantum computer to implementing a quantum computing algorithm.