A method of error correction for a quantum computer includes identifying each of a plurality of physical qubits arranged in a lattice pattern over a surface in a quantum processor of the quantum computer as a one of a data qubit, an ancilla qubit or a flag qubit to define a plurality of data qubits, ancilla qubits and flag qubits. Each pair of interacting data qubits interact with a flag qubit and adjacent flag qubits both interact with a common ancilla qubit. The method further includes performing measurements of weight-four stabilizers, weight-two stabilizers, or both of a surface code formed using at least a sub-plurality of the plurality of physical qubits, or performing measurements of weight-four Bacon-Shor type gauge operators; and correcting fault-tolerantly quantum errors in one or more of the at least sub-plurality of physical qubits based on a measurement from at least one flag qubit.