Using flag qubits for fault-tolerant implementations of topological codes with reduced frequency collisions

A system of correcting errors in a quantum computer, comprising:

• a quantum processor having a plurality of physical qubits arranged in a lattice pattern over a surface of the quantum processor, the plurality of physical qubits comprising a plurality of data qubits, a plurality of ancilla qubits and a plurality of flag qubits, wherein each pair of interacting data qubits in said plurality of data qubits interact with a flag qubit in said plurality of flag qubits and adjacent flag qubits both interact with a common ancilla qubit in said plurality of ancilla qubits;
• wherein at least a sub-plurality of said plurality of physical qubits are selected for performing measurements of weight-four stabilizers or weight-two stabilizers, or both of a surface code formed using the at least sub-plurality of said plurality of physical qubits, or for performing measurements of weight-four Bacon-Shor type gauge operators, and correcting fault-tolerantly quantum errors in one or more of said at least sub-plurality of physical qubits based on a measurement from at least one flag qubit.

A non-transitory computer-executable medium which when executed by a quantum computer comprising a plurality of coupled physical qubits arranged in a hexagonal lattice or a square lattice, causes a quantum processor comprising the plurality of coupled physical qubits arranged in the hexagonal lattice or the square lattice to:

• perform measurements of weight-four stabilizers, weight-two stabilizers, or both of a surface code formed using at least a sub-plurality of a plurality of physical qubits, or perform measurements of weight-four Bacon-Shor type gauge operators, wherein said plurality of physical qubits comprise a plurality of data qubits, a plurality of ancilla qubits and a plurality of flag qubits, wherein each pair of interacting data qubits in said plurality of data qubits interact with a flag qubit in said plurality of flag qubits and adjacent flag qubits both interact with a common ancilla qubit in said plurality of ancilla qubits;
• and
• correct fault-tolerantly quantum errors in one or more of said at least sub-plurality of physical qubits based on a measurement from at least one flag qubit.