Quantum Error Correction with Only Two Extra Qubits

Noise rates in quantum computing experiments have dropped dramatically, but reliable qubits remain precious. Fault-tolerance schemes with minimal qubit overhead are therefore essential. We introduce fault-tolerant error-correction procedures that use only two extra qubits. The procedures are based on adding "flags" to catch the faults that can lead to correlated errors on the data. They work for various distance-three codes. In particular, our scheme allows one to test the [vertical bar 5; 1; 3 vertical bar] code, the smallest error-correcting code, using only seven qubits total. Our techniques also apply to the [vertical bar 7; 1; 3 vertical bar] and [vertical bar 15; 7; 3 vertical bar] Hamming codes, thus allowing us to protect seven encoded qubits on a device with only 17 physical qubits.