Reliable qubits are difficult to engineer, but standard fault-tolerance
schemes use seven or more physical qubits to encode each logical qubit, with
still more qubits required for error correction. The large overhead makes it
hard to experiment with fault-tolerance schemes with multiple encoded qubits.
The 15-qubit Hamming code protects seven encoded qubits to distance three. We
give fault-tolerant procedures for applying arbitrary Clifford operations on
these encoded qubits, using only two extra qubits, 17 total. In particular,
individual encoded qubits within the code block can be targeted. Fault-tolerant
universal computation is possible with four extra qubits, 19 total. The
procedures could enable testing more sophisticated protected circuits in
small-scale quantum devices.
Our main technique is to use gadgets to protect gates against correlated
faults. We also take advantage of special code symmetries, and use pieceable