Fault-tolerant quantum computation with few qubits

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. We give space-efficient methods for fault-tolerant error correction and computation.

For example, with the Steane seven-qubit code, we give a scheme that uses three extra qubits, arranged in two dimensions, to fault-tolerantly extract three syndromes in parallel. With a 19-qubit system, we show that one can protect and compute fault tolerantly on seven encoded qubits. The procedures could enable testing more sophisticated protected circuits in small-scale quantum devices.