ABSTRACT
The first theme of this book-quantum error correction-has now been pre sented. We carry on in this chapter with a discussion of fault-tolerant quantum computing (FTQC), which is our second theme. The discussion will take place in the context of quantum stabilizer codes and is based on Refs. [1-6]. See also Refs. [7,8]. Section 5.1 introduces the idea of fault-tolerance and makes ba sic definitions, while Section 5.2 presents a procedure for doing fault-tolerant quantum error correction. The theory of FTQC is then taken up in the fol lowing five sections. Section 5.3 examines the action of unitary operators on a quantum stabilizer code. It introduces two important groups: (i) the normalizer of the stabilizer S whose elements are encoded unitary operations; and (ii) the Clifford group which is generated by the Hadamard, phase, and CNOT gates. The discussion focuses on unitary operations that belong to the inter section of these two groups. By combining measurements on ancilla qubits (Section 5.4) with a four-qubit unitary operation introduced in Section 5.5, it will be possible to fault-tolerantly implement these gates on encoded data for stabilizer codes that encode a single qubit. Section 5.6 presents an extension of this approach that works for stabilizer codes encoding multiple qubits, and Section 5.7 completes the theoretical development by showing how a faulttolerant encoded Toffoli gate can be constructed for any quantum stabilizer code. By this point a theoretical framework is in place that can construct a set of fault-tolerant encoded quantum gates that is capable of universal quan tum computation using any quantum stabilizer code. The final two sections illustrate the theory by applying it to the [5,1,3] and [4,2,2] stabilizer codes. A brief summary of universal sets of quantum gates is given in Appendix C.
