But the exploration of QECC does not stop here! We shall discover the elegant Calderbank-Shor-Steane (CSS) codes, which have the capability of correcting any number of errors t, both bit-flip and phase-flip. As an application of the CSS code, I then describe the 7-qubit Hadamard-Steane code, which can correct up to one error on single qubits. A corresponding quantum circuit, based on an original generator-matrix example, is presented. Quantum repetition code In Chapter 11, we saw that the simplest form of error-correction code (ECC) is the repetition code, based on the principle of majority logic. The background assumption is that in a given message sequence, or bit string, the probability of a bit error is sufficiently small for the majority of bits to be correctly transmitted through the channel. It then suffices to repeat each of the bits a certain number of times, at the cost of wasting the channel resource or "bandwidth." For instance, if one repeats each of the bits four times, the original bit sequence 0111 is encoded into (underscores introduced for reading clarity): (0000_1111_1111_1111, which may yield at the channel output 0010_1111_0111_1011. Clearly, a single bit error occurred in the first and last two blocks of the transmitted message, and the simple rule of majority logic suffices to detect such errors and revert the bits to the correct values. If p is the probability of error for a single bit, the probability 24.1 Quantum repetition code that there is more than one error in a single four-bit block is = 6p2(1-p)2+4p3(1-p (24.1) =4p3 +6p2-9p4. It is easily checked that p(4) < p for p < 0.155,and p 4 < p/10 for p < 0.0165, hence, this repetition code is reliable (i.e., errors can be reliably corrected) provided p is chosen to be sufficiently small. Here, we shall apply the principle of repetition codes to quantum channels, which defines the new notion of quantum error-correction coding (QECC). We may conceptu- ally expand our view of a "quantum channel" beyond that of a quantum communication system 