Quantum Error Correction - The Bit Flip Code

The Bit Flip Code

The repetition code works in a classical channel, because classical bits are easy to measure and to repeat. However, in a quantum channel, it is no longer possible, due to the no-cloning theorem, which forbids the creation of identical copies of an arbitrary unknown quantum state. So a single qubit can not be repeated three times as in the previous example, as any measurement of the qubit will change its wave function. Nevertheless, in a quantum computer, there is another method, which is called the three qubits bit flip code. It uses entanglement and syndrome measurements, and can perform the similar results to the repetition code.

Let be an arbitrary qubit. The first step of the three qubit bit flip code is to entangle the qubit with two other qubits using two CNOT gates with input . The result will be This is just a tensor product of three qubits, and different from cloning a state.

Now these qubits will be sent through a channel where we assume that at most one bit flip may occur. For example, in the case where the first qubit is flipped, the result would be . To diagnose bit flips in any of the three possible qubits, syndrome diagnosis is needed, which includes four projection operators:

It can be obtained:

So it will be known that the error syndrome corresponding to . This three qubits bit flip code can correct one error if at most one bit-flip-error occurred in the channel. It is similar to the three bits repetition code in a classical computer.