Quantum Error Correction

The principle of quantum error correction is the same as its classical counterpart and can be performed by following the steps below:

i. Measure all three encoded qubits.

ii. Identify the one which differs from the others.

iii. Flip back the erroneous qubit.

If the three encoded qubits are transmitted one at a time through a noisy channel, the error protection mechanism will allow the detection and correction of a bit-flop error without destroying the superposition. It is assumed that only a single qubit out of the three has flipped.

1. Bit-Flip Error Correction:

The correction circuit for a single bit-flip error is illustrated in the figure. Like in the encoder circuit, two ancilla qubits are included in this circuit and the parties of the Qubit 1 and Qubit 2 well as the parity of Qubit 2 and Qubit 3 are checked using four CNOT gates. The control bits of these CNOT gates are the encoded qubits, while the target qubits are the two ancillary qubits introduced at the receiving end. A pair of CNOT gates is used to check the parity of three-qubit data.

2. Phase Error Correction:

There are other types of quantum errors that affect single qubits in a way that the repetition code can’t shield against. For example, a phase-flip error:

|k> →(-1)k> where k ε {0,1}

It may affect a qubit when it is transmitted through the channel to the receiver and could flip the relative phase of |0> and |1>. In other words, the sign between |0> and |1> in a quantum state can become inverted when a phase-flip error occurs during transmission. Thus, a phase-flip error can be represented by the Z matrix: