Quantum Error Correction - The Shor Code

The Shor Code

The error channel may induce either a bit flip, a sign flip, or both. It is possible to correct for both types of errors using one code, and the Shor code does just that. In fact, the Shor code corrects arbitrary single-qubit errors.

Let be a quantum channel that can arbitrarily corrupt a single qubit. The 1st, 4th and 7th qubits are for the sign flip code, while the three group of qubits (1,2,3), (4,5,6), and (7,8,9) are designed for the bit flip code. With the Shor code, a qubit state will be transformed into the product of 9 qubits, where

If a bit flip error happens to a qubit, the syndrome analysis will be performed on each set of states (1,2,3), (4,5,6), and (7,8,9), then correct the error.

If the three bit flip group (1,2,3), (4,5,6), and (7,8,9) are considered as three inputs, then the Shor code circuit can be reduced as a sign flip code. This means that the Shor code can also repair sign flip error for a single qubit.

The Shor code also can correct for any arbitrary errors (both bit flip and sign flip) to a single qubit. If an error is modeled by a unitary transform U, which will act on a qubit, then can be described in the form

where ,,, and are complex constants, I is the identity, and the Pauli matrices are given by

\sigma_x=\biggl( \begin{matrix} 0&1\\1&0 \end{matrix} \biggr);
\sigma_y=\biggl( \begin{matrix} 0&-i\\i&0 \end{matrix} \biggr);
\sigma_z=\biggl( \begin{matrix} 1&0\\0&-1 \end{matrix} \biggr)

If U is equal to I, then no error occurs. If, a bit flip error occurs. If, a sign flip error occurs. If then both a bit flip error and a sign flip error occur. Due to linearity, it follows that the Shor code can correct arbitrary 1-qubit errors.

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