Editing Quantum error correction (section)

==Shor code==

The error channel may induce either a bit flip, a sign flip (i.e., a phase flip), or both. It is possible to correct for both types of errors on a logical qubit using a well-designed QEC code. One example of a code that does this is the Shor code, published in 1995.<ref name="Shor1995" /><ref>{{Cite journal |last1=Devitt |first1=Simon J |last2=Munro |first2=William J |last3=Nemoto |first3=Kae |date=2013-06-20 |title=Quantum error correction for beginners |url=http://dx.doi.org/10.1088/0034-4885/76/7/076001 |journal=Reports on Progress in Physics |volume=76 |issue=7 |pages=076001 |doi=10.1088/0034-4885/76/7/076001 |pmid=23787909 |arxiv=0905.2794 |bibcode=2013RPPh...76g6001D |s2cid=206021660 |issn=0034-4885}}</ref>{{Rp|page=10}} Since these two types of errors are the only types of errors that can result after a projective measurement, a Shor code corrects arbitrary single-qubit errors.

[[File:Shore code.svg|upright=1.8|thumb|right|Quantum circuit to encode a single logical qubit with the Shor code and then perform bit flip error correction on each of the three blocks.]]
Let <math>E</math> 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 groups 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 <math>|\psi\rangle=\alpha_0|0\rangle+\alpha_1|1\rangle</math> will be transformed into the product of 9 qubits <math>|\psi'\rangle=\alpha_0|0_S\rangle+\alpha_1|1_S\rangle</math>, where
<math display="block">|0_{\rm S}\rangle=\frac{1}{2\sqrt{2}}(|000\rangle + |111\rangle) \otimes (|000\rangle + |111\rangle
) \otimes (|000\rangle + |111\rangle)</math>
<math display="block">|1_{\rm S}\rangle=\frac{1}{2\sqrt{2}}(|000\rangle - |111\rangle) \otimes (|000\rangle - |111\rangle) \otimes (|000\rangle - |111\rangle)</math>

If a bit flip error happens to a qubit, the syndrome analysis will be performed on each block of qubits (1,2,3), (4,5,6), and (7,8,9) to detect and correct at most one bit flip error in each block.

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 a 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 <math>|\psi\rangle</math>, then <math>U</math> can be described in the form
<math display="block">U = c_0 I + c_1 X + c_2 Y + c_3 Z</math>
where <math>c_0</math>,<math>c_1</math>,<math>c_2</math>, and <math>c_3</math> are complex constants, I is the identity, and the [[Pauli matrices]] are given by
<math display="block">\begin{align}
X &= \begin{pmatrix}
  0&1\\1&0
\end{pmatrix} ; \\
Y &= \begin{pmatrix}
  0&-i\\i&0
\end{pmatrix} ; \\
Z &= \begin{pmatrix}
  1&0\\0&-1
\end{pmatrix} .
\end{align}</math>

If ''U'' is equal to ''I'', then no error occurs. If <math>U=X</math>, a bit flip error occurs. If <math>U=Z</math>, a sign flip error occurs. If <math>U=iY</math> then both a bit flip error and a sign flip error occur. In other words, the Shor code can correct any combination of bit or phase errors on a single qubit.

More generally, the error operator ''U'' does not need to be unitary, but can be an Kraus operator from a [[quantum operation]] representing a system interacting with its environment.