The stabilizer theory of quantum error correction allows one to import some classical binary or quaternary codes for use as a quantum code. The only "catch" when importing is that the classical code must satisfy the dual-containing (or self-orthogonality) constraint. Researchers have found many examples of classical codes satisfying this constraint, but most classical codes do not. Nevertheless, it is still useful to import classical codes in this way (though, see how the entanglement-assisted stabilizer formalism overcomes this difficulty).
Let us define an stabilizer quantum error-correcting code to encode logical qubits into physical qubits. The rate of such a code is . Its stabilizer is an abeliansubgroup of the -fold Pauli group : . does not contain the operator . The simultaneous -eigenspace of the operators constitutes the codespace. The codespace has dimension so that we can encode qubits into it. The stabilizer has a minimal representation in terms of independent generators
One of the fundamental notions in quantum error correction theory is that it suffices to correct a discrete error set with support in the Pauli group. Suppose that the errors affecting an encoded quantum state are a subset of the Pauli group:
Because and are both subsets of , an error that affects an encoded quantum state either commutes or anticommutes with any particular element in . The error is correctable if it anticommutes with an element in . An anticommuting error is detectable by measuring each element in and computing a syndrome identifying . The syndrome is a binary vector with length whose elements identify whether the error commutes or anticommutes with each . An error that commutes with every element in is correctable if and only if it is in . It corrupts the encoded state if it commutes with every element of but does not lie in . So we compactly summarize the stabilizer error-correcting conditions: a stabilizer code can correct any errors in if
where is the centralizer of (i.e., the subgroup of elements that commute with all members of , also known as the commutant).
Let be the set of phase-free Pauli operators where . Define the map as
Suppose . Let us employ the shorthand and where , , , . For example, suppose . Then . The map induces an isomorphism because addition of vectors in is equivalent to multiplication of Pauli operators up to a global phase:
The symplectic product gives the commutation relations of elements of :
The symplectic product and the mapping thus give a useful way to phrase Pauli relations in terms of binary algebra. The extension of the above definitions and mapping to multiple qubits is straightforward. Let denote an arbitrary element of . We can similarly define the phase-free -qubit Pauli group where
It forms the commutative group with operation defined as binary vector addition. We employ the notation to represent any vectors respectively. Each vector and has elements and respectively with similar representations for and . The symplectic product of and is
By comparing quantum error correcting codes in this language to symplectic vector spaces, we can see the following. A symplectic subspace corresponds to a direct sum of Pauli algebras (i.e., encoded qubits), while an isotropic subspace corresponds to a set of stabilizers.
An example of a stabilizer code is the five qubit stabilizer code. It encodes logical qubit into physical qubits and protects against an arbitrary single-qubit error. Its stabilizer consists of Pauli operators:
The above operators commute. Therefore the codespace is the simultaneous +1-eigenspace of the above operators. Suppose a single-qubit error occurs on the encoded quantum register. A single-qubit error is in the set where denotes a Pauli error on qubit . It is straightforward to verify that any arbitrary single-qubit error has a unique syndrome. The receiver corrects any single-qubit error by identifying the syndrome and applying a corrective operation.
A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev. Lett., vol. 77, no. 5, pp. 793–797, Jul 1996.
A. Calderbank, E. Rains, P. Shor, and N. Sloane, “Quantum error correction via codes over GF(4),” IEEE Trans. Inf. Theory, vol. 44, pp. 1369–1387, 1998. Available at http://arxiv.org/abs/quant-ph/9608006