A unitary encoding circuit rotates the global state into a subspace of a larger Hilbert space.
This highly entangled, encoded state corrects for local noisy errors.
A quantum error-correcting code makes quantum computation and quantum communication practical by providing a way for a sender and receiver to simulate a noiseless qubit channel given a noisy qubit channel whose noise conforms to a particular error model.
The stabilizer theory of quantum error correction allows one to import some classical binary or quaternary codes for use as a quantum code.
However, when importing the classical code, it must satisfy the dual-containing (or self-orthogonality) constraint.
Nevertheless, it is still useful to import classical codes in this way (though, see how the entanglement-assisted stabilizer formalism overcomes this difficulty).
The stabilizer formalism exploits elements of the Pauli group
consists of the Pauli operators: The above operators act on a single qubit – a state represented by a vector in a two-dimensional Hilbert space.
plays an important role for both the encoding circuit and the error-correction procedure of a quantum stabilizer code over
stabilizer quantum error-correcting code to encode
function in the same way as a parity check matrix does for a classical linear block code.
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
that affects an encoded quantum state either commutes or anticommutes with any particular element
It corrupts the encoded state if it commutes with every element of
physical qubits and protects against a single-bit flip error in the set
If there is a bit-flip error on the first encoded qubit, operator
If there is a bit-flip error on the second encoded qubit, operator
If there is a bit-flip error on the third encoded qubit, operator
physical qubits and protects against an arbitrary single-qubit error.
Suppose a single-qubit error occurs on the encoded quantum register.
It is straightforward to verify that any arbitrary single-qubit error has a unique syndrome.
This mapping gives a simplification of quantum error correction theory.
is equivalent to multiplication of Pauli operators up to a global phase: Let
thus give a useful way to phrase Pauli relations in terms of binary algebra.
-dimensional vector space It forms the commutative group
The symplectic product captures the commutation relations of any operators
: The above binary representation and symplectic algebra are useful in making the relation between classical linear error correction and quantum error correction more explicit.
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.