It extends the standard stabilizer formalism by including shared entanglement (Brun et al. 2006).
The advantage of entanglement-assisted stabilizer codes is that the sender can exploit the error-correcting properties of an arbitrary set of Pauli operators.
The sender can make clever use of her shared ebits so that the global stabilizer is Abelian and thus forms a valid quantum error-correcting code.
We review the construction of an entanglement-assisted code (Brun et al. 2006).
Application of the fundamental theorem of symplectic geometry (Lemma 1 in the first external reference) states that there exists a minimal set of independent generators
into the above minimal generating set determines that the code requires
The code requires an ebit for every anticommuting pair in the minimal generating set.
and thus corresponds to ancilla qubits: The elements of the entanglement subgroup
come in anticommuting pairs and thus correspond to ebits: The two subgroups
play a role in the error-correcting conditions for the entanglement-assisted stabilizer formalism.
The unencoded state is a simultaneous +1-eigenstate of the following Pauli operators: The Pauli operators to the right of the vertical bars indicate the receiver's half of the shared ebits.
The encoding unitary transforms the unencoded Pauli operators to the following encoded Pauli operators: The sender transmits all of her qubits over the noisy quantum channel.
The receiver then possesses the transmitted qubits and his half of the ebits.
He measures the above encoded operators to diagnose the error.
We can interpret the rate of an entanglement-assisted code in three different ways (Wilde and Brun 2007b).
Suppose that an entanglement-assisted quantum code encodes
We present an example of an entanglement-assisted code that corrects an arbitrary single-qubit error (Brun et al. 2006).
Suppose the sender wants to use the quantum error-correcting properties of the following nonabelian subgroup of
The error-correcting properties of the generators are invariant under these operations.
The encoding unitary then rotates the canonical stabilizer to the following set of globally commuting generators: The receiver measures the above generators upon receipt of all qubits to detect and correct errors.
We detail an algorithm for determining an encoding circuit and the optimal number of ebits for the entanglement-assisted code---this algorithm first appeared in the appendix of (Wilde and Brun 2007a) and later in the appendix of (Shaw et al. 2008).
The algorithm consists of row and column operations on the above matrix.
Row operations do not affect the error-correcting properties of the code but are crucial for arriving at the optimal decomposition from the fundamental theorem of symplectic geometry.
Three CNOT gates implement a qubit swap operation.
A CNOT, swap, Hadamard, or combinations of these operations can achieve this result.
Perform a phase gate to clear the leftmost entry in the first row of the
They need one ebit to compensate for their anticommutativity or their nonorthogonality with respect to the symplectic product.
Now we perform a "Gram-Schmidt orthogonalization" with respect to the symplectic product.
End by performing a Hadamard on qubit three: The above matrix now corresponds to the canonical Pauli operators.
Adding one half of an ebit to the receiver's side gives the canonical stabilizer whose simultaneous +1-eigenstate is the above state.