In the theory of quantum communication, the entanglement-assisted stabilizer formalism is a method for protecting quantum information with the help of entanglement shared between a sender and receiver before they transmit quantum data over a quantum communication channel. 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's Pauli operators do not necessarily have to form an
Abelian subgroup of the Pauli group formula_1 over formula_2 qubits.
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.
Definition.
We review the construction of an entanglement-assisted code (Brun "et al." 2006). Suppose that
there is a nonabelian subgroup formula_3 of size formula_4.
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
formula_5
for formula_6 with the following commutation relations:
formula_7
formula_8
formula_9
formula_10
The decomposition of formula_6 into the above minimal generating set
determines that the code requires formula_12 ancilla qubits and formula_13 ebits. The code
requires an ebit for every anticommuting pair in the minimal generating set.
The simple reason for this requirement is that an ebit is a simultaneous
formula_14-eigenstate of the Pauli operators formula_15. The second qubit
in the ebit transforms the anticommuting pair formula_16 into a
commuting pair formula_15. The above decomposition also
minimizes the number of ebits required for the code---it is an optimal decomposition.
We can partition the nonabelian group formula_6 into two subgroups: the
isotropic subgroup formula_19 and the entanglement subgroup
formula_20. The isotropic subgroup formula_19 is a commuting
subgroup of formula_6 and thus corresponds to ancilla
qubits:
formula_23.
The elements of the entanglement subgroup formula_20 come in
anticommuting pairs and thus correspond to ebits:
formula_25.
Entanglement-assisted stabilizer code error correction conditions.
The two subgroups formula_19 and formula_20 play a role in the
error-correcting conditions for the entanglement-assisted stabilizer
formalism. An entanglement-assisted code corrects errors in a set
formula_28 if for all formula_29,
formula_30
Operation.
The operation of an entanglement-assisted code is as follows. The sender
performs an encoding unitary on her unprotected qubits, ancilla qubits, and
her half of the ebits. The unencoded state is a simultaneous +1-eigenstate of
the following Pauli operators:
formula_31
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:
formula_32
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. The last step is
to correct the error.
Rate of an entanglement-assisted code.
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 formula_33 information
qubits into formula_2 physical qubits with the help of formula_13 ebits.
Which interpretation is most reasonable depends on the context in which we use
the code. In any case, the parameters formula_2, formula_33, and formula_13 ultimately govern
performance, regardless of which definition of the rate we use to interpret
that performance.
Example of an entanglement-assisted code.
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 formula_46:
formula_47
The first two generators anticommute. We obtain a modified third generator by
multiplying the third generator by the second. We then multiply the last
generator by the first, second, and modified third generators. The
error-correcting properties of the generators are invariant under these
operations. The modified generators are as follows:
formula_48
The above set of generators have the commutation relations given by the
fundamental theorem of symplectic geometry:
formula_49
The above set of generators is unitarily equivalent to the following canonical
generators:
formula_50
We can add one ebit to resolve the anticommutativity of the first two
generators and obtain the canonical stabilizer:
formula_51
The receiver Bob possesses the qubit on the left and the sender Alice
possesses the four qubits on the right. The following state is an eigenstate
of the above stabilizer
formula_52
where formula_53 is a qubit that the sender wants to
encode. The encoding unitary then rotates the canonical stabilizer to the following set of globally commuting
generators:
formula_54
The receiver measures the above generators upon receipt of all qubits to
detect and correct errors.
Encoding algorithm.
We continue with the previous example. 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 operators in
the above example have the following representation as a binary
matrix (See the stabilizer code article):
formula_55
Call the matrix to the left of the vertical bar the "formula_56
matrix" and the matrix to the right of the vertical bar the
"formula_57 matrix."
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. The operations available for manipulating columns of
the above matrix are Clifford operations. Clifford
operations preserve the Pauli group formula_1 under conjugation. The
CNOT gate, the Hadamard gate, and the Phase gate generate the Clifford group.
A CNOT gate from qubit formula_59 to qubit formula_60 adds column formula_59 to column formula_60 in the
formula_57 matrix and adds column formula_60 to column formula_59 in the formula_56 matrix. A Hadamard
gate on qubit formula_59 swaps column formula_59 in the formula_56 matrix with column formula_59 in the
formula_57 matrix and vice versa. A phase gate on qubit formula_59 adds column formula_59 in the
formula_57 matrix to column formula_59 in the formula_56 matrix. Three CNOT gates implement a
qubit swap operation. The effect of a swap on qubits
formula_59 and formula_60 is to swap columns formula_59 and formula_60 in both the formula_57 and formula_56 matrix.
The algorithm begins by computing the symplectic product between the first row
and all other rows. We emphasize that the symplectic product here is the
standard symplectic product. Leave the matrix as it is if the first row is not
symplectically orthogonal to the second row or if the first row is
symplectically orthogonal to all other rows. Otherwise, swap the second row
with the first available row that is not symplectically orthogonal to the
first row. In our example, the first row is not symplectically orthogonal to
the second so we leave all rows as they are.
Arrange the first row so that the top left entry in the formula_57 matrix is one. A
CNOT, swap, Hadamard, or combinations of these operations can achieve this
result. We can have this result in our example by swapping qubits one and two.
The matrix becomes
formula_84
Perform CNOTs to clear the entries in the formula_57 matrix in the top row to the
right of the leftmost entry. These entries are already zero in this example so
we need not do anything. Proceed to the clear the entries in the first row of
the formula_56 matrix. Perform a phase gate to clear the leftmost entry in the first
row of the formula_56 matrix if it is equal to one. It is equal to zero in this case
so we need not do anything. We then use Hadamards and CNOTs to clear the other
entries in the first row of the formula_56 matrix.
We perform the above operations for our example. Perform a Hadamard on qubits
two and three. The matrix becomes
formula_89
Perform a CNOT from qubit one to qubit two and from qubit one to qubit three.
The matrix becomes
formula_90
The first row is complete. We now proceed to clear the entries in the second
row. Perform a Hadamard on qubits one and four. The matrix becomes
formula_91
Perform a CNOT from qubit one to qubit two and from qubit one to qubit four.
The matrix becomes
formula_92
The first two rows are now complete. 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.
Add row one to any other row that has one as the leftmost entry in its formula_56
matrix. Add row two to any other row that has one as the leftmost entry in its
formula_57 matrix. For our example, we add row one to row four and we add row two to
rows three and four. The matrix becomes
formula_95
The first two rows are now symplectically orthogonal to all other rows per the
fundamental theorem of symplectic geometry.
We proceed with the same algorithm on the next two rows. The next two rows are
symplectically orthogonal to each other so we can deal with them individually.
Perform a Hadamard on qubit two. The matrix becomes
formula_96
Perform a CNOT from qubit two to qubit three and from qubit two to qubit
four. The matrix becomes
formula_97
Perform a phase gate on qubit two:
formula_98
Perform a Hadamard on qubit three followed by a CNOT from qubit two to qubit
three:
formula_99
Add row three to row four and perform a Hadamard on qubit two:
formula_100
Perform a Hadamard on qubit four followed by a CNOT from qubit three to qubit
four. End by performing a Hadamard on qubit three:
formula_101
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.
The above operations in reverse order
take the canonical stabilizer to the encoded
stabilizer.