The theory of quantum error correction plays a prominent role in the practical realization and engineering of
quantum computing and quantum communication devices. The first quantum
error-correcting codes are strikingly similar to classical block codes in their
operation and performance. Quantum error-correcting codes restore a noisy,
decohered quantum state to a pure quantum state. A
stabilizer quantum error-correcting code appends ancilla qubits
to qubits that we want to protect. 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. 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).
The Stabilizer formalism exploits elements of
the Pauli group in formulating quantum error-correcting codes. The set
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. Operators in have eigenvalues and either commute
or anti-commute. The set consists of -fold tensor products of
Elements of act on a quantum register of qubits. We
occasionally omit tensor product symbols in what follows so that
The -fold Pauli group
plays an important role for both the encoding circuit and the
error-correction procedure of a quantum stabilizer code over qubits.
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 abelian subgroup 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
The generators are
independent in the sense that none of them is a product of any other two (up
to a global phase). The operators function in the same
way as a parity check matrix does for a classical linear block code.
Stabilizer error-correction conditions
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).
Relation between Pauli group and binary vectors
A simple but useful mapping exists between elements of and the binary
vector space . This mapping gives a
simplification of quantum error correction theory. It represents quantum codes
with binary vectors and binary operations rather than with Pauli operators and
matrix operations respectively.
We first give the mapping for the one-qubit case. Suppose
is a set of equivalence classes of an operator that have the same phase:
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:
Let denote the symplectic product between two elements :
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
The group operation for the above equivalence class is as follows:
The equivalence class forms a commutative group
under operation . Consider the -dimensional vector space
It forms the commutative group with
operation defined as binary vector addition. We employ the notation
to represent any vectors
vector and has elements and respectively with
similar representations for and .
The symplectic product of and is
where and . Let us define a map as follows:
so that and belong to the same
The map is an isomorphism for the same
reason given as in the previous case:
where . The symplectic product
captures the commutation relations of any operators and :
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.
Example of a stabilizer code
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. It has code distance . 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.
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