In quantum computing and quantum communication, a stabilizer code is a class of quantum codes for performing quantum error correction. The toric code, and surface codes more generally,[1] are types of stabilizer codes considered very important for the practical realization of quantum information processing.
Conceptual background
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 first quantum error-correcting codes are strikingly similar to classical block codes in their operation and performance.
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).
Mathematical background
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 Pauli operators:

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.
Definition
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
independent generators

The generators are independent in the sense that none of them are 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

or

where
is the centralizer of
(i.e., the subgroup of elements that commute with all members of
, also known as the commutant).
Simple example of a stabilizer code
A simple example of a stabilizer code is a three qubit
stabilizer code. It encodes
logical qubit into
physical qubits and protects against a single-bit flip error in the set
. This does not protect against other Pauli errors such as phase flip errors in the set
.or
. This has code distance
. Its stabilizer consists of
Pauli operators:

If there are no bit-flip errors, both operators
and
commute, the syndrome is +1,+1, and no errors are detected.
If there is a bit-flip error on the first encoded qubit, operator
will anti-commute and
commute, the syndrome is -1,+1, and the error is detected. If there is a bit-flip error on the second encoded qubit, operator
will anti-commute and
anti-commute, the syndrome is -1,-1, and the error is detected. If there is a bit-flip error on the third encoded qubit, operator
will commute and
anti-commute, the syndrome is +1,-1, and the error is detected.
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 via a parity measurement and applying a corrective operation.
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:
![{\displaystyle \left[A\right]=\left\{\beta A\ |\ \beta \in \mathbb {C} ,\ \left\vert \beta \right\vert =1\right\}.}](./_assets_/fd49593693fe70823827513bbb79051321af4131.svg)
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:
![{\displaystyle \left[N\left(u+v\right)\right]=\left[N\left(u\right)\right]\left[N\left(v\right)\right].}](./_assets_/84c72186bd8d9741f99407d37dd32e804bf4a0e5.svg)
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
![{\displaystyle \left[\mathbf {A} \right]=\left\{\beta \mathbf {A} \ |\ \beta \in \mathbb {C} ,\ \left\vert \beta \right\vert =1\right\}.}](./_assets_/6552c351391c46e4a10ec670ffdcea07011c2a83.svg)
The group operation
for the above equivalence class is as follows:
![{\displaystyle \left[\mathbf {A} \right]\ast \left[\mathbf {B} \right]\equiv \left[A_{1}\right]\ast \left[B_{1}\right]\otimes \cdots \otimes \left[A_{n}\right]\ast \left[B_{n}\right]=\left[A_{1}B_{1}\right]\otimes \cdots \otimes \left[A_{n}B_{n}\right]=\left[\mathbf {AB} \right].}](./_assets_/ce39a8417c2e35337f743ae919d79a2141e7c490.svg)
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
respectively. Each vector
and
has elements
and
respectively with similar representations for
and
. The symplectic product
of
and
is

or

where
and
. Let us define a map
as follows:

Let

so that
and
belong to the same equivalence class:
![{\displaystyle \left[\mathbf {N} \left(\mathbf {u} \right)\right]=\left[\mathbf {Z} \left(\mathbf {z} \right)\mathbf {X} \left(\mathbf {x} \right)\right].}](./_assets_/ea6db5042a87808bdf9e86fb42ecdb90f8ee3ee1.svg)
The map
is an isomorphism for the same reason given as in the previous case:
![{\displaystyle \left[\mathbf {N} \left(\mathbf {u+v} \right)\right]=\left[\mathbf {N} \left(\mathbf {u} \right)\right]\left[\mathbf {N} \left(\mathbf {v} \right)\right],}](./_assets_/891c51f5b7994f30b80c859f641f9f4dd2d0301a.svg)
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.
References
- D. Gottesman, "Stabilizer codes and quantum error correction," quant-ph/9705052, Caltech Ph.D. thesis. https://arxiv.org/abs/quant-ph/9705052
- Shor, Peter W. (1995-10-01). "Scheme for reducing decoherence in quantum computer memory". Physical Review A. 52 (4). American Physical Society (APS): R2493 – R2496. Bibcode:1995PhRvA..52.2493S. doi:10.1103/physreva.52.r2493. ISSN 1050-2947. PMID 9912632.
- Calderbank, A. R.; Shor, Peter W. (1996-08-01). "Good quantum error-correcting codes exist". Physical Review A. 54 (2). American Physical Society (APS): 1098–1105. arXiv:quant-ph/9512032. Bibcode:1996PhRvA..54.1098C. doi:10.1103/physreva.54.1098. ISSN 1050-2947. PMID 9913578. S2CID 11524969.
- Steane, A. M. (1996-07-29). "Error Correcting Codes in Quantum Theory". Physical Review Letters. 77 (5). American Physical Society (APS): 793–797. Bibcode:1996PhRvL..77..793S. doi:10.1103/physrevlett.77.793. ISSN 0031-9007. PMID 10062908.
- 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 https://arxiv.org/abs/quant-ph/9608006