Stabilizer code

The stabilizer formalism is based on the -qubit Pauli group , and extensively uses the facts that Hermitian operators (ones with scalar factors instead of ) in have eigenvalues \pm1, and that two operators in either commute or anti-commute. A stabilizer of a stabilizer code with physical qubits is an -qubit Pauli operator such that all valid code states |\psi\rangle lie in the -eigenspace of , i.e., . A stabilizer code is defined by its stabilizers in the sense that the converse is also true: A state |\psi\rangle is a valid code state for the stabilizer code if and only if holds for each stabilizer . Therefore the simultaneous -eigenspace of the stabilizers constitutes the codespace of the stabilizer code. Any two stabilizers and must commute and must also be a stabilizer. Therefore the stabilizers of a code form the stabilizer group \mathcal{S}, an abelian subgroup of . Conversely, any abelian subgroup of that does not contain -I^{\otimes n} is a valid stabilizer group that defines a stabilizer code. The number of logical qubits encoded in a stabilizer code is determined by the size of the codespace, which is in turn determined by the number of physical qubits and the size of the stabilizer group. For a -qubit stabilizer code encoding logical qubits (denoted as an code), the codespace has dimensions, and the stabilizer group \mathcal{S} has elements. Since all non-unit Hermitian elements of have order 2, \mathcal{S} can be generated by independent generators: :\mathcal{S} = \left\langle g_{1},\ldots,g_{n-k}\right\rangle. The generators must be independent in the sense that none of them are a product of any number of other generators, or the negation thereof (otherwise they would generate -I^{\otimes n}). They are analogous to the rows of the parity-check matrix of a classical linear block code. ==Examples==

Classical repetition code As a simple example, the 3-bit classical repetition code can be regarded as a quantum stabilizer code. It encodes logical qubit into physical qubits, and protects against a single bit-flip error (represented as a Pauli operator in a quantum information context). However, since it does not protect against single-qubit phase-flip errors , its code distance as a quantum code is . The stabilizer group of the 3-qubit repetition code has generators: : \begin{array} [c]{ccc} g_{1} & = & Z & Z & I\\ g_{2} & = & I & Z & Z\\ \end{array} The stabilizer indicates that, if the first and second qubits in a valid code state are both measured in the basis, the results will always be the same (i.e., the product of eigenvalues will always be ). Similarly, indicates that the basis measurement on the second and third qubits always yield the same result. Unsurprisingly, the codespace of this code is :\text{Span}(|000\rangle, |111\rangle). Usually, the physical state |000\rangle and |111\rangle are identified with the |0\rangle and |1\rangle states of the logical qubit respectively (often written as |\overline{0}\rangle and |\overline{1}\rangle to distinguish them from physical states). As a quantum code, the codespace also includes superpositions of |\overline{0}\rangle and |\overline{1}\rangle, such as |\overline{+}\rangle = |000\rangle + |111\rangle and |\overline{-}\rangle = |000\rangle - |111\rangle. Note that a phase-flip error on any one physical qubit will change |\overline{+}\rangle to |\overline{-}\rangle, and vice versa. Five-qubit code An example of a stabilizer code is the five qubit stabilizer code. It encodes logical qubit into physical qubits. Its stabilizer group has generators: : \begin{array} [c]{ccccccc} g_{1} & = & X & Z & Z & X & I\\ g_{2} & = & I & X & Z & Z & X\\ g_{3} & = & X & I & X & Z & Z\\ g_{4} & = & Z & X & I & X & Z \end{array} As we will show later, this code protects against an arbitrary single-qubit error, and thus it has code distance . == Logical operators ==

There are many ways to decompose a -dimension codespace into logical qubits, and one way to specify such a decomposition is by giving Pauli and operators for each logical qubit. For stabilizer codes, there exists decompositions where these logical and operators are also elements of . By definition, a logical operator must map a valid code state |\psi\rangle into a valid code state . This means that for each stabilizer , SP|\psi\rangle = P|\psi\rangle = PS|\psi\rangle (the second equality holds since |\psi\rangle is itself a valid code state), which is always true when and commute and never true when they anti-commute. Therefore the set of valid logical operators in is {{tmath|C(\mathcal{S})}}, the centralizer of \mathcal{S} (i.e., the subgroup of elements that commute with all members of {{tmath|\mathcal{S}}}, also known as the commutant). However, not all these logical operators act non-trivially on the logical qubit. In particular, since \mathcal{S} is an abelian subgroup, \mathcal{S} is also contained in {{tmath|C(\mathcal{S})}}. In fact, when {{tmath|S \in \mathcal{S}}}, , meaning that implements the logical identity operator I^{\otimes k}. Furthermore, for any other logical operator , acts identically to on the code state, and thus they implement the same logical operator. Factoring out this equivalence gives the quotient group {{tmath|C(\mathcal{S}) / \mathcal{S}}}, which is isomorphic to . Therefore all -qubit logical Pauli operators can be chosen to be -qubit physical Pauli operators. To explicitly specify the logical qubit decomposition, one usually chooses logical operators . Each of these operators or is a representative for the equivalence class \overline{Z_i}\mathcal{S} or \overline{X_i}\mathcal{S} implementing the same logical operator. These operators need to satisfy the following conditions: • are all independent: The product of any nonempty subset of these operators cannot be a scalar multiple of I^{\otimes n}. • Among , the only pairs that anti-commute are and for the same . Other pairs — two logical operators on different logical qubits, one logical operator and one stabilizer generator, or two stabilizer generators — all commute. Examples For the 3-qubit repetition code described above, the stabilizer generators (repeated for convenience) and logical operator representatives can be chosen as: : \begin{array} [c]{ccc} g_{1} & = & Z & Z & I\\ g_{2} & = & I & Z & Z\\ \overline{Z} & = & Z & I & I\\ \overline{X} & = & X & X & X \end{array} Other single-qubit operators are alternative implementations of the logical operator: , . This is consistent with the above observation that any single-qubit phase-flip error changes |\overline{+}\rangle to |\overline{-}\rangle, and vice versa. It is also straightforward to check that applying , i.e., bit-flipping all three qubits, changes |\overline{0}\rangle to |\overline{1}\rangle, and vice versa. For the five-qubit code, the logical operator representatives are usually chosen as: : \begin{array} [c]{ccccccc} g_{1} & = & X & Z & Z & X & I\\ g_{2} & = & I & X & Z & Z & X\\ g_{3} & = & X & I & X & Z & Z\\ g_{4} & = & Z & X & I & X & Z\\ \overline{Z} & = & Z & Z & Z & Z & Z\\ \overline{X} & = & X & X & X & X & X \end{array} Note, however, that these are not the logical operator representative candidates with the lowest weight (number of non- Pauli factors). For example, has weight 3. == 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 \Pi^{n}. Suppose that the errors affecting an encoded quantum state are a subset \mathcal{E} of the Pauli group \Pi^{n}: :\mathcal{E}\subset\Pi^{n}. Because \mathcal{E} and \mathcal{S} are both subsets of \Pi^{n}, an error E\in\mathcal{E} either commutes or anti-commutes with any particular element S \in \mathcal{S}. If anti-commutes with an element , then SE|\psi\rangle = -ES|\psi\rangle = -E|\psi\rangle which means that E|\psi\rangle is in the -eigenspace of rather than the -eigenspace, and thus is detectable by measuring . Actually it suffices to measure each stabilizer generator , since if commutes with every , then will also commute with the product of any number of them. In this case E \in C(\mathcal{S}) is a logical operator and thus cannot be detected by the code. However, E \in \mathcal{S} is again a special case, where implements the logical identity operator: Even though it is not detectable, it also does not corrupt the encoded state. This also holds for any scalar multiple of E \in \mathcal{S}, since a global phase has no physical effect. We define an undetectable logical error as one that is undetectable but does corrupt the encoded state, i.e., :E \in C(\mathcal{S}) \setminus \{+1, +i, -1, -i\} \otimes \mathcal{S} = C(\mathcal{S}) \setminus C(C(\mathcal{S})). The equality above gives an alternative characterization of an undetectable logical error : must commute with all stabilizers, but not with all logical operators. This characterization is often more convenient since one only need to check commutativity with the generators , instead of solving a system of linear equations to determine if is the product of some subset of {{math|{gi}}} up to global phase. Operationally, each stabilizer generator can be measured via a parity measurement without disturbing states in the codespace. The combination of results of measuring each is known as the syndrome \mathbf{r}, represented as a binary vector \mathbf{r} with length n-k whose elements indicate whether the error commutes or anti-commutes with each stabilizer generator . Knill–Laflamme conditions When using a stabilizer code as an error correction code, one must also choose a correction for each syndrome. If there exists another possible error with the same syndrome as , then after correction there may be a residual error . The condition that has the same syndrome as is equivalent to that is undetectable, i.e., E_1^\dagger E_2 \in C(\mathcal{S}). However, if does not corrupt the logical qubits, then the error correction will be successful anyway. Therefore, a stabilizer code can perfectly correct a set of Pauli errors \mathcal{E} as long as there does not exist E_1, E_2 \in \mathcal{E} such that is an undetectable logical error. Examples The 3-qubit repetition code can correct single-qubit bit-flip errors, which means that it satisfies the error-correction conditions for \mathcal{E} = \{I, X_1, X_2, X_3\}. Indeed, the only undetectable logical error that consists only of and is with weight 3, and the product of two errors in \mathcal{E} . This can also be verified by explicitly checking the corrections corresponding to each syndrome: The five-qubit code can correct any single-qubit error, i.e., it satisfies the error-correction conditions for \mathcal{E} = \{I, X_i, Y_i, Z_i\} ( distinct errors). This can be verified either by showing that all undetectable logical errors of this code has weight at least 3, or by explicitly checking the syndromes. For the five-qubit code again each syndrome corresponds to one error in \mathcal{E}, although this is not typical for stabilizer codes: For codes like the surface code with high code distances and relatively low-weight stabilizers, one syndrome will usually correspond to many correctable errors that differ from each other by stabilizers. == Relation between Pauli group and binary vectors ==

The Pauli group has a binary vector representation based on the following mapping: : This mapping maps to vectors in {{tmath|(\mathbb{Z}_{2})^{2n} }}, such that multiplication of Pauli operators is equivalent to addition of binary vectors up to a global phase. Furthermore, {{tmath|(\mathbb{Z}_{2})^{2n} }} can be equipped with a symplectic algebra, such that the symplectic product of two binary vectors indicate whether the corresponding Pauli operators commute. The above binary representation and symplectic algebra are especially useful in making the relation between classical linear error correction and quantum stabilizer codes more explicit. In the language of symplectic vector spaces, 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. == Notes ==