The stabilizer formalism

In this section we present what is now known as the stabilizer formalism, developed by Gottesman, which has become the primary language for discussing quantum error correct- ing codes. Many important concepts are summarized here, for a more detailed description of the founding ideas see Ref. [58].

Definition 16. Given a subspace _{Q ⊆ H, a stabilizer S is a Pauli operator such that:}

S_{|ψi = |ψi, ∀ |ψi ∈ Q.} (1.76)

Lemma 17. Given a subspace _{Q ⊆ H, the set of stabilizers form a multiplicative Abelian} group _S.

Proof. The identity element of the group is clear as IH∈ S, where IHis the identity element

on the Hilbert spaceH. Since any Pauli operator has the property P2 _{= I}

H, any stabilizer S

is its own inverse. Let _{|ψi ∈ Q, given two stabilizers S}1 and S2, S1S2|ψi = S1|ψi = |ψi,

and as such (S1S2) is a stabilizer. Finally, given now a third stabilizer S3, S1(S2S3)|ψi =

|ψi = (S1S2)S3|ψi, showing associativity. Therefore the set of stabilizers are a group,

what remains to be shown is that the group is Abelian. Since by definition stabilizers are Pauli operators, we can use the following fact about Pauli operators: S1S2 =±S2S1.

However, we cannot have that S1S2 =−S2S1 else we would conclude that|ψi = S1S2|ψi =

−S2S1|ψi = −|ψi which is a contradiction.

We have demonstrated that the stabilizers of a given subspace Q form an Abelian subgroup _{S ⊂ P}n of the n-qubit Pauli group (assuming dim(H) = 2n), however the

converse is also true, any Abelian subgroup _{S ⊂ P}n such that −I, iI /∈ S will form a

subspace Q such that dim(Q) = dim(H) − dim(S).

Definition 18. An-qubit stabilizer group is an Abelian subgroup_{S ∈ P}nsuch that−I, iI /∈

Definition 19. A stabilizer group _{S ∈ P}n and recovery operator RS define a stabilizer

code (QS,RS), where QS ={|ψi ∈ H | S|ψi = |ψi ∀ S ∈ S}. Moreover, given dim(S) =

2n−k_{, dim(Q}

S) = 2k, that is the error correcting code is composed of k logical qubits.

The stabilizer group by definition leaves the codespace invariant and is logically equiv- alent to the identity operator. Given a codespace_QS encoding k logical qubits, there must

be k logical Pauli X and Z operators. By definition, a logical Pauli operator PL must map

any element of QS to another element of QS, that is for all |ψi ∈ QS, PL|ψi = |ψ0i ∈ QS,

where _|ψ0

i does not have to necessarily be different from |ψi. As such, we must have the following for all|ψi ∈ QS and S ∈ S, PLS|ψi = PL|ψi = |ψ0i = S|ψ0i = SPL|ψi ⇒ PLS =

SPL. Therefore, the set of logical Pauli operators are the Pauli operators that commute

with the stabilizer group _{S, that is the centralizer of S. However, in the case of Pauli} operators, the centralizer ofS is equivalent to the normalizer of S and typically we define the logical Pauli operators as the elements from this group.

Definition 20. Given a stabilizer code composed from the stabilizer group _{S, the non-} trivial (non-identity) logical Pauli operators of the code are given by the elements of the normalizer of _{S that are not in S, that is N (S)\S, where}

N (S) = {P ∈ Pn | P SP† = S0 ∈ S}. (1.77)

The normalizer gives a very nice characterization of the form of logical operators that a stabilizer code can have. Moreover, it is very indicative of how well a code protects against noise and defines a very important parameter for stabilizer codes.

Definition 21. The distance of a stabilizer group _{S is defined as:}

d = min_{{wt(P )|P ∈ P}n, P ∈ N (S)\S}, (1.78)

where wt(P ) is the weight of the Pauli operator P = ⊗n

i=1Pi, that is the total number of

qubits where Pi 6= I.

Theorem 22. Given an qubit stabilizer group_{S, encoding k logical qubits with distance d,} there always exists a recovery operator _RS such that the associated stabilizer code can

correct any error of weight less than or equal to _{b(d − 1)/2c. The associated stabilizer code} is denoted a [[n, k, d]] stabilizer code.

Proof. Let E = {Ei} be a set of errors of weight less than or equal to b(d − 1)/2c. Then

we must have that hψk|E

†

since else E_i†Ej would have overlap with a logical operator, yet we know that E †

iEj has

weight at most (d− 1), thus this would contradict the definition of the normalizer. There- fore, for any set of errors_{E satisfying the statement of the Theorem, the codespace satisfies} the Knill-Laflamme Theorem14and the decoder _RS exists.

Theorem 23. Given a [[n, k, d]] stabilizer code and let Hd−1 be a subset of at most (d−

1) qubits. Then there always exists a recovery operator _RS that corrects for any set of

errors_{E = {E}i} ∈ Hd−1.

Proof. Since for any Eiand Ej, the support of both is contained inHd−1, as such wt(Ei†Ej) ≤ d−

1 and the Knill-Laflamme conditions must trivially hold.

Therefore, the distance of a quantum error correcting code provides two important notions for correctable errors. Firstly, if the error set is of weight less than half the size of the distance then such errors must be correctable. Secondly, if the location of the errors are known, the quantum error correcting code will be able to correct for up to (d− 1) errors.