A stronger characterization

Consider an XS-stabilizer state |ψi. We have shown that|ψi has the form given in theorem 2.9.6.

However, not all covariant phasesf ∈ F are valid amplitudes. Here we characterize precisely the subclass of valid amplitudes.

First we note that the groupPn^S is closed under conjugation byX,√

S and CZ, hence we can make two assumptions about the state:

1. We can assume ~µ = 0 in theorem 2.9.6, since we can apply X to the corresponding qubits and update the stabilizers by conjugation.

2.9. CONSTRUCTING A BASIS OF A REGULAR XS-STABILIZER CODE 35 2. We can restrict ourselves to studying covariant phases of the form

f(x) = i^Pj<kζjkxjxk(−1)^Pj<k<lζjklxjxkxl (2.18) whereζ_jk andζ_jkltake values in Z2. Indeed, iff is a valid amplitude, then so is

f(x)α^l(x)i^l0(x)(−1)^q(x)

for all linear polynomialsl,l⁰ and quadratic polynomialsq. After all, we can always generate these additional phases by applying suitable combinations of the gates √

S and CZ to the state|ψi.

We will treat this class off as a vector spaceV1 overZ2, with a natural basis given by the functions i^xjxk and (−1)^xjxkxl. Similarly, we consider the set of all functions of the form

(−1)^Pj<kηjkxjxk

which can also be viewed as a vector space V₂ overZ2 with basis functions (−1)^xjxk. We define a set of linear mappings {Fh |h= 1, . . . , t}from V1 toV2 by the rules

F_h(−1)^xjxkxl =













(−1)^xkxl ifh=j, (−1)^xjxl ifh=k, (−1)^xjxk ifh=l,

1 otherwise,

(2.19)

and

F_hi^xjxk =

((−1)^xjxk ifh=j orh=k,

1 otherwise. (2.20)

Recall the matrixW that appears in equation (2.12). Let w~j = (wj1, . . . , wjt) denote thej-th row ofW, for everyj= 1, . . . , n−t. We define the quadratic functionsγ_j ∈V₂ by

γj(x) = (−1)^{Pk<lwjkwjlxkxl}. (2.21) This definition stems from the fact that when we apply the S gate on the single-qubit standard basis state described by |w_j1x₁⊕ · · · ⊕w_jtx_ti, we obtain the phase

i^{wj1x1⊕···⊕wjtxt} = i^Pkwjkxkγj(x), (2.22) where we have used lemma 2.9.1. Then we set

Γ = span{γj}. (2.23)

We will prove the following theorem

Theorem 2.9.7. Consider any state of the form

|ψi= 1

√2^t X

x∈Z^t2

f(x)|x, W xi

with f ∈V₁. If f satisfies

Fh(f)∈Γ for all 1≤h≤t (2.24)

then |ψi is an XS-stabilizer state.

Proof. Assuming that f satisfies condition (2.24), we will show how to construct a set of XS-stabilizers that uniquely stabilize|ψi. Consider XS-operators

gj =XjX(~aj)S(~bj), 1≤j≤t,

where Xj denotes the Pauli matrix X acting on the j-th qubit, where the X(~aj) are X-type operators that only act on qubits t+ 1 to nwith~a_j thej-th column of the matrixW. The strings

~bj are at the moment unspecified. Furthermore, define

gj =Z(~cj) t+ 1≤j≤n

where {~ct+1, . . . , ~cn} ⊆Zⁿ2 form a basis of the orthogonal complement of the subspace V ={(x, W x)|x∈Z^t2} ⊆Zⁿ2.

For every y, z ∈Zⁿ₂ we have

Z(z)|yi= (−1)^zTy|yi.

This implies that gj|ψi=|ψi for everyj =t+ 1, . . . , n. Next we show that, for a suitable choice of ~bj, the operator gj stabilizes |ψi for every j = 1, . . . , t. This last condition is equivalent to X_jX(~a_j)|ψi=S(~bj)|ψi. Note that

XjX(~aj)|ψi=X

x

f(x)|x+ej, W(x+ej)i=X

x

f(x+ej)|x, W xi

since~aj is thej-th column ofW and thus~aj =W ej. Thus, the conditiongj|ψi=|ψi is equivalent to

f(x+e_j)|x, W xi=S(~bj)f(x)|x, W xi.

By using the fact that f has the form (2.24) and by applying the definition of Fj, it is easy to check that we have

f(x+ej)

f(x) = i^lj(x)(Fj◦f)(x).

for some linear function lj. Summarizing so far, we find that gj|ψi=|ψi if and only if

S(~bj)|x, W xi= i^lj(x)F_j(f)|x, W xi. (2.25) Since, by assumption, we have

Fj(f)∈Γ, we can find a vector~b⁰_j ∈Z^n−t₂ , such that

Y

1≤k≤n−t

γ^b

0 jk

k =Fj(f).

This in turn means that (2.25) is equivalent to S(~bj)|x, W xi= i^lj(x) Y

1≤k≤n−t

γ_k(x)^b0jk|x, W xi. (2.26) We now claim that a string~bj satisfying this condition exists. To see this, first recall (2.22) and the surrounding discussion, which implies that

S(~b⁰_j)|W xi= i^l0j(x) Y

1≤k≤n−t

γ_k(x)^b0jk|W xi

for some linear function l_j⁰. Second, there exists a string b⁰⁰_j ∈ {0,1,2,3}^t such that S(~b⁰⁰_j)|xi = i^{lj(x)−l0j(x)}|xi. This shows that

S(~bj) :=S(~b⁰⁰_j)⊗S(~b⁰_j)

2.9. CONSTRUCTING A BASIS OF A REGULAR XS-STABILIZER CODE 37 satisfies the condition (2.26). We have shown that the operators g_j (j= 1, . . . , n) stabilize|ψi.

Finally we show that|ψiis uniquely stabilized by these operators. LetGbe the group generated by {g₁, . . . , g_n}. Since the~c_k form a basis of V^⊥, any string v ∈ Zⁿ₂ belongs to V if and only if

~z_j^Tv = 0. Thus we have

v∈V ⇔ g_j|vi=|vi for every j=t+ 1, . . . , n. (2.27) Let G_D be the diagonal subgroup of G. Let U be the set of all u∈ Zⁿ2 satisfying D|ui =|ui for all D∈G_D. According to the argument above lemma 2.9.4,U is the disjoint union of cosets ofV; the number of such cosets is the dimension of the code stabilized by G. We show that in fact U =V, implying that this dimension is one, so that|ψi is the unique stabilized state. Since each g_j (j =t+ 1, . . . , n) belongs to G_D, every u ∈U must satisfyg_j|ui=|ui for all j=t+ 1, . . . , n.

With (2.27) this shows that U ⊆V. Furthermore, sinceD|ψi =|ψi for every D∈ GD and since

|ψi has the form

|ψi=X

v∈V

αv|vi

for some coefficientsαv, it follows from that everyv∈V satisfiesD|vi=|vifor all D∈G⁰_D. This shows that V ⊆U and thus V =U.

Remark 2.9.8. SinceFh are linear transformations from V1 toV2, and Γis a linear subspace, the condition F_h(f)∈Γ can be written as linear equations.

Note that there will be multiple ways to write down the same XS-stabilizer state. Let us consider the following example with five qubits:

|ψi=X

x

f(x)|x1, x2, x3, x1⊕x2, x2⊕x3i.

Equally, we can set (x⁰₁, x⁰₂, x⁰₃) = (x1, x1⊕x2, x2⊕x3), and the state becomes

|ψi=X

x⁰

f⁰(x⁰)|x⁰₁, x⁰₁⊕x⁰₂, x⁰₁⊕x⁰₂⊕x⁰₃, x⁰₂, x⁰₃i.

We can define Γ⁰ in the same way as we defined Γ. We will show that if F_xjf(x) ∈ Γ, then F_x⁰

jf⁰(x⁰)∈Γ⁰.

More formally, we consider the state |ψi =P

xf(x)|x, W xi. For an invertible matrix R over Z2 andx⁰ =R⁻¹x, we have

|ψi=X

x

f(x)|RR⁻¹x, W RR⁻¹xi

=X

x⁰

f(Rx⁰)|Rx⁰, W Rx⁰i

≡X

x⁰

f⁰(x⁰)|W⁰x⁰i.

In the above equation, we can change the summation from over x to x⁰ because R is invertible.

Then we define Γ⁰ from W⁰ in the same way as (2.21) and (2.23). We have the following theorem Theorem 2.9.9. Let |ψi be an XS-stabilizer state in the form given in theorem 2.9.6. Then f satisfies the condition (2.24). What is more, for any invertible matrix R over Z2, the function f⁰ defined by f⁰(x⁰) :=f(Rx⁰) also satisfies

F_x⁰

jf⁰ ∈Γ⁰.

Proof. First, we note that we have slightly abused the notation here, since f⁰ is in general not a function in V1 (which is defined in section 2.9.4). However, we can simply ignore the terms

α^l(x0)(−1)^q(x0) in f⁰, again by the reasoning in section 2.9.4. After the transformationx⁰ =R⁻¹x,

where~e1is the first canonical basis vector. By the same reasoning we used in the proof of theorem 2.9.7, we know that

To illustrate how theorem 2.9.7 works, we give two examples here, which demonstrate extreme cases. First consider the state

|ψi=X

x

f(x)|x1, x2, . . . , xni.

By definition, Γ is a trivial vector space. It is then straightforward to check thatf(x) has to be of the form f(x) =i^l(x)(−1)^q(x), and thus|ψi is a Pauli stabilizer state. On the other hand, consider

It is easy to check Γ is the full vector space V₂. Thus the condition (2.24) becomes trivial, which meansf(x) can be an arbitrary function inF.

2.10 Entanglement

Im Dokument Quantum memory: design and optimization  (Seite 34-38)