LU-inequivalence of XS- and Pauli stabilizer states

Here we show that the XS-stabilizer state (2.2) is not LU-equivalent to any Pauli stabilizer state, i.e. there does not exist any Pauli stabilizer state|φisatisfying|ψi=U|φi for anyU :=U1⊗ · · · ⊗ U6 ∈ U(2)^⊗6. This result demonstrates that there exist XS-stabilizer states whose multipartite entanglement (w.r.t LU-equivalence) is genuinely different from that of any Pauli stabilizer state.

In order to prove the claim, we consider the classification of 6-qubit stabilizer (graph) states under LU-equivalence as given in Fig. 4 and Table II of Ref. [HEB04]. In the latter figure, 11 distinct LU-equivalence classes are shown to exist for fully entangled 6-qubit stabilizer states; the classes are labeled from 9 to 19. A representative of each class is given in Fig 4. Furthermore each class is uniquely characterized by its list of Schmidt ranks, i.e. the Schmidt ranks for all possible bipartitions of the system. In table II it is shown that, for any of the classes 9–17, there is at least one bipartition of the form (two qubits – rest) for which the state is not maximally entangled. This shows that the XS-stabilizer state |ψi cannot be LU-equivalent to any of the states in the classes 9–17, since |ψi is maximally entangled for all such bipartitions. Furthermore, for class 19, the entanglement is maximal for all bipartitions of the form (3 qubits – rest); since this is not the case for|ψi, the latter cannot be LU-equivalent to any state in class 19. This leaves class 18. Consider the state

|φi= X1 xj=0

|x₁, x₂, x₃, x₁⊕x₂, x₂⊕x₃, x₃⊕x₁i

2.11. EFFICIENT ALGORITHMS 41 which is a Pauli stabilizer state. By direct computation of all Schmidt ranks, one verifies that this state belongs to class 18. We prove by contradiction that |ψi is not LU-equivalent to|φi. First, it is straightforward to show that for both |φi and |ψi, the 3-qubit reduced density matrix of the 1st, 2nd and 4th qubits is

ρ₁₂₄= 1

4(|000ih000|+|011ih011|+|101ih101|+|110ih110|)

= 1

8(I+Z1⊗Z2⊗Z4).

If there is a local unitary transformation U from |φi to |ψi, then U1 ⊗U2⊗U4 must leave ρ124

unchanged. This implies thatU₁⊗U₂⊗U₄must leaveZ₁⊗Z₂⊗Z₄unchanged, so thatU₁Z₁U₁^†∝Z₁ and similarly forU2 and U4. This implies that U1, U2 and U4 must have the form Dj or DjX for some diagonal matrix D_j. Analogously, we can show that the same holds for all other j. Thus U = D X(~a) for some diagonal operator D := D₁ ⊗ · · · ⊗D₆ and some ~a ∈ Z⁶₂. Note that |ψi and |φi have the form

|φi=X

v∈V

|vi and |ψi=X

v∈V

αv|vi, for some linear subspace V ⊆Z⁶₂ and real coefficientsαv. Then

D X(~a)|φi=X

v∈V

β_v|v+~ai

for some coefficients β_v. Thus U|φi =|ψi implies that V =V +~a. This shows that~a∈V. But then X(~a)|φi=|φi. The identity U|φi =|ψi thus implies that D|φi=|ψi. It is straightforward to verify that this cannot be true. We have thus shown that|ψiand |φiare not LU-equivalent. In conclusion, |ψi does not belong to any LU-equivalence class of Pauli stabilizer states.

2.11 Efficient algorithms

In this section we will give a list of problems that can be solved with efficient classical algorithms for regular XS-stabilizer states (codes). We consider an arbitrary n-qubit regular XS-stabilizer stabilizer codeHGspecified in terms of a generating set ofmstabilizers in the standard form given in Corollary 2.8.5. Then the following holds:

1. The degeneracy dof the code can be computed in poly(n, m) time (recall section 2.9.2).

2. An efficient algorithm exists todetermine d basis states|ψ₁i, ,|ψ_di, each of which is an XS-stabilizer state with regular stabilizer group and each state having the form

|ψ_ii=X

x∈Z^t₂

f_i(x)|x, W x+~µ_ii with f_i∈ F.

The matrixW (which is the same for all|ψii) can be computed in poly(n, m) time. The list {~µ1, , ~µ_d}can be computed in poly(n, m, d) time. Given a specific~µi, a complete generating set of stabilizer operators having|ψ_iias unique stabilized state can be computed in poly(n, m) time. Furthermore, given ~µi, the functionx7→fi(x) can be computed in poly(n, m) time as well. See section 2.9.2.

3. The logical operatorsof HG can be computed in poly(n, m, d) time. See section 2.9.3.

4. The commuting Hamiltonian described in section 2.5 can be computed in poly(n, m) time.

On input of~µi the following holds in addition:

5. The von Neumannentanglement entropyof any|ψ_ii with regular stabilizer group can be computed, for any bipartition, in poly(n, m) time. This claim holds since we have shown in section 2.10 how to efficiently compute the description of a Pauli stabilizer state with the same entanglement as|ψ_ii; furthermore an efficient algorithm to compute the von Neumann entanglement entropy of Pauli stabilizer states is known [FCY⁺04].

6. A poly(n) size quantum circuit to generate any |ψii can be computed for in poly(n, m) time. This circuit can always be chosen to be a Clifford circuit followed by a circuit composed of the diagonal gates CCZ (controlled-CZ), CS := diag(1,1,1,i) (controlled-S) and T. To see this, we recall theorem 2.9.6. This implies that the state |ψiican be prepared as follows:

• Using a Clifford circuit C1, prepare the state P

|x, W x+~µ_ii. In fact, this can be done using a circuit composed of Hadamard,X and CNOT gates.

• Since the functionfi belongs to the classF, it has the form α^l(x)i^q(x)(−1)^c(x).

Note that the we have the following gate actions on the standard basis:

T:|xi 7→α^x|xi, S:|xi 7→i^x|xi, CS:|x, yi 7→i^xy|xi, CZ:|x, yi 7→(−1)^xy|x, yi, CCZ:|x, y, zi 7→(−1)^xyz|x, y, zi.

Therefore, the phase fi(x) can be generated by first applying a suitable circuit C2 of Clifford gates CZ and S to generate the quadratic part of c(x) and the linear part of q(x), and by subsequently applying a suitable circuitU composed of the (non-Clifford) gates T, CS and CCZ to generate l(x), the quadratic part of q(x) and the cubic part of c(x), respectively. Since the functionfi can be computed efficiently, the descriptions of C2 and U can be computed efficiently. The overall circuit is UC2C1.

7. Given any |ψii and Pauli operatorP, we cancompute the expectation value hψi|P|ψii in poly(n, m) time. This implies in particular that the expectation of any local observable (i.e. an observable acting on a subset of qubits of constant size) can be computed efficiently as well, since every such observable can be written as a sum of poly(n) Pauli observables.

To see thathψi|P|ψii can be computed efficiently, recall from point 6 above that|ψiican be decomposed as |ψ_ii = U|ψ⁰_ii where U is a circuit composed of T, CS and CCZ, and where

|ψ_i⁰i=C2C1|0i is a Pauli stabilizer state. Then

hψi|P|ψii=hψ⁰_i|U^†PU|ψ_i⁰i.

Its is easily verified thatU^†PU =:C⁰⁰ is a Clifford operation, for every circuitU composed of T, CS and CCZ (for example T XT ∝S). Thus we have

hψi|P|ψii=hψ_i⁰|C⁰⁰|ψ_i⁰i. Recall that |ψ_i⁰i=C2C1|ψii, we know that

hψi|P|ψii=h0|C⁰⁰⁰|0i,

where C⁰⁰⁰ = C1^†C2^†CC2C1. Note that h0|C⁰⁰⁰|0i is simply the coefficient of the basis |0i in the Pauli stabilizer state C⁰⁰⁰|0i, which can be computed efficiently according to [VDN10].