Entanglement-assisted stabilizer formalism

[Fe-r-oz]

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Brun et. al makes the following remark: "The entanglement-assisted quantum codes don't require the dual-containing constraint necessary for standard quantum error-correcting codes, enabling us to effectively “quantize” all of classical linear coding theory". In addition, they present an answer to this interesting question: What if we are given a non-commuting set of operators? Can we still construct a QECC?

This approach allows 'modern' classical linear codes, even those that don't fit the traditional quantum constraints (self-orthogonal constraint, etc.), to be transformed into valid quantum codes using shared entanglement. Thus, providing integration of modern classical coding techniques into quantum error correction, expanding the range of quantum code design.

Summary: If classical codes are not dual-containing and their stabilizer generators do not commute (S is non-abelian), shared entanglement can be used to incorporate these generators into a larger set of commuting generators, thus defining well-defined quantum code space. Decompose the non-abelian S subgroup into two subgroups, isotropic subgroup Sᵢ and entanglement-assisted/symplectic Sₑ subgroup. Thus, S = ⟨Sₑ, Sᵢ⟩ . Then follow the additional procedure outlined in the paper (page 6 onwards).

Describe the solution you’d like

Implement Lemma 1 and Lemma 2 from this paper. Alternative construction method is given in the next section. These two methods can serve as a correctness check against one another. These lemmas present group theoretic tools for dealing with non-abelian groups. Instead of a commuting set of operators from Pauli group, we are presented with a non-commuting set of operators that generates a non-abelian group S.

Short Summary: Decompose non-abelian S into isotropic and symplectic (entanglement-assisted) subgroups, relate S to a simpler group B, and analyze error-correcting conditions. The code space for B can be determined using an extended abelian group and relates to the error-correcting properties of S.

Suppose you are given non-abelian S generators:

   M₁ = ZXZI
   M₂ = ZZIZ
   M₃ = XYXI
   M₄ = XXIX

1. Find a minimal_generating_set of generators with the following commutation relations:

   • [Zᵢ, Zⱼ] = 0 ∀ i, j
     [Xᵢ, Xⱼ] = 0 ∀ i, j
     [Xᵢ, Zⱼ] = 0 ∀ i ≠ j
     {Xᵢ, Zᵢ} = 0 ∀ i

2. Decompose S into isotropic and symplectic subgroups:
   • Isotropic Subgroup Sᵢ: generated by 'commuting' generators.
   • Symplectic/entanglement-assisted subgroup Sₑ: generated by 'anti-commuting' generator pairs.
   • Suppose the following is different set of generators for S that follows the commutation relations mentioned above:

   • 
      Z₁ = ZXZI
      X₁ = ZZIZ
      Z₂ = YXXZ
      Z₃ = ZYYX

   • Thus, S = ⟨ Sᵢ, Sₑ⟩ with Sᵢ = ⟨Z₂, Z₃⟩ and Sₑ = ⟨Z₁, X₁⟩.

Suppose B is generated by

 Z₁ = ZIII
 X₁ = XIII
 Z₂ = IZII
 Z₃ = IIZI

3. Relate Group S to Group B: From previous results, B = ⟨Bᵢ, Bₑ⟩ where Bᵢ = ⟨Z₂, Z₃⟩ & Bₑ = ⟨B₁, B₁⟩. Thus, Groups B and S are isomorphic (B ≈ S). (Lemma 1)

4. Unitary Transformation (Lemma 2): If B ≈ S, then there exists a unitary U such that for all B in $B$, there exists an S in $S$ where B = USU⁻¹ up to an overall phase (Brun et. al). Thus, the error-correcting power of C(B) and C(S) are related by this unitary transformation.

5. Code Space C(B):

   • B is not a commuting group, so usual code space definitions do not apply directly.
   • "Extend" B to an abelian group:

     Z₁' = ZIII|Z
     X₁' = XIII|X
     Z₂' = IZII|I
     Z₃' = IIZI|I

   • Suppose extended group Bₑ is generated by { Z₁', X₁', Z₂', Z₃' }.
   • Code space C(B) is the +1eigenspace of all elements of Be: C(B) = { |Φᴬᴮ⟩ |0⟩|0⟩|ψ⟩ } where |Φᴬᴮ⟩ is a maximally entangled state shared between Alice and Bob and |ψ⟩ is an arbitrary single-qubit pure state. This represents an entanglement-assisted quantum error-correcting code (EAQECC) with notation [[n, k; c]] indicating k qubits encoded into n qubits with c ebits (Brun et. al). Note: c ebits == number of anti-commuting pairs of generators in Bₑ, s == number of independent generators Bᵢ

Surprisingly, Wikipedia seems to have a good writeup on Entanglement-assisted stabilizer formalism as well. It presents lemma 1 and its commutation relations nicely. ECC Zoo: EA qubit stabilizer code   Describe alternatives you’ve considered

Symplectic Gram-Schmidt orthogonalization procedure (SGSOP) is an alternative and 'natural' way presented in this paper for finding the logical operators of the EAQECs.  Applying the SGSOP to the generators in non-abelian ⟨S⟩ partitions them into two distinct subgroups: the entanglement generators, denoted as ⟨Sₑ⟩ , and the isotropic generators, denoted as ⟨Sᵢ⟩ .

This paper highlights that the SGSOP is useful for determining logical operators when Gottesman's method (converting to standard form) cannot be applied to non-abelian Pauli groups. I think SGSOP is implemented as symplecticGS, in enumeration.jl and there are many applications (low-hanging fruits) of SGSOP presented in this paper, including its use in constructing EAQECs as well as finding logical operators for these codes.   Additional context

To be added soon.

[Krastanov]

What does it mean to "implement lemma 1 and lemma 2"?

I would suggest trying to keep this type of suggestion an order of magnitude shorter to make it easier for other readers to understand what it is about. Maybe a short example of desired capability. This currently reads more as a discussion (which is not a problem at all, but it should probably be posted as a discussion, not as a feature request in that case).

This also seems related to the work on more general error correcting codes that @IsaacP1234 and @KDGoodenough are currently undertaking. See for instance #293

[Fe-r-oz]

My bad. I will improve the solution section as per your instructions.

Thanks! Indeed, the group theoretic tools such as finding minima generating set, etc. that are introduced by Kenneth and Issac come up not only in the subsystem setting, in other places as well. I think they have developed them to work on subsystem stabilizer CSS codes namely detailed in this paper: Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat’s Lemma.

P.S. Lemma 1 has to do with decomposing 'non-abelian' group S into minimal generating set according to the commutations rules described in the paper.