bibkey: eidesen2025projectiveerror
authors: Jonas Eidesen
year: 2025
title: "Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes"
doi: 10.48550/arXiv.2506.01843
url: https://arxiv.org/abs/2506.01843v3
claim: "The paper models quantum errors by projectively faithful irreducible projective representations of finite groups, compares stabilizer codes, Clifford codes and weak stabilizer codes, characterizes weak stabilizer Clifford codes for nice error bases, and asks (Question 11.3) whether a code with |G| = |L(W)| |S(W)| and a non-normal stabilizer group S(W) exists when |G| = (dim V)^2."
strata_touched:
  - D5/S3/Quantum/Information/NiceErrorBasisNonNormalStabilizer
license: citation-only
triage: anchor

Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes

Eidesen works with finite groups and unitary representations on finite-dimensional complex Hilbert spaces. A projective representation of G on V is a map π : G → U(V) whose composite with the quotient q : U(V) → PU(V) is a homomorphism, equivalently π(x)π(y) = σ(x, y)π(xy) with σ in the circle group; it is projectively faithful if q ∘ π is injective. A projective error model on V is a pair (G, π) with π a projectively faithful irreducible projective representation (Definition 3.2). For a subspace W with orthogonal projection P_W, the paper defines the logical operators and the stabilizers

L_{(G,π)}(W) := { x ∈ G : P_W π(x) = π(x) P_W }, S_{(G,π)}(W) := { x ∈ G : P_W π(x) P_W ∈ 𝕋 P_W }.

Proposition 8.1 gives, for every n ≥ 2, the projective error model G = C₂ × D_{2n} = ⟨a, b, c | a^{2n} = b² = c² = [a, c] = [b, c] = 1, bab = a⁻¹⟩ on ℂ⁴ with π(cᵏ bˡ aᵐ) the product of the k-th power of the block swap, the l-th power of X ⊕ X and the m-th power of P ⊕ (−P), where X is the Pauli matrix and P = diag(1, ζ_{2n}); for n = 2 the dihedral factor has order 8 and P = diag(1, i). Section 11 records that the author has been unable to produce a code that is both a Clifford code and a weak stabilizer code but not a stabilizer code, and restates this for |G| = (dim V)^2 as Question 11.3:

Does there exist a Hilbert space V, a subspace W ⊂ V, and a projective error model (G, π) ∈ PEM_V with |G| = (dim V)^2, such that the equation |G| = |L_{(G,π)}(W)| · |S_{(G,π)}(W)| holds, but S_{(G,π)}(W) is not normal in G?

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2506.01843
• URL: https://arxiv.org/abs/2506.01843v3
• Version and location: arXiv:2506.01843v3 (v1 2025-06-02, v3 2026-02-25), source file Projective_error_models.tex: §2 for projective representations and projective faithfulness; §3 for Definition 3.2 (projective error model); §6 for L_{(G,π)}(W) and S_{(G,π)}(W); §8 for Proposition 8.1; §11 for Question 11.3 (the question environment labelled Q3, numbered with the definition counter within sections).