Asymmetric Quantum Codes on Toric Surfaces

Asymmetric quantum error-correcting codes are quantum codes defined over biased quantum channels: qubit-flip and phase-shift errors may have equal or different probabilities. The code construction is the Calderbank-Shor-Steane construction based on two linear codes. We present families of toric surfaces, toric codes and associated asymmetric quantum error-correcting codes.

💡 Research Summary

The paper investigates the construction of asymmetric quantum error‑correcting codes (AQECC) by exploiting toric surfaces and the associated toric evaluation codes. The motivation stems from quantum channels where bit‑flip (X) and phase‑flip (Z) errors occur with different probabilities; such channels require codes that can correct different numbers of X‑ and Z‑errors. The authors adopt the Calderbank‑Shor‑Steane (CSS) framework, which builds a quantum code from two classical linear codes C₁ and C₂ satisfying the dual‑containment conditions C₁⊥ ⊆ C₂ and C₂⊥ ⊆ C₁.

The geometric foundation is a two‑dimensional lattice M ≅ ℤ² and its real extension Mℝ. For a fixed prime power q and an integer r dividing q, they define a family of integral convex polytopes Δ_b ⊂ Mℝ with vertices (0,0), (a,0), (b,q‑2), (0,q‑2), where a = b + q – 2r and 0 ≤ b ≤ q‑2. Each Δ_b lies inside the square