Coherent-Error Threshold for Surface Codes from Majorana Delocalization

Area of Science:

• Quantum Information Science
• Condensed Matter Physics
• Statistical Mechanics

Background:

• Statistical mechanics mappings are crucial for understanding quantum error correction.
• Existing models primarily address incoherent noise, neglecting coherent errors like spurious gate rotations.

Purpose of the Study:

• To develop a statistical mechanics mapping for the surface code considering coherent errors.
• To compare the error correction properties of coherent versus incoherent noise.

Main Methods:

• Mapped the surface code with coherent errors (X or Z rotations) to a 2D Ising model with complex couplings.
• Further mapped this to a 2D Majorana scattering network.
• Linked 2D networks to 1D fermions to analyze error-correcting phases.

Main Results:

• Identified commonalities and differences between coherent and incoherent error correction.
• Both error types map to a Z2-nontrivial 2D insulator in the error-correcting phase.
• Coherent errors, beyond a threshold angle (ϕth ≈ 0.14π), map to a Majorana metal, unlike the Z2-trivial insulator for incoherent errors.
• The derived threshold sin²(ϕth) ≈ 0.18 exceeds the incoherent threshold pth ≈ 0.11.

Conclusions:

• The developed mapping provides new insights into quantum error correction with coherent noise.
• Coherent errors allow for a higher storage threshold than previously considered for incoherent errors.
• The findings suggest potential improvements in the resilience of quantum error correction codes.