Hyperbolic Surface Code

Explore quantum codes built from tilings and topology on hyperbolic surfaces.

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Question

How can a hyperbolic tiling protect quantum information through topology?

An established square surface-code lattice beside the computed two-three-seven hyperbolic surface-code model.
Motivation: compare the familiar Euclidean surface-code layout with the actual computed (2,3,7) hyperbolic model used in this project.
Two disjoint boundary-to-boundary X-error chains in the computed four-four-four hyperbolic surface code.
Logical structure: the red and purple paths are two computed, data-and-check-disjoint chains joining different logical boundary classes in the (4,4,4) layer-0 code.
Three views of surface codes, a curved geometric cell, and a hyperbolic code lattice.
Euclidean, spherical, and hyperbolic structures considered in the research project.
A hyperbolic surface-code lattice contained in a circular model.
A computed layer of a hyperbolic surface code.
Logical error-rate curves for hyperbolic surface-code layers.
Threshold behaviour measured across several code layers.

Goals

  • Build a surface code from a finite hyperbolic tiling.
  • Study its stabilizers, logical operators, code rate, and distance.

Method

Place qubits and parity checks on a tiling whose non-trivial loops carry logical information.

  • Construct a regular tiling and identify edges to form a closed hyperbolic surface.
  • Build the boundary maps or stabilizer-check matrices from vertices, edges, and faces.
  • Find logical cycles and test error patterns or decoders on the resulting code.