Constructions and Performance of Hyperbolic and Semi-Hyperbolic Floquet Codes

We construct families of Floquet codes derived from color-code tilings of closed hyperbolic surfaces. These codes have weight-two check operators, a finite encoding rate and can be decoded efficiently with minimum-weight perfect matching. We also construct semi-hyperbolic Floquet codes, which have improved distance scaling, and are obtained via a fine-graining procedure. Using a circuit-based noise model that assumes direct two-qubit measurements, we show that semi-hyperbolic Floquet codes can be <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><a:mn>48</a:mn></a:math> times more efficient than planar honeycomb codes and therefore over <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><d:mn>100</d:mn></d:math> times more efficient than alternative compilations of the surface code to two-qubit measurements, even at physical error rates of 0.3% to 1%. We further demonstrate that semi-hyperbolic Floquet codes can have a teraquop footprint of only 32 physical qubits per logical qubit at a noise strength of 0.1%. For standard circuit-level depolarizing noise at <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><g:mi>p</g:mi><g:mo>=</g:mo><g:mn>0.1</g:mn><g:mi mathvariant="normal">%</g:mi></g:math>, we find a <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><k:mn>30</k:mn></k:math> times improvement over planar honeycomb codes and a <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><n:mn>5.6</n:mn></n:math> times improvement over surface codes. Finally, we analyze small instances that are amenable to near-term experiments, including a Floquet code derived from the Bolza surface that encodes four logical qubits into 16 physical qubits.