Toward Self-Correcting Quantum Codes for Neutral Atom Arrays

Discovering low-overhead quantum error-correcting codes is of significant interest for fault-tolerant quantum computation. For hardware capable of long-range connectivity, the bivariate bicycle codes offer significant overhead reduction compared to surface codes with similar performance. In this work, we present “Z semidirect Z (ZSZ) codes,” a simple nonabelian generalization of the bivariate bicycle codes based on the group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mrow> <a:mi mathvariant="double-struck">Z</a:mi> </a:mrow> <a:mi>ℓ</a:mi> </a:msub> <a:mo>⋊</a:mo> <a:msub> <a:mrow> <a:mi mathvariant="double-struck">Z</a:mi> </a:mrow> <a:mi>m</a:mi> </a:msub> </a:math> . We numerically demonstrate that certain instances of this code family achieve competitive performance with the bivariate bicycle codes under circuit-level depolarizing noise using a belief-propagation and ordered-statistics decoder, with an observed threshold around <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mn>0.5</e:mn> <e:mi mathvariant="normal">%</e:mi> </e:math> . We also benchmark the performance of this code family under local “self-correcting” decoders, where we observe significant improvements over the bivariate bicycle codes, including evidence of a sustainable threshold around <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mn>0.095</h:mn> <h:mi mathvariant="normal">%</h:mi> </h:math> , which is higher than the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mn>0.06</k:mn> <k:mi mathvariant="normal">%</k:mi> </k:math> that we estimate for the four-dimensional toric code under the same noise model. These results suggest that ZSZ codes are promising candidates for scalable self-correcting quantum memories. Finally, we describe how ZSZ codes can be realized with neutral atoms trapped in movable tweezer arrays, where a complete round of syndrome extraction can be achieved using simple global motions of the atomic arrays.