Tailoring Dynamical Codes for Biased Noise: The X$^3$Z$^3$ Floquet Code

In this talk, I will present the X$^3$Z$^3$ Floquet code, a new type of dynamical quantum error-correcting code that has an improved performance under biased noise compared to other Floquet codes. Inspite of the absence of constant stabilizer operators, suprisingly, we find a persistent symmetry in the X$^3$Z$^3$ Floquet code under infinitely biased noise, which simplifies the decoding problem. I will prove that, unlike the bias-tailored static codes, dynamical codes defined on the honeycomb lattice cannot admit decoding graphs equivalent to a disjoint series of repetition codes under infinitely biased noise, thus limiting their performance compared to their static counterparts. Furthermore, I will give detailed theoretical studies on the thresholds and subthreshold performance of various Floquet codes under biased noise. The no-go theorem for the 1D decoding of Floquet codes supplemented by our comprehensive numerical studies show that the symmetry of the X$^3$Z$^3$ Floquet code most likely has rendered its performance under biased noise close to optimal among all Floquet codes. Our work, therefore, demonstrates that the X$^3$Z$^3$ code is one of the prime code candidates for quantum error correction particularly in devices with reduced connectivity such as the honeycomb and heavy-hexagonal architectures.