Floquet Abelian Multicycle Codes

Abstract

Abelian multicycle (AMC) codes are compact quantum low-density parity-check codes whose multiblock chain-complex structure provides redundant low-weight stabilizers and supports single-shot error correction. We introduce Floquet AMC codes by deriving a quotient-lattice representation of a general level-$j$, $D$-dimensional AMC complex over a finite Abelian group algebra, lifting this lattice to spacetime, and rotating the circuit-time direction in the associated ZX network. When the check and data spiders have even valence and admit a time-oriented local port matching, the network decomposes into a periodic schedule of native two-qubit $XX$ and $ZZ$ measurements. We construct generalized-bicycle and level-$2$ AMC4 examples, determine their instantaneous stabilizer groups, and compute their embedded distances by minimizing over all inequivalent circuit cuts. For AMC4 instances locally equivalent to four-dimensional toric codes, we obtain Floquet memories with parameters $[[108,6,5]]$, $[[144,6,8]]$, and $[[324,6,10]]$. Local Pauli-web detector templates and beam-search decoding under the measurement-native EM3 noise model yield an estimated pseudothreshold of approximately $1.2\%$. These results provide compact measurement-only realizations of higher-dimensional homological redundancy without directly measuring the original weight-six stabilizers.