Hyperbolic Floquet code with graph-edge syndromes

Abstract

Quantum error correction would be a primitive for demonstrating quantum advantage in a realistic noisy environment. Floquet codes are a class of dynamically generated, stabilizer-based codes in which low-weight parity measurements are applied in a time-periodic schedule. Furthermore, for several Floquet codes, the encoding rate becomes finite even for an infinitely large qubit number. However, despite these advantageous properties, existing Floquet codes require handling more intricate, often hypergraph-structured syndromes from the decoding perspective, which makes decoding comparatively demanding. We give a concrete method for solving this issue by proposing hyperbolic color Floquet (HCF) code. To this end, we simultaneously take advantage of hyperbolic Floquet and Floquet color codes. Parity measurements in our code consist of the repetitions of six-step measurements on (semi-)hyperbolic three-colorable tilings. Since each step just measures $X \otimes X$ or $Z \otimes Z$, our code on the regular $\{8,3\}$ lattice has the following three advantages: (i) parity measurements are weight-2, (ii) for the numbers $k$ and $n$ of logical and physical qubits, respectively, the encoding rate is finite, i.e., $\lim_{n \to \infty} k/n = 1/8$, and (iii) the code distance is proportional to $\log n$. From the above property, each single-fault event generally affects at most two detectors, which implies ``graph-edge'' syndromes, and hence decoding with a minimum-weight perfect matching (MWPM) decoder is efficient and virtually scales near-linearly in the number of physical qubits $n$. This is a stark contrast to several known Floquet codes because their parity measurements repeat the measurements of $X\!\otimes\!X$, $Y\!\otimes\!Y$, and $Z\!\otimes\!Z$, and thus the syndromes are represented as a hypergraph, which basically requires decoders with longer decoding time.