Quantum Tanner Codes at Moderate Blocklength

Abstract

We present explicit constructions of quantum Tanner (QT) codes with good rate and distance, obtained through two complementary approaches: the left-right Cayley complex (LRCC) description and the "lifting" perspective, in which a seed Calderbank-Shor-Steane (CSS) code is lifted by commuting left-right regular actions of a finite group $\mathcal{G}$. Through an extensive search over non-abelian groups from GAP's SmallGrp library, we investigate the moderate-blocklength regime ($n\in [500,1000]$) and identify several new code instances with distance upper bounds exceeding $20$. These include $[[480,8,(\le 21,\le 21)]]$, $[[504,4,(\le 36,\le 27)]]$, $[[672,4,(\le 48,\le 28)]]$, $[[720,6,(\le 30,\le 30)]]$, and $[[864,8,(\le 39,\le 31)]]$, with these bounds obtained using up to $350$ million trials of sQetch, a randomized distance estimator. The code instances presented have check weights ranging from $9$ to $20$. Using the Tesseract decoder, we estimate pseudo-thresholds of $3.6\%$-$4.6\%$ under phenomenological noise and $0.14\%$-$0.27\%$ under circuit-level noise, comparable to prior results at shorter code lengths. We also provide QuantumExpanders$.$jl, an open-source Julia library for constructing QT codes and explicit constructions of Ramanujan graphs.