The differences between T1 and T2 in real-world quantum computing platforms underscore the importance of studying the thresholds of quantum error-correcting codes under biased noise, also spurring active searches for error-correcting codes with thresholds exceeding the hashing bound under biased noise. Recently, new error-correcting codes such as the XZZX code and the holographic seven-qubit tailored code have exhibited a 50% threshold under pure Pauli noise. Notably, the XZZX code achieves a threshold exceeding the hashing bound in cases of high bias.

This work reports on a holographic quantum error-correcting code, the HaPPY code, which also exhibits a 50% threshold under pure Pauli noise and surpasses the hashing bound threshold under high biased noise. Additionally, this work also explores the threshold of the holographic Steane code under biased noise for comparison.

In addition to studying thresholds under biased noise, this work also investigates the thresholds of various codes, including the Hyper-Invariant Tensor-Network code (HTN code), holographic Reed-Muller code, and some heterogeneous holographic codes, under quantum erasure channels and depolarizing channels.

This work has developed an automated quantum tensor network operator push program, which supports the automated generation of stabilizers and complete logical operators for tensor network quantum error-correcting codes. This greatly enhances the research efficiency of holographic codes, and the program is now ready to be made available to the open-source community.