The high overhead of fault-tolerant measurement sequences (FTMSs) poses a major challenge for implementing quantum stabilizer codes. Here, we address this problem by constructing efficient FTMSs for the class of quantum Hamming codes $[\![2^r-1, 2^r-1-2r, 3]\!]$ with $r=3k+1$ ($k \in \mathbb{Z}^+$). Our key result demonstrates that the sequence length can be reduced to exactly $2r+1$-only one additional measurement beyond the original non-fault-tolerant sequence, establishing a tight lower bound. The proposed method leverages cyclic matrix transformations to systematically combine rows of the initial stabilizer matrix and preserving a self-dual CSS-like symmetry analogous to that of the original quantum Hamming codes. This induced symmetry enables hardware-efficient circuit reuse: the measurement circuits for the first $r$ stabilizers are transformed into circuits for the remaining $r$ stabilizers simply by toggling boundary Hadamard gates, eliminating redundant hardware. For distance-3 fault-tolerant error correction, our approach simultaneously reduces the time overhead via shorting the FTMS length and the hardware overhead through symmetry-enabled circuit multiplexing. These results provide an important advance towards the important open problem regarding the design of minimal FTMSs for quantum Hamming codes and may shed light on similar challenges in other quantum stabilizer codes.

Variable-rate coding schemes that support a variety of different rates while maintaining the same fundamental encoder/decoder architectures are of great interest in practical communication systems. Similar error-correcting code schemes are also in demand in quantum settings. However, generally, it is difficult to introduce the variable-rate coding schemes into the quantum coding domain attributed to two challenges: Obtaining new quantum codes from old ones systematically and sharing encoder-decoder components among the newly obtained quantum codes. In this correspondence, a multiple-rate coding scheme is introduced into the quantum coding domain, we achieve this goal by providing a systematic method to construct new non-homogeneous quantum LDPC codes of CSS type from an old one (named as the mother code) whose classical parity-check matrix (PCM) is row-circulant. The basic idea is to split rows of the highest-rate stabilizer elaborately to produce the stabilizer for lower rates, which ensures that the generated code family owns the same code length. More importantly, thanks to the nested stabilizer structures possessed by the resulting quantum codes, our method also enables a scalable stabilizer circuit to be shared among them.

The concept of generalized concatenated quantum codes (GCQC) provides a systematic way for constructing good quantum codes from short component codes. We introduce a stabilizer formalism for GCQCs, which is achieved by defining quantum coset codes. This formalism offers a new perspective for GCQCs and enables us to derive a lower bound on the code distance of stabilizer GCQCs from component codes parameters, for both non-degenerate and degenerate component codes. Our formalism also shows how to exploit the error-correcting capacity of component codes to design good GCQCs efficiently.

Decoding sparse quantum codes can be accomplished by syndrome-based decoding using a belief propagation (BP) algorithm. We significantly improve this decoding scheme by developing a new feedback adjustment strategy for the standard BP algorithm. In our feedback procedure, we exploit much of the information from stabilizers, not just the syndrome but also the values of the frustrated checks on individual qubits of the code and the channel model. Furthermore we show that our decoding algorithm is superior to belief propagation algorithms using only the syndrome in the feedback procedure for all cases of the depolarizing channel. Our algorithm does not increase the measurement overhead compared to the previous method, as the extra information comes for free from the requisite stabilizer measurements.