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arXiv:2601.09131

Bidirectional Decoding for Concatenated Quantum Hamming Codes

Chao Zhang, Zipeng Wu, Jiahui Wu, Shilin Huang

quant-phSubmitted Jan 14, 2026• Announced Jan 15, 2026• 0 citations
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Abstract

High-rate concatenated quantum codes offer a promising pathway toward fault-tolerant quantum computation, yet designing efficient decoders that fully exploit their error-correction capability remains a significant challenge. In this work, we introduce a hard-decision decoder for concatenated quantum Hamming codes with time complexity polynomial in the block length. This decoder overcomes the limitations of conventional local decoding by leveraging higher-level syndrome information to revise lower-level recovery decisions -- a strategy we refer to as bidirectional decoding. For the concatenated $[[15,7,3]]$ quantum Hamming code under independent bit-flip noise, the bidirectional decoder improves the threshold from approximately $1.56\%$ to $4.35\%$ compared with standard local decoding. Moreover, the decoder empirically preserves the full $3^{L}$ code-distance scaling for at least three levels of concatenation, resulting in substantially faster logical-error suppression than the $2^{L+1}$ scaling offered by local decoders. Our results can enhance the competitiveness of concatenated-code architectures for low-overhead fault-tolerant quantum computation.

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High-rate concatenated quantum codes offer a promising pathway toward fault-tolerant quantum computation, yet designing efficient decoders that fully exploit their error-correction capability remains a significant challenge. In this work, we introduce a hard-decision decoder for concatenated quantum Hamming codes with time complexity polynomial in the block length. This decoder overcomes the limitations of conventional local decoding by leveraging higher-level syndrome information to revise lower-level recovery decisions -- a strategy we refer to as bidirectional decoding. For the concatenated [[15,7,3]][[15,7,3]][[15,7,3]] quantum Hamming code under independent bit-flip noise, the bidirectional decoder improves the threshold from approximately 1.56%1.56\%1.56% to 4.35%4.35\%4.35% compared with standard local decoding. Moreover, the decoder empirically preserves the full 3L3^{L}3L code-distance scaling for at least three levels of concatenation, resulting in substantially faster logical-error suppression than the 2L+12^{L+1}2L+1 scaling offered by local decoders. Our results can enhance the competitiveness of concatenated-code architectures for low-overhead fault-tolerant quantum computation.
We introduce a neural message-passing framework for decoding general concatenated stabilizer codes. Soft beliefs propagate bidirectionally across concatenation levels, and lightweight neural networks learn only to aggregate incoming messages. For the concatenated [[15,7,3]][[15,7,3]][[15,7,3]] quantum Hamming code, the resulting decoder achieves substantially higher thresholds than the state-of-the-art bidirectional hard-decision decoder under both bit-flip and depolarizing noise. In particular, the depolarizing pseudo-threshold nearly doubles, from 6.5%6.5\%6.5% to 12.3%12.3\%12.3%. For many-hypercube codes, a decoder fine-tuned on circuit-level errors in Knill's teleportation-based error correction can achieve lower logical-CNOT failure rates than their dedicated decoder, using a fixed number of message-passing iterations instead of extensive combinatorial search. Our framework provides a generic decoding tool for exploring the design space of concatenated codes, including non-CSS constructions, toward low-overhead fault tolerance.
Designing quantum error correcting codes that promise a high error threshold, low resource overhead, and efficient decoding algorithms is crucial to achieve large-scale fault-tolerant quantum computation. The concatenated quantum Hamming code is one of the potential candidates that allows for constant space overhead and efficient decoding. We study the concatenation of surface codes with quantum Hamming codes as a quantum memory, and estimate its error threshold, resource overhead, and decoding time. A high error threshold is achieved, which can in principle be pushed up to the threshold of the surface code. Furthermore, the concatenated codes can suppress logical errors to a much lower level than the surface codes, under the assumption of comparable amount of resource overhead. The advantage in suppressing errors starts to show for a quantum memory of intermediate scale. Concatenating surface codes with quantum Hamming codes therefore provides a promising avenue to demonstrate small-scale fault-tolerant quantum circuits in the near future, and also paves a way for large-scale fault-tolerant quantum computation.
Qudits provide a larger local Hilbert space than qubits and have recently attracted interest as building blocks for efficient fault-tolerant quantum computation (FTQC). In this work, we extend many-hypercube (MHC) codes, high-rate concatenated quantum codes, from qubits to qudits of prime dimension qqq. We construct the level-2 and level-3 MHC codes with parameters [ ⁣[62,42,22] ⁣]q\left[\!\left[6^2,4^2,2^2\right]\!\right]_q[[62,42,22]]q​ and [ ⁣[63,43,23] ⁣]q\left[\!\left[6^3,4^3,2^3\right]\!\right]_q[[63,43,23]]q​, respectively, and generalize the level-by-level minimum-distance (LLMD) decoder originally proposed for the qubit MHC codes. In a qudit XXX-error model, we evaluate the code-capacity performance of the qudit MHC codes for dimensions up to q=13q=13q=13. We find that the error threshold increases monotonically with the local dimension qqq, reaching 10.0%10.0\%10.0% for q=13q=13q=13, nearly double the value of 5.1%5.1\%5.1% for qubits. Furthermore, for sufficiently large local dimensions, we observe a pronounced waterfall regime in which the logical error rate decreases substantially faster than expected from conventional distance-based estimates. To understand the origin of the observed qudit advantage, we analyze the decoding performance in terms of the entropy of the underlying qqq-ary noise channel. This analysis reveals a competition between the increased syndrome information available at larger qqq and the simultaneous growth of the typical physical error weight at fixed entropy. These results demonstrate that higher-dimensional quantum systems can substantially improve the performance of high-rate concatenated quantum codes and motivate further development of qudit-based FTQC architectures.
Fault-tolerant capacities quantify the ability of a quantum channel to reliably transmit information when every component of the encoding and decoding procedure is noisy. Earlier work analyzed achievable communication rates under such noise using fault-tolerant implementations based on concatenated codes with a single logical qubit. In this work, we develop an alternative approach using concatenations of quantum Hamming codes, which offer constant space overhead by encoding many logical qubits simultaneously. We introduce modular techniques for implementing fault-tolerant circuits with quantum input/output interfaces using the concatenated quantum Hamming code. These tools enable an analysis of fault-tolerant entanglement-assisted communication that is not only simpler, but also yields substantially higher achievable communication rates than previous methods, owing to the limited noise correlations in syndrome qubits of high-rate quantum Hamming codes.
We consider the problem of optimally decoding a quantum error correction code---that is, to find the optimal recovery procedure given the outcomes of partial ``check'' measurements on the system. In general, this problem is NP hard. However, we demonstrate that for concatenated block codes, the optimal decoding can be efficiently computed using a message-passing algorithm. We compare the performance of the message-passing algorithm to that of the widespread blockwise hard decoding technique. Our Monte Carlo results using the five-qubit and Steane's code on a depolarizing channel demonstrate significant advantages of the message-passing algorithms in two respects: (i) Optimal decoding increases by as much as 94% the error threshold below which the error correction procedure can be used to reliably send information over a noisy channel; and (ii) for noise levels below these thresholds, the probability of error after optimal decoding is suppressed at a significantly higher rate, leading to a substantial reduction of the error correction overhead.
Fault-tolerant quantum computation (FTQC) is expected to address a wide range of computational problems. To realize large-scale FTQC, it is essential to encode logical qubits using quantum error-correcting codes. High-rate concatenated codes have recently attracted attention due to theoretical advances in fault-tolerant protocols with constant-space-overhead and polylogarithmic-time-overhead, as well as practical developments of high-rate many-hypercube codes equipped with a high-performance level-by-level minimum-distance decoder (LMDD). We propose a general, high-performance decoder for high-rate concatenated stabilizer codes that extends LMDD by leveraging the Chase algorithm to generate a suitable set of candidate errors. Our simulation results demonstrate that the proposed decoder outperforms conventional decoders for high-rate concatenated Hamming codes under bit-flip noise.