Bidirectional Decoding for Concatenated Quantum Hamming Codes

TL;DR

This work addresses decoding high-rate concatenated quantum Hamming codes by introducing a bidirectional hard-decision decoder that uses higher-level syndrome information to revise lower-level recoveries. The core method combines a recursive Decode with two subroutines, Reassign and FlipCost, to greedily minimize the total physical recovery cost while satisfying global syndromes, achieving polynomial-time complexity. Empirically, the approach yields a dramatic threshold improvement (e.g., from $1.56\%$ to $4.35\%$ for the $[[15,7,3]]$ code) and preserves the full code distance $d=3^L$ over multiple concatenation levels, resulting in substantially faster logical-error suppression than local decoding. These results enhance the viability of high-rate, low-overhead quantum computation by enabling scalable, effective decoding across concatenated architectures.

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.

Figures (5)

• Figure 1: Decoding of a weight-$4$ error in a two-level concatenation of the $[[15,7,3]]$ quantum Hamming code. The level-$0$ qubits form a $15 \times 15$ array, while the level-$1$ qubits form a $15 \times 7$ array. (a) The four physical errors occur on the level-$0$ qubits $(1,1)_0$, $(1,2)_0$, $(2,1)_0$, and $(2,2)_0$. (b) Level-$1$ local decoding applies recoveries on the level-$0$ qubits $(1,3)_0$ and $(2,3)_0$ (shown in blue). (c) The resulting level-$1$ errors are concentrated on the first two rows (subblocks) and share the same pattern. (d) Level-$2$ local decoding propagates this error pattern to the third level-$1$ subblock (shown in red). (e) At the physical level, level-$2$ local decoding introduces an additional weight-$3$ recovery (shown in red), resulting in a total recovery weight of $5$. (f) By reassigning the level-$1$ recoveries on the first and second subblocks in a manner consistent with the level-$1$ syndrome, the error can be corrected. (g) The reassigned recovery has physical weight $6 - 2 = 4$. This example illustrates the core mechanism of the bidirectional decoder: by using higher-level syndrome information to revise lower-level decisions (reassignment), it successfully corrects error patterns that cause the local hard-decision decoder to fail.
• Figure 2: The information flow in bidirectional hard-decision decoding for a 3-level concatenated quantum Hamming code. Unlike standard local decoding which flows strictly bottom-up (blue arrows), the bidirectional decoder incorporates a top-down "reassignment" pass (red arrows). This allows the decoder to update lower-level recovery choices based on global constraints, significantly improving error correction performance.
• Figure 3: Performance comparison between the local hard-decision decoder (cross markers) and the bidirectional decoder (dot markers) for concatenated quantum Hamming codes with two to four concatenation levels under independent bit-flip errors with probability $p$. Power-law fits $p_L \propto p^{\alpha}$ are shown for each curve, where $p_L$ denotes the logical error rate and $\alpha$ indicates the error suppression exponent. The fitted exponents for the bidirectional decoder are indicated by boxed annotations; these values are close to the theoretical limits $\lceil 3^L/2 \rceil$ expected for distance-$3^L$ codes. The threshold is estimated to be $1.56\%$ for the local hard-decision decoder and $4.35\%$ for the bidirectional decoder. The bidirectional decoder not only improves the threshold by nearly $3\times$ but also achieves much steeper error suppression (higher $\alpha$) than the local decoder.
• Figure 4: Logical error rates for $\mathrm{CQHC}(15,15,n)$ with $n = 15, 31, 63, 127$. In all cases, the level-1 and level-2 local codes are fixed to the $[[15,7,3]]$ quantum Hamming code, while the level-3 local quantum Hamming code determines both the encoding rate and the resulting decoding performance. Although increasing the top-level block size $n$ slightly increases the logical error rate, it significantly boosts the encoding rate while maintaining a high effective distance (slope $\alpha \approx 16$), demonstrating the flexibility of the architecture.
• Figure 5: Logical error rates for concatenated quantum Hamming codes with heterogeneous structures. The codes $\mathrm{CQHC}(15,15,31)$, $\mathrm{CQHC}(15,31,15)$, and $\mathrm{CQHC}(31,15,15)$ each encode 1029 logical qubits into 6975 physical qubits by concatenating two $[[15,7,3]]$ QHCs with one $[[31,21,3]]$ QHC, arranged at different concatenation levels. The configuration $\mathrm{CQHC}(15,15,31)$, which places the largest code block at the highest level, achieves the lowest logical error rate. This indicates that to maximize performance in heterogeneous concatenations, larger code blocks should be prioritized for higher concatenation levels.