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arXiv:2610.11225

High-Rate Concatenated Quantum Error-Correcting Codes for Qudits

Takanori Nishi, Hayato Goto

quant-phSubmitted Oct 8, 2026• Announced Oct 9, 2026• 0 citations
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Abstract

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 $q$. We construct the level-2 and level-3 MHC codes with parameters $\left[\!\left[6^2,4^2,2^2\right]\!\right]_q$ and $\left[\!\left[6^3,4^3,2^3\right]\!\right]_q$, respectively, and generalize the level-by-level minimum-distance (LLMD) decoder originally proposed for the qubit MHC codes. In a qudit $X$-error model, we evaluate the code-capacity performance of the qudit MHC codes for dimensions up to $q=13$. We find that the error threshold increases monotonically with the local dimension $q$, reaching $10.0\%$ for $q=13$, nearly double the value of $5.1\%$ 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 $q$-ary noise channel. This analysis reveals a competition between the increased syndrome information available at larger $q$ 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.

Comments
11 pages, 2 figures

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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.
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Quantum error-correcting codes (QECCs) require high encoding rates in addition to high thresholds unless a sufficiently large number of physical qubits are available. Many-hypercube (MHC) codes, defined as concatenations of the [ [ 6 , 4 , 2 ] ] quantum error-detecting code, have been proposed as high-performance and high-encoding-rate QECCs. However, concatenated codes have the disadvantage that the syndrome weight grows exponentially with respect to the concatenation level. To address this issue, here we propose subsystem quantum codes based on MHC codes. In particular, we study the smallest subsystem MHC codes, namely, subsystem codes derived from concatenated [ [ 4 , 2 , 2 ] ] error-detecting codes. The resulting codes have a constant syndrome-measurement weight of 4, while keeping high encoding rates. We build block-MAP and neural-network decoders and show that they demonstrate superior performance to the bounded-distance decoder.
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.
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