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.
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.
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 [[62,42,22]]q and [[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 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.
To maximize the potential of quantum error-correcting codes, it is essential to develop high-performance decoders. The subsystem many-hypercube (MHC) codes have been developed to achieve both high encoding rates and low-weight syndrome-measurements, but the introduction of gauge degrees of freedom makes decoding more challenging. In this work, we develop neural-network-based decoders for the subsystem MHC codes in a circuit-level noise model. We demonstrate that even the gauge-measurement information can be utilized for decoding by carefully arranging the syndrome-measurement sequence, improving the decoding performance. We further show that recurrent neural decoders outperform simple fully connected neural decoders, and can decode syndrome-measurement sequences longer than those used during training.
Many-hypercube codes, concatenated [[n,n−2,2]] quantum error-detecting codes (n is even), have recently been proposed as high-rate quantum codes suitable for fault-tolerant quantum computing. While the original many-hypercube codes with n=6 can achieve remarkably high encoding rates (about 30% and 20% at concatenation levels 3 and 4, respectively), they have large code block sizes at high levels (216 and 1296 physical qubits per block at levels 3 and 4, respectively), making not only experimental realization difficult but also logical error rates per code block high. Toward earlier experimental realization and lower logical error rates, here we comprehensively investigate smaller many-hypercube codes with [[6,4,2]] and/or [[4,2,2]] codes, where, e.g., D6,4,4 denotes the many-hypercube code using [[6,4,2]] at level 1 and [[4,2,2]] at levels 2 and 3. As a result, we found a counterintuitive fact that D6,4,4 (D6,6,4,4) can achieve lower logical error rates per code block than D4,4,4 (D4,4,4,4), despite its higher encoding rate and larger code block size. Focusing on level 3, we also developed efficient fault-tolerant encoders realizing about 60% overhead reduction while maintaining or even improving the performance, compared to the original design. Using them, we numerically confirmed that D6,4,4 also achieves the best performance for logical controlled-NOT gates in a circuit-level noise model. These results are important for targeting a high-rate code toward early experimental realization of efficient fault-tolerant quantum computing.
Concatenated quantum error-correcting codes have recently gained popularity because they improve the effective distance of a joint code without requiring the discovery of new codes with desirable parameters. A common building block for such studies is the [[4,1,2]] Iceberg code, the smallest error-detecting code and therefore the one with the lowest resource overhead. However, it has an asymmetric number of X and Z checks, and concatenating it produces skewed logical error rates (LERs) between the X and Z observables. This asymmetry can be mitigated by Clifford-deforming the individual blocks of the concatenation, but the space of possible Hadamard deformations grows doubly exponentially with the number of concatenation levels. We formalize this deformation space and investigate it by proposing six concrete deformation strategies, which we compare under a code capacity (with and without bias) noise model using a concatenated maximum-likelihood soft-information decoder. Among the strategies we study, one one of them achieves near-identical X and Z LERs in a logical-memory experiment while simultaneously reducing the total LER (the sum of the X and Z error rates).
Standard approaches to quantum error correction for fault-tolerant quantum computing are based on encoding a single logical qubit into many physical ones, resulting in asymptotically zero encoding rates and therefore huge resource overheads. To overcome this issue, high-rate quantum codes, such as quantum low-density parity-check codes, have been studied over the past decade. In this case, however, it is difficult to perform logical gates in parallel while maintaining low overheads. Here, we propose concatenated high-rate small-size quantum error-detecting codes as a family of high-rate quantum codes. Their simple structure allows for a geometrical interpretation using hypercubes corresponding to logical qubits. We thus call them many-hypercube codes. They can realize both high rates, e.g., 30% (64 logical qubits are encoded into 216 physical ones), and parallelizability of logical gates. Developing dedicated decoder and encoders, we achieve high error thresholds even in a circuit-level noise model. Thus, the many-hypercube codes will pave the way to high-performance fault-tolerant quantum computing.
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 3L code-distance scaling for at least three levels of concatenation, resulting in substantially faster logical-error suppression than the 2L+1 scaling offered by local decoders. Our results can enhance the competitiveness of concatenated-code architectures for low-overhead fault-tolerant quantum computation.