Estimating the logical error rate (LER) of a quantum error-correcting (QEC) code by Monte Carlo sampling takes
O(p−⌈d/2⌉/ε2) samples at physical error rate
p, code distance
d and relative error
ε, which becomes intractable at large distances and low error rates, and especially for concatenated codes, whose distance grows as
dℓ with the number of levels
ℓ. The fail fast method reduces this cost by sampling the failures of each weight separately, but it only works for unconcatenated codes, and does not model the soft information that the levels of a concatenated code pass to each other. We propose a predictor that takes the stabilizers, the minimum-weight logical operators and, for odd
d, the logical operators of weight
d+1 of a code, and computes, without sampling, the leading monomial
cpo of the LER when expressed as a polynomial in
p, at every level of concatenation, in time polynomial in the code parameters and linear in
ℓ for fixed budgets. We prove that our prediction is exact for any unconcatenated stabilizer code, for every logical observable and arbitrary per-qubit error rates, whenever no budget is exhausted. Furthermore, for concatenated codes, we construct a recursive approach that computes the leading monomial exactly in exponential time, and our predictor approximates this approach. Our predictions match exact enumeration at level 1 on every code we evaluate, and up to three levels for the iceberg code. Using our predictor, we evaluate and compare
37 code configurations under four noise biases in
48 core-hours, where Monte Carlo sampling would need more than
1012 shots at
p=10−3 for
113 of these
148 cases.
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.
Biased-noise qubits, in which one type of error (e.g.
X- and
Y-type errors) is significantly suppressed relative to the other (e.g.
Z-type errors), can significantly reduce the overhead of quantum error correction. Codes such as the rectangular surface code or XZZX code substantially reduce the qubit overhead under biased noise, but they still face challenges. The rectangular surface code suffers from a relatively low threshold, while the XZZX code requires twice as many physical qubits to maintain the same code distance as the surface code. In this work, we introduce a 2D local code construction that outperforms these codes for noise biases
η≥7×104, reducing the qubit overhead by over 50% at
pZ=10−3 and
η=2×106 to achieve a logical error rate of
10−12. Our construction relies on the concatenation of two classical codes. The inner codes are repetition phase-flip codes while the outer codes are high-rate bit-flip codes enabled by their implementation at the logical level, which circumvents device connectivity constraints. These results indicate that under sufficiently biased noise, it is advantageous to address phase-flip and bit-flip errors at different layers of the coding scheme. The inner code should prioritize a high threshold for phase-flip errors, while the bit-flip outer code should optimize for encoding rate efficiency. In the strong biased-noise regime, high-rate outer codes keep the overhead for correcting residual bit-flip errors comparable to that of the repetition code itself, meaningfully lower than that required by earlier approaches.
Concatenating quantum error correction codes scales error correction capability by driving logical error rates down double-exponentially across levels. However, the noise structure shifts under concatenation, making it hard to choose an optimal code sequence. We automate this choice by estimating the effective noise channel after each level and selecting the next code accordingly. In particular, we use learning-based methods to tailor small, non-additive encoders when the noise exhibits sufficient structure, then switch to standard codes once the noise is nearly uniform. In simulations, this level-wise adaptation achieves a target logical error rate with far fewer qubits than concatenating stabilizer codes alone--reducing qubit counts by up to two orders of magnitude for strongly structured noise. Therefore, this hybrid, learning-based strategy offers a promising tool for early 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.
We introduce heterogeneous quantum error-correcting codes composed of qubit types with distinct error channels and study their performance in the code-capacity regime using maximum-likelihood tensor network decoding. In the regime where both qubit types share the same noise bias but differ in physical error rate, placing noisier qubits in the bulk -- where each error triggers more syndrome bits -- and cleaner qubits on the boundary yields thresholds exceeding 0.4 (compared to ~0.2 for the reverse placement) and improvements exceeding three orders of magnitude in logical error rate at high bias, with the advantage growing exponentially with code distance. In the regime where both types share the same error rate but differ in bias, the optimal strategy reverses: placing high-bias (more predictable) qubits on the boundary increases the threshold from 0.292(5) to 0.360(9) at a bias ratio of 100, and from 0.29(1) to 0.398(4) at a bias ratio of 1000. We also observe a striking bias-inversion property: the logical error channel becomes strongly XX X- and YY Y-biased despite the physical noise being ZZ Z-biased. We propose a stabilizer-ratio hypothesis that provides a unified information-theoretic explanation for both placement rules and predicts even larger advantages for code families such as color codes.