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.
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.
In some quantum computing architectures, Pauli noise is highly biased. Tailoring Quantum error-correcting codes to the biased noise may benefit reducing the physical qubit overhead without reducing the logical error rate. In this paper, we propose a family of quantum XYZ cyclic codes, which are the only one family of quantum cyclic codes with code distance increasing with code length to our best knowledge and have good error-correcting performance against biased noise. Our simulation results show that the quantum XYZ cyclic codes have 50% code-capacity thresholds for all three types of pure Pauli noise and around 13% code-capacity threshold for depolarizing noise. In the finite-bias regime, when the noise is biased towards Pauli Z errors with noise bias ratios ηZ=1000, the corresponding code-capacity threshold is around 49%. Besides, we show that to reach the same code distance, the physical qubit overhead of XYZ cyclic code is much less than that of the XZZX surface code.
Quantum computers are highly sensitive to noise due to interactions with the environment, which severely limits the utility of near-term devices. Quantum error correction addresses this challenge by encoding logical information over multiple qubits in order to preserve information in the presence of noise. This facilitates the realization of the theoretical advantages of quantum computing such as exponential speed-up compared to classical computers for certain algorithms. Physical realizations of qubits may have different noise profiles, including biased noise, where phase-flip errors may be more likely than bit-flip errors. We propose a quantum error-correcting code, the XYZ² code, which is tailored to biased noise. The code has high error thresholds against biased noise, below which logical errors are exponentially suppressed with the linear dimension of the code. The important classical task of deriving corrections given partial information about errors is called decoding. We investigate several decoding strategies tailored to specific challenges, ranging from a novel maximum-likelihood sampling decoder, to a model-free, data-driven, neural-network-based decoder, to an efficient multi-step decoder for concatenated codes. We also study the link between quantum error-correcting codes and their statistical-mechanical counterparts by mapping to generalized random-bond Ising models and subsequently deriving the exact results for finite-size corrections under biased noise for moderate code sizes. Quantum error correction is still in its developing stages, and this work provides a useful contribution to the field by bridging a gap between theory and practice. By adapting well-studied topological codes to error models that are closer to reality, this work paves the way for the realization of a fault-tolerant quantum architecture.
We find that the substantial advantages predicted for bias-tailored quantum error correction (QEC) under code-capacity noise are strongly reduced once realistic syndrome extraction and circuit-level noise models are considered. We start by comparing XZZX codes to rectangular surface codes with a bias-dependent optimised anisotropy. Although code-capacity simulations predict an advantage of rectangular surface codes in the limit of high noise bias, this actually disappears under circuit-level noise, making the XZZX codes the preferred and simplest choice even for platforms that allow for a flexible variation of the code layout adapted to changes in noise calibration. Our results identify bias degradation during syndrome extraction under circuit-level noise as the central limitation of biased-tailored QEC. To partially mitigate this effect, we introduce a bias-filtering CNOT gadget that temporarily encodes the ancillary target qubit during syndrome extraction in a repetition code and, upon measurement and feed forward, manages to reduce the bias degradation. In a regime of high-bias and low-idle errors, this bias-filtering gadget yields a few-percent relative improvement of the XZZX code error threshold, demonstrating that lightweight bias-filtering strategies can recover part of the lost bias-tailoring advantage for realistic circuit-level noise.
A crucial insight for practical quantum error correction is that different types of errors, such as single-qubit Pauli operators, typically occur with different probabilities. Finding an optimal quantum code under such biased noise is a challenging problem, related to the (generally unknown) maximum capacity of the corresponding noisy channel. A benchmark for this capacity is given by the hashing bound, which describes the performance of random stabilizer codes and leads to the matter of identifying codes that come close to the bound while also being efficiently decodable. In this work, we perform a comprehensive analysis of asymptotically zero-rate holographic codes under biased noise. We show that many representatives from such models of this code class fulfill both the channel optimality and efficient decoding guarantees for tensor-network codes. In fact, all tested holographic codes are found to reach the hashing bound in some bias regime, while several built from the [[ 5 , 1 , 2 ]] surface code and [[ 6 , 1 , 3 ]] code exceed state-of-the-art code performance in the 2-Pauli noise regime. Furthermore, we consider Clifford deformations which allow all considered codes to reach the hashing bound for 1-Pauli noise as well. Our results establish that holographic codes, which were previously shown to possess efficient tensor-network decoders, also exhibit competitive thresholds under biased noise.
Studying the threshold behavior of surface codes under biased noise models is an active area of research. Previous work Tuckett et al. (2018), using an optimal tensor-network decoder, demonstrated that replacing Z-type stabilizers with Y-type stabilizers significantly improves the surface code threshold under code-capacity level dephasing noise. In this work, we construct and study a ZY surface code by replacing the X-type stabilizers with Y-type stabilizers. We compare it with the standard ZX surface code under circuit-level Pauli-X biased noise, with and without an additional gate-based XX crosstalk noise. We find that for the ZX surface code, the X-memory threshold increases monotonically with bias while the Z-memory threshold decreases and saturates. For the ZY surface code, the Y-memory threshold is nearly constant across all bias values. The Z-memory thresholds of the ZX and ZY codes are consistent within the uncertainty. Adding XX crosstalk reduces the Z-memory threshold beyond the fitting uncertainty while leaving the X memory threshold largely unaffected. The choice of CNOT ordering redistributes threshold performance between the two logical memories. Our work extends prior observations from code-capacity level noise to circuit-level noise. It also indicates the need for decoders capable of jointly reasoning over correlated syndrome information so that tailored stabilizer structures could be fully utilized for quantum error correction.
Quantum error correction codes (QECCs) are essential for reliable quantum computing as they protect quantum states against noise and errors. Limited research has explored the resilience of QECCs to biased noise, critical for selecting optimal codes. We examine how different noise types impact QECCs, considering the varying susceptibility of quantum systems to specific errors. Our goal is to identify opportunities to minimize the resources—or overhead—needed for effective error correction. We conduct a detailed study on two QECCs—rotated and unrotated surface codes—under various noise models using simulations. Rotated surface codes generally perform better due to their simplicity and lower qubit overhead. They exceed the noise threshold of current quantum processors, making them more effective at lower error rates. This study highlights a hierarchy in surface code implementation based on resource demand, consistently observed across both code types. Our analysis ranks the code-capacity model as the most pessimistic and the circuit-level model as the most realistic, mapping error thresholds that show surface code advantages. Additionally, higher code distances improve performance without excessively increasing qubit overhead. Tailoring surface codes to align with the target logical error rate and the biased physical error profile is crucial for optimizing reliability and resource use.