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.
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.
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.
Tailoring quantum-error-correction codes (QECCs) to biased noise has demonstrated significant benefits. However, most of the prior research on this topic has focused on code-capacity noise models. Furthermore, a no-go theorem prevents the construction of controlled- () gates for two-level qubits in a bias-preserving manner which may, in principle, imply that noise bias cannot be leveraged in such systems. In this work, we show that a residual bias up to η ∼ 5 can be maintained in gates under certain conditions. Moreover, we employ controlled- () gates in syndrome-extraction circuits and show how to natively implement these in a bias-preserving manner for a broad class of qubit platforms. This motivates the introduction of what we call a (HBD) circuit-level noise model that captures these features. We numerically study the performance of the X Z Z X surface code and observe that bias-preserving gates are critical for leveraging biased noise. Accounting for the residual bias present in the gates, we observe an increase in the code threshold up to a 1.27 % physical error rate, representing a 90 % improvement. Additionally, we find that the required qubit footprint can be reduced by up to 75 % at relevant physical error rates.
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.
Quantum computing is one of the most promising technological revolutions of the last few decades. Despite its potential, one problem it suffers from is that quantum bits (or qubits), its basic units of information, are intrinsically susceptible to noise. To counter the effects of such noise, we need fault-tolerant quantum computing, and for that, we need a systematic method to protect quantum information. This is the role of quantum error correction (QEC). However, most QEC codes are built on the assumption that X- and Z-type errors occur with equal frequency, which is not always the case for some realistic quantum hardware platforms. Among the several QEC protocols currently developed, surface codes are the leading implementation due to their high thresholds and local connectivity. Previous results have shown that modified variants of these codes, such as the XZZX surface code, can perform well under biased noise. Despite being realizable in two-dimensional real architectures, surface codes have poor encoding rates, leading to large overheads that prevent them from being scalable in fault-tolerant quantum architectures. Quantum low-density parity-check (qLDPC) codes provide a promising alternative, since they can achieve higher encoding rates. In this thesis, we study the code-capacity performance of bias-tailored qLDPC codes under both X- and Z-bias Pauli noise. We construct bias-tailored qLDPC codes, including bivariate bicycle codes, generalized toric codes, and clustered-cyclic codes, by applying Hadamard deformations that transform the Calderbank-Shor-Steane (CSS) codes into non-CSS codes. Using Monte Carlo simulations with minimum-weight perfect matching and belief propagation with localized-statistics decoding, we evaluate the word error rate of these bias-tailored qLDPC codes across different bias regimes. We show that Hadamard-deformed qLDPC codes can yield substantial performance improvements under biased noise and can match or even surpass those performances obtained from XZZX surface code implementations. In particular, we find that generalized toric codes perform consistently the best under bias noise in comparison to the bivariate and clustered-cyclic codes. These results indicate that bias tailoring can be extended to high-rate qLDPC constructions, leading to future work to study their performance in more realistic settings, such as circuit-level noise.
Quantum error correction (QEC) for generic errors is challenging due to the demanding threshold and resource requirements. Interestingly, when physical noise is biased, we can tailor our QEC schemes to the noise to improve performance. Here we study a family of codes having XZZX-type stabilizer generators, including a set of cyclic codes generalized from the five-qubit code and a set of topological codes that we call generalized toric codes (GTCs). We show that these XZZX codes are highly qubit efficient if tailored to biased noise. To characterize the code performance, we use the notion of effective distance, which generalizes code distance to the case of biased noise and constitutes a proxy for the logical failure rate. We find that the XZZX codes can achieve a favorable resource scaling by this metric under biased noise. We also show that the XZZX codes have remarkably high thresholds that reach what is achievable by random codes, and furthermore they can be efficiently decoded using matching decoders. Finally, by adding only one flag qubit, the XZZX codes can realize fault-tolerant QEC while preserving their large effective distance. In combination, our results show that tailored XZZX codes give a resource-efficient scheme for fault-tolerant QEC against biased noise.
Fault-tolerant Quantum Computing (FTQC) relies on Quantum Error Correction (QEC) codes that encode logical qubits across many physical qubits to detect and correct errors. The surface code is among the most widely studied codes due to its high error threshold, the existence of efficient decoders, and hardware-friendly properties: a planar, two-dimensional layout with nearest-neighbor connectivity. In practice, however, the fabrication of solid-state quantum processors introduces hardware defects, resulting in defective qubits and couplers that must be discarded. Adapting the surface code to these defects often requires measuring the X- and Z-type checks in separate rounds rather than simultaneously. In this work, we investigate the optimal X-to-Z checks round-scheduling ratio under biased noise systems. Our results characterize how key architectural parameters, such as noise bias, code distance, and defect rate, impact the logical error rate. We provide insights into how to determine the optimal scheduling ratio directly from device calibration data, enabling manufacturers to maximize performance without extensive simulations. Our approach reduces the logical error rate by up to 4.25× at a 1% defect rate and up to 8.46× at a 2% defect rate for a distance-13 surface code under moderately biased noise. Furthermore, we demonstrate that the benefits of round-scheduling extend beyond the defective-hardware setting. In biased-noise architectures subject to CNOT crosstalk, separating X and Z measurement rounds yields up to 4.5× reduction in logical error rate.