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.
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.
Abstract Quantum error correction codes (QECCs) are crucial for quantum computing, protecting quantum states from noise and errors. This paper compares rotated and unrotated surface codes in different error types and noise models using simulations. Rotated codes outperform with higher thresholds and lower qubit overhead. The study finds QECC noise thresholds surpassing error rates in current quantum processors. Surface code implementation shows a hierarchy based on error or noise types, emphasizing resource demands. Our analysis ranks code-capacity as the most pessimistic and circuit-level as the most realistic model. The research underscores customizing surface codes to hardware needs and balancing reliability with qubit resources, highlighting the importance of addressing qubit efficiency and surface code overhead challenges.
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.
For quantum computers to reach their full potential will require error correction. We study the surface code, one of the most promising quantum error correcting codes, in the context of predominantly dephasing (Z-biased) noise, as found in many quantum architectures. We find that the surface code is highly resilient to Y-biased noise, and tailor it to Z-biased noise, whilst retaining its practical features. We demonstrate ultrahigh thresholds for the tailored surface code: ~39% with a realistic bias of = 100, and ~50% with pure Z noise, far exceeding known thresholds for the standard surface code: ~11% with pure Z noise, and ~19% with depolarizing noise. Furthermore, we provide strong evidence that the threshold of the tailored surface code tracks the hashing bound for all biases. We reveal the hidden structure of the tailored surface code with pure Z noise that is responsible for these ultrahigh thresholds. As a consequence, we prove that its threshold with pure Z noise is 50%, and we show that its distance to Z errors, and the number of failure modes, can be tuned by modifying its boundary. For codes with appropriately modified boundaries, the distance to Z errors is O(n) compared to O(n1/2) for square codes, where n is the number of physical qubits. We demonstrate that these characteristics yield a significant improvement in logical error rate with pure Z and Z-biased noise. Finally, we introduce an efficient approach to decoding that exploits code symmetries with respect to a given noise model, and extends readily to the fault-tolerant context, where measurements are unreliable. We use this approach to define a decoder for the tailored surface code with Z-biased noise. Although the decoder is suboptimal, we observe exceptionally high fault-tolerant thresholds of ~5% with bias = 100 and exceeding 6% with pure Z noise. Our results open up many avenues of research and, recent developments in bias-preserving gates, highlight their direct relevance to experiment.
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.
Realizing the full potential of quantum computation requires quantum error correction (QEC), with most recent breakthrough demonstrations of QEC using the surface code.QEC codes use multiple noisy physical qubits to encode information in fewer logical qubits, enabling the identification of errors through a decoding process.This process increases the logical fidelity (or accuracy) making the computation more reliable.However, most fast (efficient run-time) decoders neglect important noise characteristics, thereby reducing their accuracy.In this work, we introduce decoders that are both fast and accurate, and can be used with a wide class of QEC codes including the surface code.Our decoders, named belief-matching and belief-find, exploit all noise information and thereby unlock higher accuracy demonstrations of QEC.Using the surface code threshold as a performance metric, we observe a threshold at 0.94% error probability for our decoders, outperforming the 0.82% threshold for a standard minimum-weight perfect matching decoder.We also test our belief-matching decoders in a theoretical case study of codes tailored to a biased noise model.We find that the decoders lead to a much higher threshold and lower qubit overhead in the tailored surface code with respect to the standard, square surface code.Surprisingly, in the wellbelow-threshold regime, the rectangular surface code becomes more resource efficient than the tailored surface code due to a previously unnoticed phenomenon that we call "fragile boundaries."Our decoders outperform all other fast decoders in terms of threshold and accuracy, enabling better results in current quantum-error-correction experiments and opening up new areas for theoretical case studies.
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.