We introduce the domain wall color code, a new variant of the quantum error-correcting color code that exhibits exceptionally high code-capacity error thresholds for qubits subject to biased noise. In the infinite bias regime, a two-dimensional color code decouples into a series of repetition codes, resulting in an error-correcting threshold of 50%. Interestingly, at finite bias, our color code demonstrates thresholds identical to those of the noise-tailored XZZX surface code for all single-qubit Pauli noise channels. The design principle of the code is that it introduces domain walls which permute the code's excitations upon domain crossing. For practical implementation, we supplement the domain wall code with a scalable restriction decoder based on a matching algorithm. The proposed code is identified as a comparably resource-efficient quantum error-correcting code highly suitable for realistic noise.
Bias-tailored codes such as the XZZX surface code and the domain wall color code achieve high dephasing-biased thresholds because, in the infinite-bias limit, their
Z syndromes decouple into one-dimensional repetition-like chains; the
X3Z3 Floquet code shows an analogous strip-wise structure for detector events in spacetime. We capture this common mechanism by defining strip-symmetric biased codes, a class of static stabilizer and dynamical (Floquet) codes for which, under pure dephasing and perfect measurements, each elementary
Z fault is confined to a strip and the Z-detector--fault incidence matrix is block diagonal. For such codes the Z-detector hypergraph decomposes into independent strip components and maximum-likelihood
Z decoding factorizes across strips, yielding complexity savings for matching-based decoders. We characterize strip symmetry via per-strip stabilizer products, viewed as a
Z2 1-form symmetry, place XZZX, the domain wall color code, and
X3Z3 in this framework, and introduce synthetic strip-symmetric detector models and domain-wise Clifford constructions that serve as design tools for new bias-tailored Floquet codes.
We introduce the domain wall color code, a new variant of the quantum error-correcting color code that exhibits exceptionally high code-capacity error thresholds for qubits subject to biased noise. In the infinite bias regime, a two-dimensional color code decouples into a series of repetition codes, resulting in an error-correcting threshold of 50%. Interestingly, at finite bias, our color code demonstrates thresholds identical to those of the noise-tailored XZZX surface code for all single-qubit Pauli noise channels. The design principle of the code is that it introduces domain walls which permute the code's excitations upon domain crossing. For practical implementation, we supplement the domain wall code with a scalable restriction decoder based on a matching algorithm. The proposed code is identified as a comparably resource-efficient quantum error-correcting code highly suitable for realistic noise.
Performing large calculations with a quantum computer will likely require a fault-tolerant architecture based on quantum error-correcting codes. The challenge is to design practical quantum error-correcting codes that perform well against realistic noise using modest resources. Here we show that a variant of the surface code-the XZZX code-offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favourable sub-threshold resource scaling that can be obtained by specialising a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation.
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.
Abstract We show that a variant of the surface code—the XZZX code—offers remarkable performance for fault-tolerant quantum computation. The error threshold of this code matches what can be achieved with random codes (hashing) for every single-qubit Pauli noise channel; it is the first explicit code shown to have this universal property. We present numerical evidence that the threshold even exceeds this hashing bound for an experimentally relevant range of noise parameters. Focusing on the common situation where qubit dephasing is the dominant noise, we show that this code has a practical, high-performance decoder and surpasses all previously known thresholds in the realistic setting where syndrome measurements are unreliable. We go on to demonstrate the favorable sub-threshold resource scaling that can be obtained by specializing a code to exploit structure in the noise. We show that it is possible to maintain all of these advantages when we perform fault-tolerant quantum computation. We finally suggest some small-scale experiments that could exploit noise bias to reduce qubit overhead in two-dimensional architectures
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.