We can design efficient quantum error-correcting (QEC) codes by tailoring them to our choice of quantum architecture. Useful tools for constructing such codes include Clifford deformations and appropriate gauge fixings of compass codes. In this work, we find Clifford deformations that can be applied to elongated compass codes resulting in QEC codes with improved performance under noise models with errors biased towards dephasing commonly seen in quantum computing architectures. These Clifford deformations enhance decoder performance by introducing symmetries, while the stabilizers of compass codes can be selected to obtain more information on high-rate errors. As a result, the codes exhibit thresholds that increase with bias and lower logical error rates under both code capacity and phenomenological noise models. One of the Clifford deformations we explore yields QEC codes with better thresholds and logical error rates than those of the XZZX surface code at moderate biases under code capacity noise.
We can design efficient quantum error-correcting (QEC) codes by tailoring them to our choice of quantum architecture. Useful tools for constructing such codes include Clifford deformations and appropriate gauge fixings of compass codes. In this work, we find Clifford deformations that can be applied to elongated compass codes resulting in QEC codes with improved performance under noise models with errors biased towards dephasing commonly seen in quantum computing architectures. These Clifford deformations enhance decoder performance by introducing symmetries, while the stabilizers of compass codes can be selected to obtain more information on high-rate errors. As a result, the codes exhibit thresholds that increase with bias and lower logical error rates under both code capacity and phenomenological noise models. One of the Clifford deformations we explore yields QEC codes with better thresholds and logical error rates than those of the XZZX surface code at moderate biases under code capacity noise.
Quantum error correction (QEC) is often implemented on hardware that experiences biased noise, where dephasing errors occur more frequently than other errors. This has motivated many recent efforts to develop bias-tailored QEC codes, such as the Clifford-deformed compass codes: a family of codes that achieve high thresholds under biased dephasing noise. We perform circuit-level simulations of the Clifford-deformed elongated compass codes under a biased noise model and evaluate code thresholds using standard minimum weight perfect matching (MWPM) and correlated MWPM. We find that correlated decoding enhances thresholds for all noise biases relative to standard MWPM under circuit-level noise. Our results demonstrate that correlated decoding leads to a higher relative gain in thresholds compared to standard MWPM when applied to codes with asymmetric stabilizers under biased noise.
Applying single-qubit Clifford unitaries to a Pauli stabilizer code produces a Clifford-deformed variant whose stabilizers remain Pauli operators, but with locally rotated Pauli axes. Such deformations provide a simple way to tailor a fixed code to anisotropic noise, and have enabled unusually high thresholds under strongly biased dephasing. In this work, we discuss zero-rate quantum low-density parity-check (LDPC) codes, for which there exist Clifford-deformed variants where the number of biased logical operators scales slower than the distance, or there exists a basis of logical operators whose overlap satisfies certain scaling conditions; in this case, the code-capacity threshold for the Clifford-deformed variant under i.i.d. pure dephasing noise approaches 50%. This property provably explains previously known code examples with 50% biased noise thresholds, such as XY surface code, XZZX surface code, color code, as well as some 3D Clifford-deformed codes. As a concrete new example, we study Clifford deformations of the tile codes of Ref. [1]. Similar to the phase diagram of 50% thresholds for random Clifford deformations of the surface code in Ref. [2], we find a similar phase diagram for the tile codes. We also construct several translationally invariant deformations of the tile code with 50% thresholds, and present numerical evidence for improved performance at finite bias and under circuit-level noise. In the circuit-level setting, performance is governed by the residual bias after a full syndrome-extraction cycle, linking our simulations to phenomenological models commonly used to study Clifford-deformed codes. We estimate this residual bias for different qubit platforms by modeling microscopic implementations of tile-code syndrome extraction.
Practical fault-tolerant quantum computing requires qubit-efficient quantum error correction (QEC). Clifford deformations (CDs) modify the stabilizers of a QEC code to lower its logical error rate at no additional resource cost. However, the number of possible CDs grows exponentially with the number of qubits, making exhaustive search infeasible beyond small codes. In this work, we further show that under circuit level noise the optimal CD depends strongly on both the noise model and decoder. To make systematic CD optimization possible, we introduce Evolution-Based Best-arm Identification (EBBI), a shot-efficient search algorithm that treats the circuit, noise model and decoder as a black box. We benchmark EBBI's performance on the rotated surface code, under two strongly Z-biased circuit-level noise models with the PyMatching decoder. At distance 3, EBBI finds the optimal CD using up to ∼1100× fewer shots than random search. At distance 5, when run at the target error rate, the median EBBI run converges to the CDs we believe to be optimal, while random search never converges. Under a Z-biased SI1000-like noise model, EBBI discovers a new CD, the B-XZZX code, which we generalized to larger distances. It differs from XZZX on only a fraction O(d−1) of the qubits at code distance d, yet its relative advantage grows from ≈5% at d=3 to ≈12% at d=11. We thus show that the B-XZZX pattern found by EBBI can be systematically scaled to larger code distances. Our work represents a first step towards scalable CD optimization for general QEC codes.
Concatenated quantum error-correcting codes have recently gained popularity because they improve the effective distance of a joint code without requiring the discovery of new codes with desirable parameters. A common building block for such studies is the [[4,1,2]] Iceberg code, the smallest error-detecting code and therefore the one with the lowest resource overhead. However, it has an asymmetric number of X and Z checks, and concatenating it produces skewed logical error rates (LERs) between the X and Z observables. This asymmetry can be mitigated by Clifford-deforming the individual blocks of the concatenation, but the space of possible Hadamard deformations grows doubly exponentially with the number of concatenation levels. We formalize this deformation space and investigate it by proposing six concrete deformation strategies, which we compare under a code capacity (with and without bias) noise model using a concatenated maximum-likelihood soft-information decoder. Among the strategies we study, one one of them achieves near-identical X and Z LERs in a logical-memory experiment while simultaneously reducing the total LER (the sum of the X and Z error rates).
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.