In fault-tolerant quantum computing systems with biased noise, Clifford deformation can substantially reduce the logical error rate (LER) without additional physical hardware overhead, such as extra qubits, syndrome extraction rounds, or code distance. Although Google Willow calibration data shows that $43\%$ of qubits exhibit strong $X/Z$ bias, existing calibration-aware deformation techniques remain impractical: (1) global searches over the $6^n$ deformation choices rely on computing-intensive simulations, and (2) local heuristics often underperform undeformed baselines.
We present Chameleon, a fast, high-performance, and code-agnostic Clifford deformation compiler. We utilize our approximation to tackle a deformation problem based on an analytical bound on the LER. By minimizing this surrogate, Chameleon finds an optimized deformation that empirically reduces the LER with substantially lower computational overhead. In our evaluation, using calibration models derived from real superconducting devices, Chameleon demonstrates that improvements in our surrogate are strongly correlated with actual LER reductions, with an average rank correlation of $ρ=0.8$ and $ρ=0.89$-$0.94$ on the most strongly biased system. It also reduces classical computational time from $1.2$ days to $3.1$ minutes for the BB72 code. Chameleon achieves maximum LER reductions of $19\%$ ($13\%$ on average) for surface codes, $16\%$ ($7\%$) for color codes, and $10\%$ ($4\%$) for bivariate bicycle codes relative to competing baselines. The maximum gains for all code families are observed on the most strongly biased system.
In fault-tolerant quantum computing systems with biased noise, Clifford deformation can substantially reduce the logical error rate (LER) without additional physical hardware overhead, such as extra qubits, syndrome extraction rounds, or code distance. Although Google Willow calibration data shows that 43% of qubits exhibit strong X/Z bias, existing calibration-aware deformation techniques remain impractical: (1) global searches over the 6n deformation choices rely on computing-intensive simulations, and (2) local heuristics often underperform undeformed baselines.
We present Chameleon, a fast, high-performance, and code-agnostic Clifford deformation compiler. We utilize our approximation to tackle a deformation problem based on an analytical bound on the LER. By minimizing this surrogate, Chameleon finds an optimized deformation that empirically reduces the LER with substantially lower computational overhead. In our evaluation, using calibration models derived from real superconducting devices, Chameleon demonstrates that improvements in our surrogate are strongly correlated with actual LER reductions, with an average rank correlation of ρ=0.8 and ρ=0.89-0.94 on the most strongly biased system. It also reduces classical computational time from 1.2 days to 3.1 minutes for the BB72 code. Chameleon achieves maximum LER reductions of 19% (13% on average) for surface codes, 16% (7%) for color codes, and 10% (4%) for bivariate bicycle codes relative to competing baselines. The maximum gains for all code families are observed on the most strongly biased system.
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.
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.
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.
Fault-tolerant quantum computing requires understanding how error-correcting codes perform on diverse physical hardware. This is typically assessed via noisy stabilizer simulation of logical circuits at HPC scale, combined with a noise model that yields a logical error rate for the relevant code distances and depths. The uniform depolarizing model is the standard baseline, but its homogeneous assumptions fail to capture the heterogeneity, asymmetries, and correlations of real devices, where Pauli, measurement, and spatio-temporal errors are not weakly coupled. Yet these same structured features create opportunities for joint code-hardware co-design, motivating noise models that more faithfully reflect target hardware while remaining tractable to simulate.
We introduce FTPrimitiveBench, a systematic benchmarking approach for studying how logical primitives interact with hardware-motivated noise. It supports both custom specifications and representative structured noise families: Pauli bias, measurement bias, and spatial or spatio-temporal non-uniformity -- together with generators for core surface-code Clifford primitives: logical memory, lattice surgery, transversal logical Hadamard, and the logical phase gate via lattice surgery. We find that structured noise affects these primitives in qualitatively distinct ways, with outcomes shaped by the interplay between noise model, primitive, and decoder choice. These results extend memory benchmarks to active logical computation, where the interaction between noise structure and primitive implementation matters. By standardizing the link between noise-model specification and primitive construction, FTPrimitiveBench enables reproducible comparative studies of QEC protocols and decoders, supporting hardware-aware co-design of fault-tolerant architectures. Code: https://github.com/ShuwenKan/FTPrimitiveBench.
Various realizations of Kitaev’s surface code perform surprisingly well for biased Pauli noise. Attracted by these potential gains, we study the performance of Clifford-deformed surface codes (CDSCs) obtained from the surface code by the application of single-qubit Clifford operators. We first analyze CDSCs on the 3×3 square lattice and find that, depending on the noise bias, their logical error rates can differ by orders of magnitude. To explain the observed behavior, we introduce the effective distance d′ , which reduces to the standard distance for unbiased noise. To study CDSC performance in the thermodynamic limit, we focus on random CDSCs. Using the statistical mechanical mapping for quantum codes, we uncover a phase diagram that describes random CDSC families with 50% threshold at infinite bias. In the high-threshold region, we further demonstrate that typical code realizations outperform the thresholds and subthreshold logical error rates, at finite bias, of the best-known translationally invariant codes. We demonstrate the practical relevance of these random CDSC families by constructing a translation-invariant CDSC belonging to a high-performance random CDSC family. We also show that our translation-invariant CDSC outperforms well-known translation-invariant CDSCs, such as the XZZX and XY codes. Published by the American Physical Society 2024
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.