Computationally Efficient Optimization of Per-Qubit Clifford Deformation for Non-uniform Biased Noise

Abstract: 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.

• The paper introduces Chameleon, a decoder-free method for optimizing per-qubit Clifford deformation that minimizes logical error rates (LER) in quantum error correction codes by focusing on frame selection based on ambiguity set constructions and analytic Bhattacharyya bounds, but struggles with weakly biased maps and correlated/multi-qubit noise.
• As compared to existing calibration-aware approaches, Chameleon showed significant LER reductions in surface, color, and BB codes, averaging 13%, 7%, and 4% respectively on calibration maps, and up to 357% on synthetic fields.
• The method achieves computational efficiency: frame selection takes seconds to minutes versus days for other methods.

Motivation and problem statement

Physical noise on superconducting quantum processors is both Pauli-biased and spatially non-uniform: Google Willow calibration data shows that 43% of qubits exhibit a strong $X/Z$ imbalance ($\eta_q > 1.5$) whose magnitude varies across the chip, and this bias propagates to the logical level — undeformed surface-code logical $X$ and $Z$ failure rates differ by up to $6.6\times$ at distance 7. Clifford deformation offers a zero-overhead remedy: conjugating each physical qubit by a single-qubit Clifford operator permutes its noise vector $(p_X, p_Y, p_Z)$ without changing the code's rate or distance. The paper identifies two failure modes of existing calibration-aware approaches. Global fixed deformations (e.g., XZZX) are worse than the undeformed CSS code in 29% of the 1,756 code-map instances studied, while the local greedy rule of Tiurev et al. underperforms the undeformed code in 27% of instances because it ignores decoding ambiguity. Randomized search with decoder validation (CDSC), though effective, requires Monte Carlo estimation of each candidate's LER — on the order of a day per calibration map even for 75,000 candidates with an idealized 1 µs-per-shot decoder.

Method

Chameleon is a compile-time frame-selection pass built on two observations. First, only Type-A failures (decoding ambiguity, where the decoder picks a same-syndrome pattern at least as likely as the true error) are addressable at the code level; Type-B failures are decoder artifacts. Second, since ambiguity likelihoods dominate the LER at low physical error rates, frame selection should minimize the probability of the rare event in each low-weight ambiguity operator $\ell = ee'$ — a product of same-syndrome errors differing in logical class.

The pipeline has three stages:

• Ambiguity-set construction (offline): Low-weight ambiguity operators are enumerated once per code up to weight cutoff $W=d+2$ (surface/color/small BB codes) or $W=d+10$ with randomized Gaussian elimination plus translation orbits for BB72, reducing storage from GiB–PiB scale to MiB-scale lookup tables (e.g., BB72: $4.40\times10^{12}$ complete class-$\eta_q > 1.5$0 operators reduced to 45,483 retained operators, 50 MiB).
• Analytic bound: For each operator, exact rare-event likelihood evaluation is exponential in operator weight; Chameleon replaces it with a Bhattacharyya bound, $\eta_q > 1.5$1 with $\eta_q > 1.5$2, reducing per-operator cost from $\eta_q > 1.5$3 to $\eta_q > 1.5$4 under independent per-qubit noise.
• Frame search: A cross-entropy method over binary frames $\eta_q > 1.5$5 minimizes $\eta_q > 1.5$6, aggregated as either worst-axis ($\eta_q > 1.5$7) for memory or single-axis for targeted workloads; a second CEM over full frames $\eta_q > 1.5$8, warm-started from the binary result and deployed only if it improves the objective beyond margin $\eta_q > 1.5$9, handles $X$0-dominant qubits.

Online scoring of one frame costs $X$1 — microseconds for geometric codes, milliseconds for BB72 — so the entire search is decoder-free.

Surrogate fidelity and computational overhead

The central empirical claim is that the surrogate preserves LER rankings precisely where deformation matters most. On strongly biased Willow maps, Spearman rank correlation between $X$2 and measured worst-axis LER reaches $X$3–$X$4 across all three code families, whereas on weakly biased Miami it degrades to $X$5 (surface) and $X$6 (color); interestingly, the delocalized BB36 code retains high fidelity even there ($X$7). This implies the method is self-limiting in the right way: gains vanish as bias vanishes, but ranking reliability does not fail silently in the biased regime where deployment occurs.

The latency reduction is substantial: frame selection takes 9.4 s (surface $X$8), 39 s (color $X$9), and 3.1 min (BB72), versus 7.2 days, 4.2 days, and 1.2 days respectively for pricing all 75,000 candidates by decoding under an idealized decoder favorable to the baseline (the real BP+OSD decoder is ~$Z$0 slower per shot). In a head-to-head search-effectiveness comparison at $Z$1, CDSC cannot match Chameleon's LER even after validating $Z$2 candidates with the decoder, while Chameleon achieves its optimum with roughly $Z$3 decoder-free evaluations.

Logical performance

In memory experiments using Type-A LER against the best prior baseline, Chameleon reduces worst-axis LER by an average of 13% for surface codes, 7% for color codes, and 4% for BB codes across calibration-derived maps, with maxima of 19%, 16%, and 10% (BB72 on Willow: +32%). Gains are largest on synthetic xyz fields with strong uniform bias — up to +357% for surface $Z$4 at $Z$5 — and grow monotonically with bias strength, reaching $Z$6 (surface) and $Z$7 (color) LER reduction at $Z$8. The paper attributes the smaller color-code gains to self-duality: class-$Z$9 and class-$6.6\times$0 operators occupy identical supports, so Hadamard frames trade one axis's problem for the other's rather than resolving either. For BB codes, gains relative to the undeformed CSS frame are substantial (19–58%), indicating limited deformation headroom rather than absent opportunity.

Two additional case studies extend applicability. Single-axis optimization (relevant to magic-state preparation) yields mean relative $6.6\times$1 of $6.6\times$2–$6.6\times$3 versus two-axis optimization. Across 332 admissible surface-code placements on Berlin, Miami, and Willow respecting connectivity constraints, Chameleon improves 79% of patches with median gain 5.2%; losses concentrate on nearly uniform patches, consistent with the fidelity analysis. Under circuit-level Si1000 noise with a biased idle component derived from Willow, gains persist but shrink to 5.2% average (14% maximum), reflecting diluted overall bias — the authors note honestly that validating a more precise biased circuit-level model would require hardware access they lack.

Limitations and open questions

Several limitations are conceded directly. All main results use phenomenological noise models; circuit-level gains are demonstrated only once and are materially smaller, and the biased-noise structure of real devices may not match the resampled calibration maps used here. The surrogate's independence assumption (factorized per-qubit noise) excludes correlated errors, and its fidelity collapses on weakly biased maps, meaning Chameleon offers no benefit — and no reliable guidance — when bias is absent. Full-frame refinement is useless on current transmon devices lacking $6.6\times$4-dominant qubits, useful only synthetically or hypothetically. The ambiguity set is truncated by weight cutoff; while sensitivity studies show convergence at $6.6\times$5 for BB72, the completeness of the retained set for large qLDPC codes relies on randomized Gaussian elimination heuristics without formal guarantees. Finally, the deployment margin $6.6\times$6 exhibits nontrivial sensitivity: always deploying the full-frame candidate ($6.6\times$7) yields only +12.0% over baselines versus +23.5% at $6.6\times$8 on BB72, leaving open how to choose $6.6\times$9 adaptively per map. Whether the surrogate remains faithful under correlated or time-varying noise is unresolved.

Conclusion

The paper contributes a principled surrogate objective — a Bhattacharyya-type upper bound on aggregate rare-event likelihood over precomputed low-weight ambiguity operators — that decouples Clifford-deformation optimization from decoder simulation. Combined with offline ambiguity-set construction and CEM-based search over binary and full frame spaces, it reduces frame-selection latency from days to seconds-to-minutes while delivering consistent, bias-scaled LER improvements across surface, color, and bivariate bicycle codes. The approach's dependence on spatial bias strength is presented as a feature rather than a defect, but its validation remains bounded by phenomenological noise modeling and independence assumptions, both explicitly flagged as open directions requiring hardware access.