When Does Adaptation Win? Scaling Laws for Meta-Learning in Quantum Control

TL;DR

This work addresses calibration bottlenecks in scalable quantum processors by deriving a scaling law for meta-learning-based adaptation in differentiable control. It proves that the adaptation gain after $K$ gradient steps satisfies $G_K \ge A_\infty\left(1 - e^{-\beta K}\right)$ with $A_\infty \propto \sigma_\tau^2$ and $\beta = \eta\mu_{\min}$, requiring a local Polyak-Łojasiewicz condition. The theory is validated on single- and two-qubit gates and a classical LQR, showing exponential saturation of gains and a linear dependence on task variance, with substantial fidelity improvements under extreme out-of-distribution noise for CZ gates. The results yield a transferable framework for deciding when adaptation justifies its overhead and suggest practical pathways to reduce per-device calibration time on cloud quantum processors, while highlighting the broader applicability to other differentiable control systems.

Abstract

Quantum hardware suffers from intrinsic device heterogeneity and environmental drift, forcing practitioners to choose between suboptimal non-adaptive controllers or costly per-device recalibration. We derive a scaling law lower bound for meta-learning showing that the adaptation gain (expected fidelity improvement from task-specific gradient steps) saturates exponentially with gradient steps and scales linearly with task variance, providing a quantitative criterion for when adaptation justifies its overhead. Validation on quantum gate calibration shows negligible benefits for low-variance tasks but $>40\%$ fidelity gains on two-qubit gates under extreme out-of-distribution conditions (10$\times$ the training noise), with implications for reducing per-device calibration time on cloud quantum processors. Further validation on classical linear-quadratic control confirms these laws emerge from general optimization geometry rather than quantum-specific physics. Together, these results offer a transferable framework for decision-making in adaptive control.

Figures (11)

• Figure 1: Overview. (a) In the loss landscape over control parameters $\theta_1$ and $\theta_2$, the robust (non-adaptive) controller ($\theta^\star_{\text{rob}}$, star) minimizes average loss but is suboptimal for individual devices (colored dots); gradient-based adaptation traces distinct trajectories toward device-specific optima, yielding different controls $u(t)$ (inset). (b) The adaptation gap follows $G_K = A_\infty(1 - e^{-\beta K})$, where $\beta = \eta\mu$ captures adaptation rate and $A_\infty \propto \sigma^2_\tau$ scales with task variance. $K^*$ marks diminishing returns.
• Figure 2: Theory Validation (a) PL condition: Optimization trajectories for two representative tasks (different colors) plotting gradient norm $\frac{1}{2}\|\nabla L\|^2$ versus optimality gap $L - L^*$. The PL inequality (Eq. \ref{['eq:pl']}) holds when points lie above the line with slope $\mu$. In the shaded adaptation regime (near the optimum, where $L - L^* < 0.14$) for optimum $L^\star$, the relationship is approximately linear with $\mu \approx 0.03$, confirming that gradient descent makes consistent progress. Outside this regime, the PL bound is violated as trajectories traverse saddle regions. (b) Lipschitz continuity: Lindbladian distance $\|f_\xi - f_{\xi'}\|$ versus task distance $\|\xi - \xi'\|$, where $\xi = (\Gamma_{\mathrm{deph}}, \Gamma_{\mathrm{relax}})$ are the dissipation rates defining each task. The linear bound $\|L_\xi - L_{\xi'}\| \leq C_L \|\xi - \xi'\|$ holds with $C_L \approx 2.8$, confirming Lemma \ref{['lem:2']}. (c) Control separation: Distance between task-optimal controls $\|\theta^*_\xi - \theta^*_{\xi'}\|$ versus task parameter distance $\|\xi - \xi'\|$. The linear relationship ($R^2 = 0.98$) demonstrates consistency with Lemma \ref{['lem:3']}: when tasks differ in their dissipation rates, their optimal controls differ proportionally, ensuring that task variance creates meaningful opportunity for adaptation
• Figure 3: Scaling law validation. (a) Adaptation gap $G_K$ versus inner-loop steps $K$ exhibits exponential saturation ($R^2 > 0.999$) for small task variance (in-distribution), consistent with Theorem \ref{['thm:1']}. The fitted curve $G_K = c(1 - e^{-\beta K})$ captures the transition from rapid gains to diminishing returns. (b) Asymptotic gap $G_\infty$ scales linearly with actual task variance $\sigma^2_\tau = \mathrm{Var}(\Gamma_{\mathrm{deph}}) + \mathrm{Var}(\Gamma_{\mathrm{relax}})$ ($R^2 = 0.94$), evaluated across ${\sim}0.01$--$5\times$ the nominal training variance. This is consistent with the predicted proportionality $A_\infty \propto \sigma^2_\tau$.
• Figure 4: Adaptation dynamics (mild out-of-distribution (OOD), $\sim$1.1$\times$ training noise). (a) Fidelity versus adaptation steps $K$ for two initialization strategies: meta-learned (FOMAML) starts higher and converges faster than the fixed-average baseline. (b) Fidelity distributions across strategies: fixed-average baseline achieves $\mathcal{F} = 0.881$ before adaptation and improves substantially to $\mathcal{F} = 0.935$ after $K = 10$ steps; meta-initialization achieves $\mathcal{F} = 0.959$ before adaptation, improving only marginally to $\mathcal{F} = 0.960$ after adaptation. (c) Learned pulse sequences: FOMAML pulses before and after adaptation are nearly indistinguishable, confirming the initialization already generalizes well. Fixed-average optimized pulses after $K=10$ adaptation steps (orange) show qualitatively different structure.
• Figure 5: Two-qubit CZ gate adaptation under high noise (strong OOD, $10\times$ training noise). (a) Meta-initialized control pulses ($K=0$) yield $\mathcal{F}=54.2\%$ gate fidelity in a high-noise environment. (b) After $K=10$ adaptation steps, task-specific corrections achieve $\mathcal{F}=95.7\%$. (c) Adaptation gap $G_K$ follows the predicted exponential saturation $G_K = c(1-e^{-\beta K})$ with $R^2=0.986$ and $\beta=0.333$.

Theorems & Definitions (15)

• Lemma 4.4: Trap-Free Landscape
• proof
• Lemma 4.5: Lipschitz Task Dependence
• proof
• Definition 4.6: Task Variance
• Lemma 4.7: Control Separation
• proof
• Theorem 4.8: Adaptation Gap Scaling Law
• proof
• Corollary 4.9: Diminishing Returns Threshold