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

TL;DR

This paper establishes a quantitative scaling law for meta-learning in quantum control, showing when adaptation improves fidelity and how it scales with task variance and gradient steps, validated on quantum and classical control tasks.

Contribution

It derives a universal lower bound for adaptation gains in meta-learning, validated on quantum gate calibration and classical control, and introduces a few-shot pre-adaptation protocol.

Findings

Adaptation gain saturates exponentially with gradient steps.

Significant fidelity improvements (>40%) in out-of-distribution quantum tasks.

The scaling laws are derived from general optimization geometry, not quantum physics.

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. We further introduce a few-shot pre-adaptation protocol that estimates the optimal adaptation budget from $N{=}3$-5 probe steps within 3-19% relative error across out-of-distribution regimes.