A Symmetry-Breaking Dial for Equivariant Quantum Machine Learning

Symmetry-preserving circuits encode conservation laws into quantum machine learning models, but a fully equivariant circuit cannot distinguish states related by that symmetry. This creates a structural blind spot on mirror-pair tasks, where the label identifies one of two symmetry-related ground-state branches. We introduce a continuous symmetry-breaking dial that rescales symmetry-breaking gate angles while keeping the parameter count fixed, interpolating between equivariant and unconstrained ansatze. We prove that the equivariant endpoint has exactly $50\%$ accuracy on every mirror pair and that the class separation is odd in the dial, with a linear coefficient determined in closed form by the initialization and the data. With a preconditioned optimizer, the smallest dial value reaching $99.9\%$ accuracy follows ${\lambda }^{\star }(T)\approx 0.3/T$ where the grid resolves it, and extends to a calibrated joint law in budget and task strength. On transverse-field mirror-pair tasks, this prescription keeps the dial below $10^{-2}$, breaking only a fraction of a percent of the symmetry, while reaching full accuracy. The transport law persists across depth, register size, and a decade of task strength, with recalibrated constants. We also identify a zero-transport boundary where the circuit cannot propagate the mirror distinction to the readout. Beyond quantum models, whenever the label is defined by the symmetry, full equivariance is a provable ceiling rather than a safe default.