Universal scaling framework for parameterized quantum evolutions at criticality

Abstract

Variational ansätze are a cornerstone of quantum many-body physics, providing compact approximations to complex ground states using finite resources. Recent quantum-technology advances have introduced a new class based on layered parameterized evolutions. Assessing whether they can represent critical ground states is challenging: correlations span all length scales, while finite circuit depth limits how far they extend. Building on finite-resource scaling from tensor networks, we assign each ansatz an emergent correlation length $ξ_D$, the longest range over which it faithfully captures critical correlations. Its growth with refinement parameter $D$, $ξ_D \propto D^κ$, defines an exponent $κ$ measuring how efficiently an architecture converts resources into long-distance correlations. Applying this framework to the critical transverse-field Ising model, with $D$ the circuit depth of parameterized evolutions, we compare ansätze with nearest-neighbor and long-range generators, and layers where generators act separately or combined. All ansätze are compatible with algebraic growth, but fitted exponents range from $κ\simeq1$ to $κ\simeq3$. Exponential interactions give the largest exponents, while power-law interactions stay close to nearest-neighbor behavior, showing long-range support alone gives no scaling advantage. Combined-generator layers generally outperform separable ones, so layer organization matters alongside interaction range. Finally, a quasiparticle analysis shows finite depth leaves an unresolved window of width $ξ_D^{-1}$ around the low-energy modes responsible for long-distance correlations. The emergent correlation length thus acts as an infrared resolution scale, providing a benchmark for critical-state preparation and a guide for designing resource-efficient variational architectures.