Callan-Symanzik-like equation in information theory

TL;DR

The paper studies the complexity growth rate (CGR) in the generalized holographic framework $\mathcal{C}_{Any}$ and shows that CGR can undergo phase-transition–like jumps governed by bulk field dynamics that map, via fluid-gravity duality, to the boundary energy-momentum tensor. Near these transitions, CGR exhibits universal scaling and satisfies a Callan-Symanzik–like equation with a beta function $\beta = -\nu\gamma$ and an anomalous dimension $\Delta$, with regime-dependent exponents, implying that changing the energy scale can alter computational speed. The results establish a concrete link between bulk geometry, boundary RG flow, and information-processing capabilities, suggesting new levers for tuning quantum information tasks in holographic settings. The findings point to universal critical behavior of CGR across dimensions and provide a framework for exploring energy-scale control of computation in strongly coupled systems.

Abstract

Within the "complexity=anything" proposal of holography, the complexity growth rate (CGR) can exhibit jumps, interpreted as phase transitions. We demonstrate that the location and amplitude of these jumps are governed by the dynamics of bulk fields, which, via the fluid-gravity correspondence, map to the boundary energy-momentum tensor. The behavior of the CGR near these critical points exhibits scaling and universality. We show that the CGR satisfies a Callan-Symanzik-like equation near the transitions. We suggest that this enables enhanced computational efficiency by tuning the energy scale.

Figures (2)

• Figure 1: a schematic behavior of CGR in terms of the boundary time. The black line shows the phase transition where the complexity has an intersection in terms of $r_{min}$phase.
• Figure 2: A schematic behavior of the potential $\tilde{U}$ in terms of radial coordinate $r$ and the horizon $R$.