Quantum Complexity Dynamics for Disjoint Subsystems

Abstract

Quantum complexity has emerged as a natural probe of chaos, thermalization, and the black hole interior on timescales long after local observables have equilibrated. However, its time evolution has been studied almost exclusively in subsystems confined to a single connected region. We show, using complementary tools from holography and random quantum circuits, that noncontiguous subsystems composed of multiple disjoint regions give rise to qualitatively new physics compared to the contiguous case. First, at finite temperature, a subsystem occupying less than half of the total system can carry high complexity at late times, even as the complexity of its larger complement remains low. This inversion of the usual hierarchy is intrinsically thermal: it vanishes in the infinite-temperature limit, which is the regime modeled by random quantum circuits. Second, if a subsystem's complexity equilibrates at an early time, then fragmenting it into $m$ disjoint components can further reduce this timescale by a factor of $m$, a phenomenon we exhibit in both holography and random quantum circuits. These results not only sharpen the correspondence between geometric and computational notions of complexity, but also motivate the search for novel complexity phenomena in quantum dynamics.