Sharp time-discontinuity in subsystem complexity at large local dimension

Establish that, in the limit of large local qudit dimension (q → ∞), for an n-qudit random quantum circuit and a contiguous subsystem A with size n_A < n/2, the subsystem complexity C(ρ_A(t)) exhibits a sharp transition in time: it rises during early evolution and then drops discontinuously at a time of order n_A (predicted to occur near t ≈ n_A/2), matching the holographically predicted “sawtooth” behavior.

Background

The paper proves upper and lower bounds showing that for less-than-half subsystems in 1D random quantum circuits, the complexity grows and then decreases, equilibrating to O(n_A). Holography predicts a discontinuous (sharp) drop at a finite time in this regime, while rigorous RQC methods currently indicate only a smooth rise-and-fall.

To reconcile the holographic sharp transition with solvable circuit models, the authors propose studying the limit of large on-site Hilbert space dimension. They argue that increasing the local dimension may capture additional holographic features and render the transition sharp rather than smooth, and they formalize this as a conjecture and target for future proof.