Growth and collapse of subsystem complexity under random unitary circuits

For chaotic quantum dynamics modeled by random unitary circuits, we study the complexity of reduced density matrices of subsystems as a function of evolution time where the initial global state is a product pure state. The state complexity is defined as the minimum number of local quantum channels to generate a given state from a product state to a good approximation. In $1+1$d, we prove that the complexity of subsystems of length $\ell$ smaller than half grows linearly in time $T$ at least up to $T = \ell / 4$ but becomes zero after time $T = \ell /2$ in the limit of a large local dimension, while the complexity of the complementary subsystem of length larger than half grows linearly in time up to exponentially late times. Using holographic correspondence, we give some evidence that the state complexity of the smaller subsystem should actually grow linearly up to time $T = \ell/2$ and then abruptly decay to zero.

💡 Research Summary

The paper investigates how the state‑complexity of subsystems evolves under chaotic dynamics modeled by random brick‑work quantum circuits in one spatial dimension plus time. Complexity is defined as the minimal number of local quantum channels (completely positive trace‑preserving maps acting on two neighboring qudits, with ancillae allowed) required to prepare a given state from a product state up to a constant trace‑distance error. Starting from a global pure product state, the authors consider reduced density matrices of intervals of length ℓ on a chain of L sites, with the local Hilbert space dimension q.

The authors first establish a rigorous linear lower bound on the complexity of “large” intervals, i.e., those with ℓ > L/2. They use the fact that random brick‑work circuits become ε‑approximate unitary k‑designs after a depth linear in k. Setting k = T/poly(L) they show that after time T the circuit generates roughly N ≈ q^{T/poly(L)} almost orthogonal reduced states on the large interval. Distinguishing these states requires at least log N ≈ T/poly(L) local channels, implying C(T) ≥ c T for some constant c. This linear growth persists up to exponentially long times (∼e^{cL}), confirming that a subsystem larger than half the system retains high complexity for a very long period.

For “small” intervals (ℓ < L/2) the situation is different. The light‑cone spreads at speed one, so after a time T ≈ ℓ/2 the entire interval has been causally influenced by the circuit. The authors compute the ensemble‑averaged purity P(T;ℓ) of the reduced state and obtain tight bounds showing that P quickly approaches the maximally mixed value 1/q^{ℓ}. When the reduced density matrix is essentially maximally mixed, its complexity drops to zero. Rigorously they prove linear growth up to T = ℓ/4 and, in the limit of large local dimension q → ∞, that the complexity becomes exactly zero at T = ℓ/2. Thus, a small subsystem thermalizes to infinite temperature and loses all computational content after a finite time.

To connect these results with information‑theoretic quantities, the paper studies mutual information I(A:B) between a subsystem A and its complement B. By invoking Stinespring dilation and strong subadditivity, they show that any increase in mutual information provides a lower bound on the number of channels needed to generate the state, reproducing the linear‑in‑time lower bound for the large interval and giving a weaker bound for the small interval.

The authors then turn to holographic duality. In AdS/CFT, subsystem complexity is conjectured to be proportional to the volume (or action) of the entanglement wedge. Using entanglement‑wedge reconstruction arguments, they argue that a small interval’s entanglement wedge disappears abruptly when the light‑cone reaches the interval’s far end, leading to a sudden drop of complexity at T = ℓ/2. This holographic picture supports the “sharp transition” scenario rather than a smooth saturation.

Section 6 explores the memory capacity of a small subsystem. By analyzing a variant of the random circuit (the q‑design brick‑work), they demonstrate that, up to times just before thermalization, the reduced state retains full information about every gate that acted on the interval. This reinforces the idea that the complexity collapse coincides with loss of circuit memory.

Finally, in Section 7 the paper addresses a combinatorial problem: how many approximately orthogonal density matrices of a given rank can exist in a d‑dimensional Hilbert space? Using unitary design techniques and random‑matrix integrals, they derive upper bounds that feed back into the complexity lower bounds, showing that the number of distinguishable reduced states grows exponentially with time for large intervals.

Overall, the work provides a mathematically rigorous picture of subsystem complexity dynamics in random quantum circuits: large subsystems exhibit sustained linear growth, while small subsystems grow linearly only up to a finite “scrambling” time and then collapse abruptly. The results are consistent with holographic expectations and illuminate the interplay between entanglement spreading, information retention, and computational complexity in chaotic quantum many‑body systems.