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Growth and collapse of subsystem complexity under random unitary circuits

Jeongwan Haah, Douglas Stanford

TL;DR

This work analyzes how the complexity of reduced density matrices in chaotic quantum dynamics grows and collapses under random brickwork circuits in 1+1 dimensions. By combining rigorous bounds from unitary designs and mutual information with holographic perspectives, it establishes a linear-in-time growth of subsystem complexity for large regions up to $T\le \ell/4$, and a rapid relaxation to near-maximal mixing for small regions around $T\approx \ell/2$. It further links complexity to mutual information profiles and shows, via replica techniques and tailored brickwork circuits, that memory of the circuit can be localized to the entanglement wedge and then abruptly vanishes. The holographic discussion suggests a sharp transition at $T=\ell/2$ for the small subsystem, consistent with a sudden collapse of complexity, while counting arguments demonstrate a rich capacity for distinguishable, high-entropy states. Together, these results illuminate the interplay between chaos, information memory, and geometry-inspired notions of complexity in quantum many-body dynamics.

Abstract

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.

Growth and collapse of subsystem complexity under random unitary circuits

TL;DR

This work analyzes how the complexity of reduced density matrices in chaotic quantum dynamics grows and collapses under random brickwork circuits in 1+1 dimensions. By combining rigorous bounds from unitary designs and mutual information with holographic perspectives, it establishes a linear-in-time growth of subsystem complexity for large regions up to , and a rapid relaxation to near-maximal mixing for small regions around . It further links complexity to mutual information profiles and shows, via replica techniques and tailored brickwork circuits, that memory of the circuit can be localized to the entanglement wedge and then abruptly vanishes. The holographic discussion suggests a sharp transition at for the small subsystem, consistent with a sudden collapse of complexity, while counting arguments demonstrate a rich capacity for distinguishable, high-entropy states. Together, these results illuminate the interplay between chaos, information memory, and geometry-inspired notions of complexity in quantum many-body dynamics.

Abstract

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 d, we prove that the complexity of subsystems of length smaller than half grows linearly in time at least up to but becomes zero after time 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 and then abruptly decay to zero.
Paper Structure (20 sections, 5 theorems, 97 equations, 4 figures)

This paper contains 20 sections, 5 theorems, 97 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Complexity growth of a larger region
  3. Late-time behavior of a smaller interval
  4. Complexity growth as seen by mutual information
  5. Mutual information growth
  6. Spatial profile
  7. A complexity lower bound
  8. Complexity growth according to holographic proposals
  9. Can a subsystem remember the circuit?
  10. Warmup problem
  11. $q,Q$ brickwork circuit
  12. Counting distinguishable states
  13. Using unitary $k$-designs
  14. Using full unitary group
  15. Deferred Proofs
  16. ...and 5 more sections

Key Result

Lemma 2.8

Let $P$ and $Q$ be orthogonal projectors of rank $p$ and $q$ on ${\mathbb{C}}^d$, respectively. Let $U \sim {\mathsf{U}}(d)$ be a Haar random unitary. For any $i$, let $x_i \sim \mathcal{N}(0,{\tfrac{1}{2}})$ be an independent real Gaussian random variable. Then, for any convex function $\phi: {\mat

Figures (4)

  • Figure 1: The graph of the mutual information profile $I(x;T,\ell)$, a trapezoid if $T\neq \ell / 4$ and the full triangle if $T = \ell/4$. Note that $I(x;T,\ell) = I(x;\frac{1}{2} \ell - T, \ell)$. The area under the graph is a lower bound on the complexity.
  • Figure 2: Two domains walls must propagate from the top to bottom and must not go up. They may or may not merge in the bulk. Each have weight $7$ in the figure, giving total weight $14$.
  • Figure 3: Coordinate system for counting the domain walls. The points $C$ and $D$ are present only if the two domain walls merge. The meeting point is neither $C$ nor $D$; these points are immediately before the merger and are used to evaluate $N(x,y;\ell)$.
  • Figure 4: With gates along $2\xi$-separated columns removed, we still have an approximate unitary $k$-design where $k$ is proportional to the depth. Starting with a pure product state, the state of an interval under (b) has much smaller entropy than that under (a).

Theorems & Definitions (16)

  • Remark 1.4
  • Remark 2.5
  • Lemma 2.8
  • Proposition 3.1
  • Remark 7.5
  • Remark 7.7
  • Remark 7.9
  • proof : Proof of \ref{['thm:randomProjectorOverlap']}
  • Lemma A.3: Lemma III.5 of Hayden2004
  • proof
  • ...and 6 more