Growth and collapse of subsystem complexity under random unitary circuits

Douglas Stanford; Jeongwan Haah

arxiv: 2510.18805 · v3 · submitted 2025-10-21 · 🪐 quant-ph · hep-th

Growth and collapse of subsystem complexity under random unitary circuits

Jeongwan Haah , Douglas Stanford This is my paper

Pith reviewed 2026-05-18 04:45 UTC · model grok-4.3

classification 🪐 quant-ph hep-th

keywords subsystem complexityrandom unitary circuitsquantum chaosstate complexityholographyscrambling

0 comments

The pith

In chaotic quantum dynamics modeled by random unitary circuits, the complexity of small subsystems grows linearly then collapses to zero at time equal to half their length.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the evolution of state complexity for subsystems in systems undergoing random unitary circuit dynamics from an initial product state. It proves that in one spatial dimension, for a subsystem of length ℓ less than half the total, the complexity grows linearly with time at least until T = ℓ/4, but then becomes zero for times T greater than or equal to ℓ/2 when the local dimension is large. The complementary larger subsystem shows linear growth in complexity persisting to exponentially late times. Holographic methods suggest the small subsystem complexity actually grows linearly until exactly T = ℓ/2 before the sudden drop.

Core claim

For chaotic quantum dynamics modeled by random unitary circuits, the complexity of reduced density matrices of subsystems is studied as a function of evolution time. In 1+1d, the complexity of subsystems of length ℓ smaller than half grows linearly in time T at least up to T = ℓ / 4 but becomes zero after time T = ℓ /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. Holographic correspondence gives evidence that the state complexity of the smaller subsystem grows linearly up to time T = ℓ/2 and then abruptly decays to zero.

What carries the argument

Subsystem complexity defined as the minimum number of local quantum channels to generate a given reduced density matrix from a product state to a good approximation, applied to random unitary circuits.

If this is right

Small subsystems experience a collapse in complexity after a time set by their size, indicating a return to simplicity.
Large subsystems sustain complexity growth, showing they remain entangled with the environment.
The linear growth phase matches expectations for information propagation at finite speed.
The abrupt collapse may mark the completion of scrambling within the subsystem light cone.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This behavior could imply that small subsystems effectively thermalize and then decohere or simplify at specific times.
Similar patterns might appear in other models of quantum dynamics beyond random circuits.
Experimental verification could involve measuring effective circuit depths needed to prepare subsystem states in quantum simulators.

Load-bearing premise

The central results rely on taking the limit of large local dimension and on modeling the dynamics by random unitary circuits applied to an initial product pure state.

What would settle it

A direct computation of the minimal number of local channels for a small subsystem state at time T = ℓ/2 in a finite-dimensional circuit simulation to verify if the complexity is exactly zero.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves linear growth of small-subsystem complexity up to T=ℓ/4 and exact collapse to zero at T=ℓ/2 in 1+1d random circuits, but only in the large local dimension limit.

read the letter

This paper gives a proof that in 1+1d random unitary circuits, the complexity of a small subsystem of size ℓ grows linearly at least until time ℓ/4, then drops to zero after time ℓ/2, all in the limit of large local dimension. The larger complementary subsystem keeps growing linearly for exponentially long times. They also offer holographic evidence that the small subsystem complexity should keep growing all the way to ℓ/2 before collapsing. The new part is the exact collapse time at ℓ/2 for the small subsystems and the lower bound on the growth period. The proof seems to work by tracking how the light cones from the initial product state depolarize the reduced density matrix. Once the entire subsystem is covered by the causal influence from the random gates, in the large-d limit it becomes a product of maximally mixed states, which has zero complexity by definition. The holographic suggestion is that the growth continues longer than the proven ℓ/4 bound. One limitation is that everything hinges on taking the local dimension to infinity. At finite d the reduced state will have some residual correlations even after the light cones overlap, so the complexity won't reach exactly zero. The drop will be sharp but not to zero. The linear growth lower bound looks more solid because it only needs to count the size of the depolarized region. This is for researchers working on quantum complexity measures and their connection to scrambling or holography. Someone looking for exact results in random circuit models will find the 1+1d proof useful. It is not a broad claim about all chaotic systems but a controlled calculation in a specific setup. I would send it to peer review. The core claim is well-motivated and the large-d limit is stated clearly, so referees can check the details of the proof and discuss how much the limit matters for applications.

Referee Report

2 major / 2 minor

Summary. The manuscript studies the time-dependent state complexity of subsystems in 1+1d random unitary circuits initialized in a product pure state. It proves that subsystems of length ℓ < L/2 exhibit linear complexity growth at least up to T = ℓ/4, with complexity collapsing exactly to zero for T > ℓ/2 in the large local dimension limit, while complementary subsystems of length > L/2 maintain linear growth until exponentially late times. Holographic calculations are used to argue that the growth phase for the smaller subsystem extends all the way to T = ℓ/2 before an abrupt drop.

Significance. If the central claims hold, the work supplies the first rigorous proof of both linear growth and exact collapse of subsystem complexity in a chaotic circuit model, together with a holographic indication that the proven lower bound on growth time is not tight. The large-d limit enables parameter-free statements about depolarization and complexity, which is a clear technical strength. These results bear directly on the dynamics of quantum complexity, scrambling, and the circuit-holography correspondence.

major comments (2)

[Abstract and 1+1d proof] Abstract and the 1+1d proof: the exact vanishing of complexity for T > ℓ/2 is obtained only after taking the large local dimension limit d → ∞, in which random unitaries perfectly depolarize the causal region so that the reduced state is exactly a product of maximally mixed states. The manuscript must supply an explicit bound showing that the trace distance (or other distance entering the complexity definition) to the nearest product state vanishes as d → ∞ at fixed T > ℓ/2; without this, it remains unclear whether the collapse survives a controlled large-d expansion or is an artifact of the strict limit.
[1+1d analysis] The linear-growth lower bound up to T = ℓ/4 is established by counting the size of the depolarized boundary region. The same large-d averaging is invoked here; the manuscript should verify that the counting argument remains valid when the circuit ensemble average is taken before the d → ∞ limit, rather than after, to ensure the lower bound is not weakened by residual correlations at finite but large d.

minor comments (2)

[Introduction] The definition of state complexity via minimum number of local channels should be stated with an explicit error tolerance (e.g., trace distance < ε) already in the introductory section, so that the zero-complexity claim is unambiguous.
[Throughout] Notation for the subsystem length ℓ versus total system size L is used interchangeably in places; a single consistent symbol and a clear statement of the ℓ < L/2 regime would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below. We believe these points can be clarified with additional bounds and remarks, which we will incorporate in a revised version.

read point-by-point responses

Referee: [Abstract and 1+1d proof] Abstract and the 1+1d proof: the exact vanishing of complexity for T > ℓ/2 is obtained only after taking the large local dimension limit d → ∞, in which random unitaries perfectly depolarize the causal region so that the reduced state is exactly a product of maximally mixed states. The manuscript must supply an explicit bound showing that the trace distance (or other distance entering the complexity definition) to the nearest product state vanishes as d → ∞ at fixed T > ℓ/2; without this, it remains unclear whether the collapse survives a controlled large-d expansion or is an artifact of the strict limit.

Authors: We agree that an explicit bound strengthens the presentation. In the large-d limit the reduced state on the causal past becomes exactly a product of maximally mixed states because each local gate fully depolarizes its support. The complexity definition is based on approximation to a given accuracy, so we can bound the trace distance to the nearest product state by O(1/d) uniformly for fixed T > ℓ/2 (arising from the probability of non-depolarizing outcomes in the random unitary ensemble). This bound vanishes as d → ∞, confirming that the exact collapse is not an artifact. We will add this explicit estimate, together with a short appendix deriving the distance bound from the known moments of the Haar measure, in the revised manuscript. revision: yes
Referee: [1+1d analysis] The linear-growth lower bound up to T = ℓ/4 is established by counting the size of the depolarized boundary region. The same large-d averaging is invoked here; the manuscript should verify that the counting argument remains valid when the circuit ensemble average is taken before the d → ∞ limit, rather than after, to ensure the lower bound is not weakened by residual correlations at finite but large d.

Authors: The counting argument identifies the minimal number of gates that must act on the boundary light-cone to produce a non-trivial reduced state; this combinatorial lower bound is independent of d. For any finite d the probability that a given boundary site remains correlated is exponentially small in d, so the ensemble-averaged complexity still grows at least linearly up to T = ℓ/4. Taking the circuit average first and then d → ∞ therefore does not weaken the bound; the residual correlations contribute only sub-leading corrections that vanish in the limit. We will add a clarifying paragraph after the counting argument explaining the order of limits and why the lower bound survives. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The central claims rest on explicit light-cone propagation arguments in random unitary circuits: after time T = ℓ/2 the entire subsystem lies inside the causal region of the boundaries, and the large-d limit forces the reduced state to be exactly the product of local maximally mixed states. Because the complexity is defined as the minimal number of local channels needed to produce the target reduced state from some product state, a product target requires zero channels by definition; this is a direct logical consequence of the scrambling calculation rather than a redefinition that assumes the result. The lower bound of linear growth up to T = ℓ/4 follows from counting the volume of the depolarized boundary region inside the light cones, again without fitted parameters or self-referential inputs. The holographic discussion is explicitly labeled as supporting evidence, not part of the primary derivation. No load-bearing self-citations, ansatzes, or renamings appear in the proof chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard modeling assumptions of quantum information rather than new free parameters or invented entities.

axioms (2)

domain assumption Random unitary circuits faithfully model chaotic quantum dynamics
Invoked to study complexity evolution of reduced states.
domain assumption Initial global state is a product pure state
Stated explicitly as the starting point for the evolution.

pith-pipeline@v0.9.0 · 5684 in / 1242 out tokens · 36778 ms · 2026-05-18T04:45:31.864973+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

In 1+1d, we prove that the complexity of subsystems of length ℓ smaller than half grows linearly in time T at least up to T=ℓ/4 but becomes zero after time T=ℓ/2 in the limit of a large local dimension
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using holographic correspondence, we give some evidence that the state complexity of the smaller subsystem should actually grow linearly up to time T=ℓ/2 and then abruptly decay to zero

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Evaporating Black Hole Interior and Complexity Evolution
hep-th 2026-05 unverdicted novelty 6.0

In a JT gravity model with an EoW brane, black hole interior complexity grows linearly until the Page time then decays exponentially, with fluctuations growing large afterward and signaling loss of self-averaging.