Dynamics of Loschmidt echoes from operator growth in noisy quantum many-body systems

Lucas S\'a; Takato Yoshimura

arxiv: 2509.01585 · v2 · submitted 2025-09-01 · ❄️ cond-mat.stat-mech · cond-mat.str-el· hep-th· quant-ph

Dynamics of Loschmidt echoes from operator growth in noisy quantum many-body systems

Takato Yoshimura , Lucas S\'a This is my paper

Pith reviewed 2026-05-18 19:36 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.str-elhep-thquant-ph

keywords Loschmidt echooperator growthnoisy quantum systemsdissipative dynamicsquantum chaosFloquet systemsout-of-time-order correlatorsrandom phase model

0 comments

The pith

The operator Loschmidt echo in noisy quantum systems matches the norm of averaged dissipative dynamics and decays in two distinct regimes.

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

The paper establishes an equivalence between the operator Loschmidt echo under noisy unitary evolution and the operator norm of the corresponding dissipative dynamics after averaging over the noise. This equivalence supports a heuristic picture that links the echoes to out-of-time-order correlators and operator growth in generic Floquet systems, holding regardless of dissipation strength. As a result, the echo exhibits Gaussian decay when the noise strength times time is much less than one, transitioning to exponential decay with a rate independent of the noise strength when that product is large. The claims are proven exactly in a solvable model known as the dissipative random phase model for chaotic quantum circuits.

Core claim

We show that the operator Loschmidt echo in noisy unitary dynamics is equivalent to the operator norm of the corresponding dissipative dynamics upon noise averaging. Using a heuristic for generic Floquet systems that connects Loschmidt echoes, out-of-time-order correlators, and operator growth at any dissipation strength, we find two regimes: Gaussian decay for pt ≪ 1 and exponential decay with noise-independent rate for pt ≫ 1. These are rigorously verified in the dissipative random phase model.

What carries the argument

The equivalence of the noise-averaged operator Loschmidt echo to the operator norm of dissipative dynamics, which maps the problem to operator growth analysis.

If this is right

The Loschmidt echo dynamics can be analyzed via dissipative evolution and norm calculations.
Short-time behavior is Gaussian in the scaled time pt.
Long-time decay rate does not depend on noise strength.
The connection between echoes, OTOCs, and operator growth persists under dissipation.

Where Pith is reading between the lines

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

The two-regime structure may appear in other models of noisy quantum dynamics beyond Floquet circuits.
This could provide a way to measure operator growth rates through echo experiments in open systems.
Extensions to systems with conservation laws might reveal different behaviors.

Load-bearing premise

The heuristic picture connecting Loschmidt echoes to out-of-time-order correlators and operator growth holds for generic Floquet systems at any dissipation strength.

What would settle it

Compute the operator Loschmidt echo and the dissipative operator norm in the random phase model and check if they match after noise averaging, or simulate the decay to see if it switches from Gaussian to exponential at pt around 1.

read the original abstract

We study the dynamics of Loschmidt echoes in noisy quantum many-body systems without conservation laws. We first show that the operator Loschmidt echo in noisy unitary dynamics is equivalent to the operator norm of the corresponding dissipative dynamics upon noise averaging. We then analyze this quantity in two complementary ways, revealing universal dynamical behavior. First, we develop a heuristic picture for generic Floquet systems that connects Loschmidt echoes, out-of-time-order correlators, and operator growth, which is valid at any dissipation strength. We assert that the Loschmidt echo has two dynamical regimes depending on the time $t$ and the strength of the noise $p$: Gaussian decay for $pt\ll1$ and exponential decay (with a noise-independent decay rate) for $pt\gg1$. Lastly, we rigorously prove all our results for a solvable chaotic many-body quantum circuit, the dissipative random phase model -- thus providing exact insight into dissipative quantum chaos.

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 gives a clean equivalence between noise-averaged Loschmidt echoes and dissipative operator norms, with an exact proof in one circuit model and a heuristic extension to generic Floquet systems that may need more checks at strong noise.

read the letter

The main point is that noise averaging converts the operator Loschmidt echo into the norm of the corresponding dissipative evolution, and this produces two clear decay regimes in a solvable model. The authors prove the equivalence directly from the noise average and then use it to connect Loschmidt echoes to out-of-time-order correlators and operator growth. They claim the link holds at any dissipation strength. In the dissipative random phase model they derive the Gaussian short-time regime for pt much less than 1 and the later exponential decay whose rate does not depend on p. That exact solution is the most solid part of the work and supplies concrete numbers without free parameters or fitting. It also shows how the same growth exponents that control OTOCs appear in the echo decay. The heuristic picture for generic Floquet circuits is presented as valid across the full range of noise, but it rests on the assumption that operator spreading remains unchanged even when pt is large. The random phase model has enough structure that this may hold there, yet nothing in the argument rules out truncation of operator support or shifts in effective velocity under strong local dephasing in other circuits. That is the main soft spot, and it is worth a referee comment rather than a fatal flaw. The paper is aimed at people who already work on open-system quantum chaos, dissipative many-body dynamics, or Loschmidt echoes as diagnostics. Readers who follow operator growth literature will see the direct tie-in and the exact results as useful. The formal grounding in the model is strong enough that the manuscript deserves a serious referee, even if the generic heuristic section will probably require extra justification or numerical tests. I would send it for peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the dynamics of operator Loschmidt echoes in noisy quantum many-body systems without conservation laws. It first establishes an equivalence between the noise-averaged operator Loschmidt echo in unitary dynamics and the operator norm of the corresponding dissipative dynamics. A heuristic picture is then developed for generic Floquet systems that relates Loschmidt echoes to out-of-time-order correlators and operator growth, asserted to hold at arbitrary dissipation strength; this yields the claim of two regimes (Gaussian decay for pt ≪ 1 and exponential decay with noise-independent rate for pt ≫ 1). All claims are rigorously proven for the exactly solvable dissipative random phase model.

Significance. If the central claims hold, the work supplies a concrete link between Loschmidt echoes, operator growth, and dissipative quantum chaos, identifying universal noise-dependent regimes that could serve as benchmarks for open-system many-body dynamics. The exact solvability in the random phase model is a clear strength, providing parameter-free insight that can be used to test the heuristic in other settings.

major comments (1)

[§3] §3: The heuristic that equates the noise-averaged Loschmidt echo to an operator-norm quantity controlled by the same growth exponents appearing in OTOCs is asserted to remain valid at any dissipation strength. However, the manuscript does not address whether strong local dephasing (pt ≫ 1) can truncate operator support or alter the effective light-cone velocity in a generic Floquet circuit, which would change the late-time decay rate and undermine the noise-independent exponential regime. This assumption is load-bearing for the two-regime claim outside the random phase model.

minor comments (2)

The notation for the operator Loschmidt echo and its relation to the dissipative superoperator should be introduced with an explicit equation in the introduction or §2 to improve readability for readers outside the immediate subfield.
[§4] §4: When presenting the exact solution for the dissipative random phase model, include a brief statement of the circuit depth and gate assumptions that enable the closed-form result, so that the scope of the rigorous proof is immediately clear.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this important point about the range of validity of our heuristic argument. We respond to the major comment below.

read point-by-point responses

Referee: [§3] §3: The heuristic that equates the noise-averaged Loschmidt echo to an operator-norm quantity controlled by the same growth exponents appearing in OTOCs is asserted to remain valid at any dissipation strength. However, the manuscript does not address whether strong local dephasing (pt ≫ 1) can truncate operator support or alter the effective light-cone velocity in a generic Floquet circuit, which would change the late-time decay rate and undermine the noise-independent exponential regime. This assumption is load-bearing for the two-regime claim outside the random phase model.

Authors: We thank the referee for this observation. The exact equivalence between the noise-averaged Loschmidt echo and the operator norm of the corresponding dissipative dynamics (derived in Section 2) holds for arbitrary dissipation strength and does not rely on any assumption about operator growth. The heuristic developed in §3 for generic Floquet circuits connects this norm to the operator growth exponents extracted from OTOCs of the underlying unitary evolution. In the strong-dephasing regime pt ≫ 1 the local dephasing damps coherences, yet after noise averaging the effective spreading of operator support remains controlled by the unitary gates; the light-cone velocity is therefore unchanged and the late-time decay rate of the norm becomes independent of p. We acknowledge that the manuscript does not explicitly discuss possible truncation effects in generic circuits. To address the referee’s concern we have added a paragraph in §3 that explains why strong local dephasing does not modify the effective light-cone or growth exponents in the noise-averaged dynamics, thereby supporting the noise-independent exponential regime. We have also clarified that the two-regime claim is rigorously established only for the dissipative random phase model, while the heuristic provides a physically motivated picture for generic systems. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence from explicit averaging and exact results in independent model

full rationale

The paper's core steps consist of an explicit noise-averaging argument establishing equivalence between the operator Loschmidt echo and the norm of the dissipative dynamics, followed by a heuristic picture (explicitly labeled as such) connecting Loschmidt echoes to OTOCs and operator growth for generic Floquet systems, and finally rigorous exact results derived in the independently solvable dissipative random phase model. None of these steps reduce a claimed prediction or regime to a fitted parameter, self-definition, or self-citation chain by construction; the two-regime assertion for generic systems is presented as a heuristic consequence rather than a tautological output, while the model-specific proofs stand on the circuit structure itself. The derivation remains self-contained against the stated external benchmarks of noise averaging and exact solvability.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The results rest on standard quantum mechanics for unitary and dissipative evolution plus two paper-specific assumptions: absence of conservation laws and validity of the heuristic connection at arbitrary dissipation.

free parameters (1)

noise strength p
The parameter controlling noise probability is varied to delineate the short-time and long-time regimes.

axioms (2)

domain assumption Quantum many-body systems without conservation laws
Explicitly stated as the setting in which the dynamics are studied.
ad hoc to paper Heuristic picture connecting Loschmidt echoes, OTOCs and operator growth holds for generic Floquet systems at any dissipation strength
Introduced to obtain the two-regime picture before the exact proof.

pith-pipeline@v0.9.0 · 5702 in / 1408 out tokens · 55588 ms · 2026-05-18T19:36:54.492487+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.

We first show that the operator Loschmidt echo in noisy unitary dynamics is equivalent to the operator norm of the corresponding dissipative dynamics upon noise averaging... heuristic picture for generic Floquet systems that connects Loschmidt echoes, out-of-time-order correlators, and operator growth
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

the Loschmidt echo has two dynamical regimes depending on the time t and the strength of the noise p: Gaussian decay for pt≪1 and exponential decay (with a noise-independent decay rate) for pt≫1

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.

Reference graph

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