Large deviations in quantum dynamics and complexity

Jorge Kurchan; Xiangyu Cao

REVIEW 3 major objections 5 minor 1 cited by

Large deviations in quantum dynamics and complexity

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper argues that large deviations in many-body quantum dynamics split into three distinct phenomena whose equilibration times differ enormously: the full distribution of an observable settles in size-independent O(1) time, monitored m

desk verdict The Mattis-model large-deviation computation is solid and the sliding-cutoff picture is attractive, but the exp(e^N) time scale is a heuristic that the numerics are far too small to support. read the letter →

arxiv 2607.20959 v1 pith:AFUTOEET submitted 2026-07-23 quant-ph cond-mat.stat-mechhep-th

Xiangyu Cao , Jorge Kurchan This is my paper

classification quant-phcond-mat.stat-mechhep-th

keywords largedeviationsquantumdynamicsequilibrationtimescalescomplexityspectralformfactorreturnprobabilityMattismodelkickedIsing

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that 'large deviations' in many-body quantum dynamics split into three distinct phenomena with vastly different equilibration times. The full distribution of an extensive observable settles to its infinite-time form in a time of order one, independent of system size, because quantum parallelism visits rare and typical configurations simultaneously. Under continuous measurement, the distribution of outcomes over a time window settles only at t_max ~ e^N, as in classical systems. The distribution of expectation values—such as the return probability or spectral form factor—settles at the Hilbert-space recurrence time t_max ~ exp(e^N). Before each of these limits, the distribution matches the infinite-time large-deviation form below a sharp, slowly drifting cutoff, and the paper proposes tracking that cutoff as a measure of quantum complexity.

What carries the argument

The argument rests on a statistical-mechanics mapping. The phase factors s_i(t)=e^{-iε_i t} built from the pseudo-energy eigenvalues of the unitary evolution are treated as XY spins living on the torus U(1)^D, where D~e^N is the Hilbert-space dimension. The return probability and the spectral form factor become Hamiltonians of a Mattis model with these spins as degrees of freedom, and the long-time distribution of their values is obtained from the model's entropy density S(-a). The finite-time distribution is then predicted by the sliding-cutoff formula P_{t_max}(a) ~ e^{D S(-a)} for a ≤ a*, with S(-a*) = -D^{-1} ln t_max, which forces the equilibration time to be double-exponential. The sam

What would settle it

Numerically compute the time distribution of the spectral form factor in an integrable (e.g., noninteracting) spin chain and check whether the cutoff a*(t_max) obeys S(-a*) = -D^{-1} ln t_max with a Mattis-model entropy; if the cutoff grows much slower or the distribution fails to match the truncated large-deviation form, the ergodicity assumption is violated and the paper's central prediction for expectation values collapses.

Watch

Extended reading notes

Core claim

The paper's central claim is a trichotomy of equilibration time scales for large deviations in generic, non-integrable many-body quantum dynamics without conservation laws. For an extensive observable A, the full distribution ⟨δ(A(t)-Na)⟩ reaches its long-time (infinite-temperature) form at t ~ O(1), independently of system size N. The distribution of measurement outcomes of a continuously monitored extensive observable over 0 ≤ t ≤ t_max reaches its infinite-time form only at t_max ~ e^N. And the distribution of expectation values ⟨A⟩_t over 0 ≤ t ≤ t_max—including return probability, spectral form factor, autocorrelation function, and operator size—reaches its infinite-time form only at t_

Load-bearing premise

The entire double-exponential time scale and sliding-cutoff prediction rest on the assumption that the sequence of phase factors s_i(t)=e^{-iε_i t} samples the full torus U(1)^D uniformly, i.e., that the time trajectory of the wavefunction is ergodic over the Hilbert-space configuration space; if this ergodicity fails, the exp(e^N) scale does not follow.

Editorial extensions

If this is right

The full distribution of an extensive quantity is not a useful complexity measure: it equilibrates in size-independent O(1) time, unlike classical deterministic systems.
Monitored quantum large deviations reproduce the classical story: atypical measurement outcomes are observed only after exponentially long times t ~ e^N.
Expectation-value spikes—brief, one-shot-observable rare fluctuations—occur on the Hilbert-space recurrence scale t ~ exp(e^N), matching known recurrence bounds.
The slowly drifting cutoff furnishes a concrete, operator-dependent measure of quantum complexity, with a single-parameter (number of independent samples) that is nearly operator independent.
For Hamiltonian systems with a conserved energy, the double-exponential scale survives as long as the initial state has nonzero entropy density, with D replaced by the effective thermal Hilbert-space dimension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the mapping is robust, the same trichotomy should appear in other chaotic quantum models, e.g., random circuits, and the sliding cutoff could be measured in current quantum simulators over times much shorter than exp(e^N) by looking at the frontier position for moderately rare events.
The expectation-value complexity, with its exp(e^N) saturation time, is a natural candidate holographic dual to continued black-hole interior growth; the paper hints at this but does not develop a gravitational dictionary.
A testable corollary: in integrable or many-body-localized systems where the XY-spin trajectory does not sample U(1)^D uniformly, the exp(e^N) cutoff growth should fail or slow dramatically, providing a clean diagnostic of quantum chaos.
The proposed 'number of independent samples' M ≈ ln(t_max/t_c) offers an operator-independent complexity that could be compared across different observables and measurement schemes; verifying that M grows as predicted would distinguish this notion from Krylov/Nielsen complexity.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [Colleague],

You should know two things about this paper before reading it. First, the Mattis-model computation for the infinite-time large deviations of the return probability and spectral form factor is clean, parameter-free, and very likely correct. Second, the paper's headline time scale, t~exp(e^N), is a heuristic extrapolation from that static computation; the numerics (N≤7) are far too small to confirm it. Read it for the analytic result and the conceptual trichotomy, not for the double-exponential claim.

What is actually new: The paper separates three notions of quantum large deviation — the full distribution of an observable, the distribution of measured outcomes, and the distribution of expectation values over a time window — and shows they equilibrate on very different time scales: O(1), e^N, and presumably exp(e^N). The distinction between "potential" and "observed" rare fluctuations via quantum parallelism is worth thinking about. The mapping of the long-time return-probability/SFF distribution to a Mattis model with D XY spins is an elegant trick, and Appendix A solves it exactly in D→∞ without free parameters. The prediction that finite-time distributions are the infinite-time ones with a sliding cutoff is concrete and testable, and the small-scale numerics in Figures 3–4 do support it.

Soft spots. The load-bearing step is Eq. (29): the time distribution of a linear quasiperiodic flow on the U(1)^D torus is assumed to sample the Haar measure with a rare-region hitting time set by the Mattis entropy. For a genuinely chaotic or stochastic process that is the standard extreme-value argument. For a torus translation it is not automatic — Diophantine properties of the eigenphases can change hitting times to small sets. The paper gives no derivation and no direct check of the exp(e^N) scale. The numerics are small (D≤128, tmax≤1e7) and without error bars, so they cannot distinguish exp(e^N) from, say, e^{cN^p}. The general-operator section is also partly circular: the rate function is extracted from the same time series used to verify the cutoff. And the correlation time t_c is set to one without discussion of N-dependence, which would affect the cutoff drift. One more minor internal inconsistency: the abstract says the expectation-value distribution equilibrates at exp(e^N), but Section 3.3 notes the cutoff never disappears—it just approaches a=1.

That said, the central analytic result is independent of the dynamical assumption. If you work on return probabilities, SFFs, or large deviations in Floquet systems, this is worth citing.

My recommendation: send it to peer review. It is coherent, original, and honest about where its evidence stops. A referee should push for a direct test of the hitting-time assumption or at least a caveat that the exp(e^N) scale is a conjecture, and for larger-N numerics if feasible. But this is the kind of paper that generates useful discussion.

Referee Report

3 major / 5 minor

Summary. The paper studies large deviations in many-body quantum dynamics and proposes a trichotomy of time scales. It distinguishes (i) the full distribution of an extensive observable, argued to equilibrate at t ~ O(1) independently of N; (ii) the time distribution of monitored measurement outcomes, whose finite-time cutoff drifts on an e^N scale; and (iii) the time distribution of expectation values such as the return probability and the spectral form factor, whose cutoff drifts on an exp(e^N) scale. The latter is derived by mapping the unitary evolution to variables s_i(t)=e^{-i ε_i t}, treating them as XY spins, and computing the infinite-time distribution from an analytically solvable Mattis model (Appendix A). A classical rare-region argument is then used to predict a sliding cutoff. The paper proposes the evolution of these cutoffs as a measure of quantum complexity. Numerical support is provided for the kicked Ising model.

Significance. If the trichotomy holds, the paper gives a new, operationally defined notion of quantum complexity with time scales distinct from Nielsen and Krylov complexity, and connects large deviations to Hilbert-space recurrence. The analytic Mattis-model solution in Appendix A is a clear strength: it is parameter-free and directly yields the infinite-time distribution for the return probability and SFF. The numerical comparison in Figs. 3-4 supports the distribution prediction. The paper is clearly written and provocative. However, the central exp(e^N) time scale rests on a heuristic hitting-time assumption that is not derived or directly tested, and the mode of convergence to the longtime limit is left ambiguous.

major comments (3)

[§3.3, Eqs. (22)–(29)] The double-exponential time scale is the central claim, but it rests on the unproved assertion that the linear flow s_i(t)=e^{-i ε_i t} samples U(1)^D uniformly and that the first-passage time to {r>a} is exp[D|S(-a)|]. For a Kronecker flow on a torus, ergodicity and hitting-time laws are controlled by Diophantine properties of the eigenphases ε_i; they are not consequences of the Haar measure. The numerics in Figs. 3–4 use D≤128 and t_max≤10^7, which is many orders of magnitude below e^D; they test the shape of P_{t_max}(a) at moderate a but cannot test the claimed exp(e^N) scaling. Please either prove or derive the hitting-time estimate under explicit assumptions, or clearly label it as a conjecture and provide a test of the D-dependence of the cutoff position (e.g., collapse of a* versus D^{-1} ln t_max for several N).
[Abstract and §3.3, Eq. (29)] The abstract states that (iii) 'reach[es] its longtime limit at t∼exp(e^N)'. However, Eq. (29) combined with Eq. (50) (S∼(1/2)ln(1−a)) implies S is unbounded below, so the cutoff a* satisfies a*<1 for every finite t_max and the support of P_{t_max} never equals that of P∞. The text itself notes that the cutoff 'always exists and approaches 1 as t_max→∞'. The mode of convergence is therefore unclear: pointwise convergence of the large-deviation rate on a fixed interval is different from convergence of the probability measure, which can occur much earlier because the missing tail has small probability. Please state precisely in what sense the longtime limit is attained and adjust the abstract accordingly.
[§3.2, Eq. (15)] The monitored-outcome result is presented through a single realization (N=18, Fig. 2) with no ensemble average or error bars. The central prediction of Eq. (15) is a cutoff with f*=N^{-1} ln t_max; to support this, the paper should extract the empirical cutoff and compare it with the predicted relation for more than one N. The same applies to the claim that the result is independent of measurement scheme, which is not tested.

minor comments (5)

[§3.3, near Eq. (29)] The text writes S(−a)∼ln(1−a), but Appendix A Eq. (50) gives (1/2) ln(1−a). Correct the typo.
[Eq. (24)] The notation '−h = r or SFF' is confusing; define h explicitly in the partition function.
[Figures 3–4] Clarify whether panel (b) combines N=5 and N=7 data or shows one; the caption is ambiguous.
[§4] The assertion that t_c scales at most as a power law for generic operators and measurement schemes is not demonstrated; add a reference or a test.
[General presentation] There are several typos ('atpyical', 'an double exponential', 'The R_ij' capital letter, 'Dlnt'). Please proofread.

Circularity Check

0 steps flagged · score 0.0 of 10

Core large-deviation predictions are independent of their inputs; no circular derivation.

full rationale

The core derivation is self-contained and not circular. For the two flagship observables (return probability and SFF), the infinite-time large-deviation rate is computed analytically from the Mattis model (Eqs. (24)-(28), Appendix A) using only the Haar-uniformity assumption for s_i(t) and Gaussian eigenvector weights; no quantity fitted to the r/SFF time series enters this calculation. The finite-time cutoff prediction (29) is parameter-free: it determines a*(t_max) from S(-a*) = -D^{-1} ln t_max and the entropy computed in (26)-(28). The numerical comparisons in Figs. 3-4 are therefore nontrivial tests. The monitored-outcome section similarly compares a histogram to the binomial rate with a t_max-dependent cutoff, with no fitted constants. Self-citations ([24], [26]) are peripheral and appear in citation clusters with independent works, so they are not load-bearing. The general-operator subsection (Fig 5) does not claim an analytic rate prediction; it extracts the rate from the same time series and observes large-deviation scaling and cutoffs. That is a weaker test and an admitted limitation ('we numerically extract its energy large deviation rate function'), but it is not a case of renaming a fitted parameter as a prediction, and the central r/SFF claim does not depend on it. The main unproven assumption—that the quasiperiodic flow s_i(t)=e^{-iε_i t} has rare-region hitting times governed by the Haar entropy—is a physical assumption subject to correctness risk, but it is not circular.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The main theoretical burden is a set of domain assumptions: ergodicity on the U(1)^D torus, ETH for matrix elements, Gaussian eigenvector statistics, and the applicability of the classical rare-region argument to quantum time series. These are stated explicitly but are not derived from the unitary dynamics.

free parameters (1)

t_c (correlation time) = 1
In the Discussion, the number of independent samples is estimated as ln M ≈ ln(t_max/t_c), and t_c is set to 1 in the numerical tests. This is an arbitrary choice that affects the quantitative definition of the complexity measure.

assumptions (6)

domain assumption The long-time dynamics of the XY spin variables s_i(t)=e^{-iε_i t} samples the U(1)^D configuration space uniformly.
Section 3.3: 'In the long time limit, for a generic U, the XY spin variables s_i=s_i(t) sample uniformly the configuration space U(1)^D'. This ergodicity assumption is the basis for using equilibrium statistical mechanics to compute the long-time distribution.
domain assumption Off-diagonal matrix elements of a local operator A in the eigenbasis of U are random with variance 1/D (ETH).
Section 3.3, Eq. (32): |J_ji| ~ D^{-3/2} for i≠j follows from ETH and the assumption D^{-1}Tr[A^2] ~ O(1). This is used to map the general expectation value to an XY spin-glass energy.
domain assumption The initial-state coefficients c_i = ⟨ε_i|ψ⟩ are real Gaussian random variables with zero mean and variance 2^{-N}.
Section 3.3: 'we expect that ⟨ε_i|ψ⟩ to be distributed as real Gaussian random variables with variance 2^{-N} in the N→∞ limit (we checked this numerically)'. This is used to specify the Mattis model for the return probability.
domain assumption The classical rare-region argument (exponential-in-N visit times) applies to the quantum time series of measurement outcomes and expectation values.
Section 2 presents the classical argument and Section 3.2/3.3 'apply the argument of Section 2' to the monitored dynamics and expectation-value dynamics. This is the bridge that converts the equilibrium distribution into a finite-time truncated distribution.
domain assumption The kicked Ising model (5) is a generic chaotic many-body system with no local conserved quantities.
Section 3: 'This is a chaotic quantum many-body system with no local conserved quantities and no small parameters'. This ensures the absence of transport-induced slow relaxation and justifies ETH assumptions.
standard math The Mattis-model partition function is evaluated in the D→∞ limit by a saddle point (Hubbard-Stratonovich).
Appendix A: standard large-D saddle-point analysis of the Mattis model, including the replacement of sums by integrals over the eigenvalue distribution p(v). This is a standard statistical-mechanics technique.

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Cite this review

Pith. "Pith review of Large deviations in quantum dynamics and complexity." pith.science (2026). https://pith.science/paper/AFUTOEET

@misc{pith2026260720959,
  author       = {Pith},
  title        = {Pith review of: Large deviations in quantum dynamics and complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AFUTOEET}},
  note         = {Machine review of arXiv:2607.20959}
}

abstract

We study three definitions of large deviation in many-body quantum dynamics: (i) via the full distribution of an extensive observable, (ii) via the distribution of measurement outcomes (from a continuous monitoring of the observable) over a time interval $t \le t_{\max}$, and (iii) via the distribution of expectation values over $t \le t_{\max}$. In generic systems without conservation laws, the large deviation function (i) reaches its longtime limit at $t \sim \mathcal{O}(1)$, independently of system size $N$. (ii) and (iii) reach their longtime limit at $t \sim e^{ N}$ and $t \sim \exp(e^{N})$, respectively. Before that, there is a {\it sharp} frontier between the explored and unexplored outcomes/expectation values; their distribution equals the longtime limit truncated at values that drift with $t_{\max}$. We propose that the evolution of these values with $t_{\max}$ provides a measure of quantum complexity.

Figures

Figures reproduced from arXiv: 2607.20959 by the authors.

**Figure 2.** Large deviation of the outcomes time sequence in the kicked Ising model [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Return probability in the kicked chaotic Ising model of size [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Spectral form factor (SFF) in the kicked chaotic Ising model of size [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Rare fluctuations of the expectation value [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Rare fluctuations of the auto-correlation function for the kicked Ising [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

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

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Rare fluctuations in quantum walks are ruled by a measurement-induced relative entropy, and applying this to stochastic inflation yields a steady state that violates detailed balance.

Reference graph

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Reviewed August 1, 2026 · model on record in the stance chip above.

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