Chaos, Complexity, and Random Matrices

Jordan Cotler; Nicholas Hunter-Jones; Junyu Liu; Beni Yoshida

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Chaos, Complexity, and Random Matrices

Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu, Beni Yoshida

arXiv:1706.05400v2Jun 16, 2017hep-thcond-mat.stat-mechquant-ph

TL;DR

The paper asks how to characterize quantum chaos, scrambling, randomness, and complexity across early and late times. It analytically studies GUE Hamiltonian evolution using OTOCs, spectral quantities, and frame potentials, and introduces k-invariance as a random-matrix criterion. The analysis captures late-time chaotic behavior qualitatively but exhibits unphysical early-time behavior and apparent loss of spatial and temporal locality.

Problem
The relation between OTOC-based early-time chaos and spectral-statistics-based late-time chaos remains unclear, limiting a unified information-theoretic definition of quantum chaos.
Method
The paper analytically studies time evolution by GUE Hamiltonians through spectral form factors, OTOCs, frame potentials, and complexity bounds, using GUE Haar-invariance for tractability.
Results
The analysis finds qualitatively correct late-time behavior, but GUE dynamics show an O(1) scrambling time, faster-decaying 4-point OTOCs, and apparent breakdown of spatial and temporal locality.
Takeaways & Limitations
k-invariance provides an information-theoretically precise criterion for when quantum dynamics admit an effective random-matrix description and may connect early- and late-time chaos.
Takeaways & Limitations
The GUE random-matrix description has unphysical early-time behavior, including an O(1) scrambling time and apparent loss of spatial and temporal locality.

Abstract

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Chaos and complexity entail an entropic and computational obstruction to describing a system, and thus are intrinsically difficult to characterize. In this paper, we consider time evolution by Gaussian Unitary Ensemble (GUE) Hamiltonians and analytically compute out-of-time-ordered correlation functions (OTOCs) and frame potentials to quantify scrambling, Haar-randomness, and circuit complexity. While our random matrix analysis gives a qualitatively correct prediction of the late-time behavior of chaotic systems, we find unphysical behavior at early times including an $\mathcal{O}(1)$ scrambling time and the apparent breakdown of spatial and temporal locality. The salient feature of GUE Hamiltonians which gives us computational traction is the Haar-invariance of the ensemble, meaning that the ensemble-averaged dynamics look the same in any basis. Motivated by this property of the GUE, we introduce $k$-invariance as a precise definition of what it means for the dynamics of a quantum system to be described by random matrix theory. We envision that the dynamical onset of approximate $k$-invariance will be a useful tool for capturing the transition from early-time chaos, as seen by OTOCs, to late-time chaos, as seen by random matrix theory.

1 Introduction

The paper uses GUE Hamiltonian dynamics to connect OTOCs, spectral statistics, Haar-randomness, frame potentials, and complexity, while introducing k-invariance as a random-matrix diagnostic. It finds qualitatively correct late-time behavior but unphysical early-time features and locality loss.

Motivation: The paper addresses the unclear relation between spectral statistics and OTOCs as part of a broader information-theoretic characterization of quantum chaos.The goal includes scrambling, chaotic correlations, complexity, approximate randomness, and random-matrix universality.
Method: GUE Hamiltonians provide analytical traction because Haar-invariance makes ensemble-averaged dynamics basis-independent.This allows averaged correlators to be expressed through spectral form factors.
Results: GUE 4-point OTOCs decay faster than 2-point correlators, unlike findings for local quantum Hamiltonians.The paper also derives relations between averaged OTOCs and spectral form factors for arbitrary quantum systems.
Results: GUE correlators are independent of operator time-ordering, implying that the ensemble ignores both spatial and temporal locality.This behavior follows from the basis independence of the GUE-averaged correlation functions.
Results: The ensemble forms an approximate k-design at intermediate times but deviates from a k-design at late times, while circuit-complexity bounds grow quadratically in time.These results motivate k-invariance as a more suitable probe of early- and late-time chaos.
Method: The paper analytically computes GUE spectral form factors, OTOCs, correlation functions, frame potentials, and complexity growth.It computes infinite- and finite-temperature spectral quantities and uses frame potentials to study approximate designs and Haar-randomness.

2 Form factors and random matrices

The paper develops analytic random-matrix diagnostics for spectral form factors, connecting their ramp, dip, and plateau structure to quantum-chaos observables and design timescales. It specializes to GUE statistics and identifies approximation boundaries, especially at finite temperature.

2 Form factors and random matrices: Random-matrix spectral statistics are used to study late-time quantum-chaos behavior, with SYK exhibiting the slope, dip, ramp, and plateau structure characteristic of GUE statistics.These features describe early decay, intermediate minimum, linear recovery, and late-time saturation of the 2-point form factor.
2.1 Random matrix theory: The GUE is selected as the least restrictive Hamiltonian symmetry class, while extensions to GOE and GSE are left for future work.The ensemble is defined by L × L random Hermitian matrices with Gaussian-distributed diagonal and off-diagonal entries.
2.1 Random matrix theory: GUE integration is simplified by unitary-conjugation invariance, allowing the Hamiltonian measure to be expressed using eigenvalues, Haar measure, and the Vandermonde determinant.The large-L spectral density follows Wigner’s semicircle law, and the spectral 2-point function separates into disconnected and sine-kernel pieces.
2.2 Spectral form factors: The 2-point spectral form factor is defined from the analytically continued partition function and, at infinite temperature, is related to the Fourier transform of the spectral 2-point function.Using the semicircle law yields a Bessel-function contribution whose oscillations decay at late times.
2.2 Spectral form factors: The analysis uses an unnormalized finite-temperature form factor because analytic control is available for separately averaged numerator and denominator, rather than for the quenched ratio.The paper notes that the quenched form factor is the correct object conceptually, but studies R2 under this tractable approximation.
2 Form factors and random matrices: The paper relates form factors to correlation functions and frame potentials, using these analytic expressions to extract timescales for approximate designs and study Haar-randomness.Its 2k-point form factors include both connected and disconnected spectral contributions.
2.3 4-point spectral form factor at infinite temperature: The GUE 4-point form factor supplies timescales relevant to k-designs, with plateau time tp = 2L and plateau value 2L2 − L.At the dip time, the form factor satisfies R4(td) ≈ L, and its late-time rise is quadratic in t rather than linear.

3 Out-of-time-order correlation functions

The paper connects averaged OTOCs to spectral form factors and shows that GUE dynamics reproduce global spectral behavior while erasing spatial and temporal locality.

3.1 Spectral form factor from OTOCs: Averaged 2k-point OTOCs provide a direct route to corresponding spectral form factors, linking spectral statistics with physical correlators.The relation also offers a practical method for estimating form factors from correlation functions.
3.1 Spectral form factor from OTOCs: A 2-point form factor can be estimated using only a few random unitary or 2-design operators because the estimation error is suppressed by 1/L.Random Clifford operators provide an example of a suitable 2-design ensemble.
3.3 Scrambling in random matrices: GUE dynamics rapidly delocalize information, causing time-ordered and out-of-time-ordered correlators to coincide and eliminating local temporal correlations.The same behavior reflects the GUE’s strong sensitivity to global-scale rather than local physics.
3.2 Correlators for GUE Hamiltonians: GUE correlators can be written solely in terms of spectral form factors because the GUE measure is Haar-invariant under unitary conjugation.This invariance is the key ingredient and does not require assumptions about Pauli-operator locality.
3.3 Scrambling in random matrices: The 2-point correlation decay time t2 is O(1), implying that information is lost on an O(1) timescale for local observers under GUE evolution.This behavior is identified as differing from strongly chaotic large-N systems studied in black hole physics.

4 Frame potentials and random matrices

The paper uses frame potentials and spectral form factors to study how GUE evolution approaches Haar-randomness. GUE ensembles can transiently approximate k-designs, but their nonmonotonic and late-time behavior exposes limitations of k-designs as chaos diagnostics.

4.2 Frame potentials for the GUE: The GUE frame potentials are computed analytically from spectral form factors, with dominant behavior governed by the 2-point and 4-point form factors.The analysis extracts the relevant timescales for approach to Haar values.
4.2 Frame potentials for the GUE: At the dip time, GUE frame potentials approach Haar values and form approximate k-designs for some k, but they later deviate from k-design behavior.The late-time deviation follows from the distinct late-time behavior of spectral form factors.
4.1 Overview of QI machinery: Frame potentials quantify how closely a unitary ensemble reproduces Haar moments and therefore how closely it forms a unitary k-design.They measure the 2-norm distance between the ensemble’s k-fold channel and the Haar k-fold twirl.
4.1 Overview of QI machinery: Equality with the Haar frame-potential value holds if and only if the ensemble is a k-design.The deviation from the Haar value therefore measures the ensemble’s failure to reproduce the target Haar moments.
4.2 Frame potentials for the GUE: Frame potentials generated by fixed Hamiltonians are not generically monotonic, unlike those of random local quantum circuits.The paper proposes an alternative quantity expected to decrease monotonically at late times.
4.4 Frame potentials at finite temperature: Finite-temperature frame potentials are defined using a thermal density matrix and normalized to recover the infinite-temperature form factor as β → 0.The resulting thermal form factor has a β-independent late-time value.
4.5 Time scales from GUE form factors: At infinite temperature, the first 1-design time is set by the first zero of J1(2t), t ≈ 1.92, while a scrambling-like deviation begins at O(1) time.The universal t ≈ 1.92 scale is identified as an artifact of working at infinite temperature.

5 Complexity and random matrices

The paper uses frame potentials and counting arguments to lower-bound the circuit complexity of GUE time evolution. The analysis finds quadratic complexity growth for an extended period, while noting that this behavior is unphysical for local dynamics.

Complexity bounds: Frame potentials quantify ensemble randomness and provide lower bounds on the number of distinguishable unitary operators, which imply circuit-complexity lower bounds.The counting argument is rigorous but can yield loose bounds.
Complexity bounds: Random local circuits are used as a contrasting model whose complexity growth captures fast scrambling and complexity growth qualitatively.The ensemble complexity and the complexity of an individual unitary are distinct, but the former approximately lower-bounds typical-unitary complexity.
Small-k behavior: At t ∼ O(1), the first dip of the frame potential yields a lower bound on complexity based on the small-k approximation.The approximation uses R_2k ≃ (R_1)^2k up to the first dip time.
Complexity growth: The GUE complexity lower bound grows at least logarithmically in t up to the thermal dip time.This estimate uses the asymptotic envelope R_1(t) ∼ 1/t^3/2.
Complexity growth: The large-k analysis predicts quadratic quantum-complexity growth, continuing until near saturation at complexity ∼ L.The estimate relies on a heuristic bound whose validity at large k remains unclear and is assumed only up to the dip time.

6 Characterization of Haar-invariance

The paper introduces k-invariance as a moment-level characterization of Haar-invariant dynamics and connects it to OTOCs, frame potentials, and spectral statistics. In a chaotic spin system, approximate 1-invariance emerges at late times, although exact or monotonic k-invariance is not generally expected.

GUE dynamics: A typical GUE Hamiltonian is non-local, so local operators delocalize essentially immediately and its OTOCs can decay faster than 2-point correlators.The GUE therefore obscures spatial and temporal locality in correlation functions.
Definition: k-invariance means that an ensemble of unitaries appears Haar-invariant up to its k-th moments.Haar invariance is recovered when an ensemble is k-invariant for every k ≥ 1.
Physical meaning: After the k-invariance time, 2k-point OTOCs can be determined solely by spectral statistics and become insensitive to operator locality and time-ordering.The same spectral-statistics reduction applies to k-th frame potentials for k-invariant ensembles.
Diagnostic: Frame potentials provide a quantitative test of k-invariance through the distance between an ensemble and its Haar-invariant extension.The stated inequality becomes an equality exactly when the ensemble is k-invariant.
Caveats: The distance measure used for k-invariance is a 2-norm quantity, whereas a more rigorous analysis would require the diamond distance.The diamond norm is difficult to compute generally, and its treatment is left for future work.
Spin-system example: In the chaotic spin system, the frame potential approaches but does not equal its Haar-invariant counterpart at late times, with the difference decreasing as system size increases.The numerics use a modest system of n = 6 spins, and the authors expect k-invariance at late times for large N.
Caveats: The k-invariance measure is not monotonic under time evolution, and realistic local Hamiltonians are not expected to become exactly k-invariant even at very late times.For local systems, late-time infinite-temperature OTOCs can remain O(1/N) for local operators overlapping with the Hamiltonian.

7 Discussion

The discussion presents Haar-invariance as the feature enabling GUE calculations and proposes k-invariance as a bridge between local early-time chaos and delocalized late-time random-matrix behavior. It also identifies gravitational systems as a future testing ground.

Discussion: Haar-invariance makes GUE ensemble dynamics basis-independent, enabling analysis of OTOCs, frame potentials, randomness, and complexity.The resulting dynamics do not respect a tensor-factor decomposition and immediately delocalize quantum information.
Discussion: GUE dynamics capture features of the long-time physics of local systems after correlations have become delocalized, but not their local early-time behavior.The paper distinguishes pre-scrambling local correlations from post-scrambling delocalized correlations.
Discussion: The proposed transition between early- and late-time regimes is the dynamical onset of approximate Haar-invariance, characterized through k-invariance.This provides a moment-level characterization of when random-matrix descriptions become applicable.
Future directions: The authors propose testing whether gravitational systems exhibit k-invariance and identifying late-time behavior produced by gravitational universality.The broader goal is to use random matrices to characterize chaos and complexity in local quantum systems.

A.1 Scrambling

The paper defines scrambling through the decay of local-operator OTOCs and relates it to information recovery in the Hayden–Preskill setting. It emphasizes that non-local operators can decay earlier than local ones.

Definition: Scrambling occurs when the OTOC becomes O(ε), with ε ≪ 1, for all pairs of local operators.The initial OTOC value at t = 0 is 1.
Scrambling time: Local-operator OTOCs are often the slowest to decay, while non-local-operator OTOCs may already be O(ε) by the scrambling time.In 0-dimensional O(1)-local systems, the scrambling time is lower-bounded by O(log(n)).
Information recovery: In the Hayden–Preskill setting, small two-point correlators imply no reconstruction of A from D, whereas small OTOCs imply reconstruction from D and M.The distinction reflects whether the early radiation memory M is available.
Information recovery: Scrambling therefore implies recovery of local quantum information through local measurements on Hawking radiation, and random unitaries typically produce the required small OTOCs.This connects OTOC decay to the Hayden–Preskill reconstruction protocol.

A.2 Unitary designs

A unitary 2-design is characterized by matching Haar-ensemble OTOCs, linking approximate Haar-randomness to scrambling while distinguishing their timescales and operator requirements.

A unitary ensemble is a 2-design when its averaged OTOCs match Haar averages for every pair of Pauli operators.
A typical 2-design scrambles because single-instance OTOCs are typically 1/L in magnitude and ensemble-averaged OTOCs are 1/L^2.
Scrambling can occur earlier than the 2-design timescale because it requires OTOC = O(ϵ), whereas 2-design behavior approaches O(1/L).
Scrambling requires small OTOCs only for local operators, while a 2-design makes OTOCs small for all Pauli-operator pairs.

A.3 Approximate 2-designs

Approximate 2-designs quantify closeness to Haar channels through norms and frame potentials, but typical small OTOCs do not by themselves guarantee local-operator scrambling.

An approximate 2-design is an ensemble whose quantum channel is close to the Haar channel under a chosen norm.
The frame-potential difference equals the squared 2-norm distance between ensemble and Haar channels.
A δ-approximate 2-design in the 2-norm implies that OTOCs are typically small, but local-operator OTOCs may decay more slowly.
Guaranteeing scrambling requires a δL-approximate 2-design in the 2-norm under an assumption about vanishing cross-OTOCs; the diamond norm is an alternative.

B Information scrambling in black holes

The paper connects correlation decay to information retrieval in black-hole dynamics, representing evolution as a four-partite state whose information becomes nonlocally distributed.

The unitary state representation maps an n-qubit evolution operator to a pure state on a 2n-qubit Hilbert space partitioned into A, B, C, and D.
Averaged two-point correlators are related to Rényi-2 mutual information between subsystems of the unitary state.
Decay of two-point correlators implies that Bob cannot reconstruct Alice’s quantum state from the considered radiation-access scenario.
OTOC decay leads to large I^(2)(A,BD), enabling the possibility of decoding Alice’s state using both early radiation B and new radiation D.
After scrambling, information injected into A is nonlocally hidden across C and D, and accessing any two of B, C, and D suffices for reconstruction.

C Spectral correlators and higher frame potentials

The paper derives spectral correlators for the GUE using kernel methods and Fourier transforms, then uses them to control form factors while tracking approximation errors and temperature limits.

The GUE n-point spectral correlation function is expressed through a kernel whose large-L limit uses the sine kernel off-diagonal and Wigner semicircle on-diagonal.
Spectral form factors are computed by expanding the kernel determinant and Fourier transforming sums of kernel products.
A cutoff regularizes the eigenvalue integral, enabling large-L higher-point calculations but introducing errors relative to the exact answer.
At infinite temperature, the box approximation is analytically controlled at early times O(1) and late times greater than O(L).
The finite-temperature modified kernel is reliable for small β but may fail when large β invalidates short-distance integral dominance.

C.1 Expressions for spectral correlators

The paper derives GUE spectral form factors at infinite and finite temperature, including the 2-point and 4-point functions and their asymptotic regimes. The 4-point form factor rises quadratically at late times, unlike the linear 2-point ramp.

2-point form factors: The GUE 2-point spectral form factor is computed at infinite temperature and extended to finite temperature using spectral-kernel methods.The finite-temperature calculation uses a short-distance-limit kernel to capture long-time correlations.
4-point form factors: The GUE 4-point spectral form factor is obtained by separating coincident eigenvalues, evaluating kernel determinants, and Fourier transforming the result.The calculation yields expressions for both R4 and its coincident-eigenvalue contribution R4,1.
Asymptotic regimes: At early and late times, the 4-point form factor is simplified by dropping terms subdominant in the large-L expansion.The resulting approximation applies at early times of O(1) and late times beyond the scale indicated in the passage.

C.2 Expressions for higher frame potentials

The paper derives higher GUE frame potentials as combinations of spectral form factors and analyzes their early-, intermediate-, and late-time behavior. It also supplies the third frame potential and the Haar-integration ingredients used in these calculations.

Second frame potential: The second frame potential is computed as a lengthy expression involving 2-point, 4-point, and higher spectral correlators.The full formula is organized as terms with different powers of L and products of spectral form factors.
Second frame potential: Keeping terms unsuppressed in 1/L gives the leading early-time behavior of the second frame potential.The early-time analysis uses the scalings R2 ∼ L2, R4 ∼ L4, R4,1 ∼ L3, and R4,2 ∼ L2.
Time regimes: At the dip time, suppression of the form-factor terms makes the second frame potential approach its Haar value.This conclusion follows from the large-L behavior of the relevant spectral form factors.
Time regimes: At late times, substituting the limiting spectral form factors gives a large-L second frame potential of approximately 10.The strict late-time limits used are R2 → L, R4 → 2L2 − L, and R4,1, R4,2 → L.
Numerical benchmark: The late-time finite-L expression determines the value to which numerical estimates should converge as the sample size increases.This provides the analytic benchmark for the frame-potential numerics.
Third frame potential: The full third frame potential is given separately in the appendix.Its leading early-time behavior is introduced, but the supplied passage does not include the complete formula.
Haar integration: The frame-potential calculation uses Haar moments expressed through unitary Weingarten functions indexed by permutations and integer partitions.The first frame potential requires explicit Weingarten values such as Wg({1,1}) and Wg({2}).
Numerical checks: The section concludes with numerical checks of the derived form-factor and frame-potential formulae.These checks connect the analytic expressions to finite-sample computations.

D.1 Form factors and numerics

Numerical tests compare analytic GUE form-factor and frame-potential expressions with finite-sample calculations. The approximations reproduce broad time-regime behavior, while the ramp remains the main source of deviation, especially at finite temperature.

D.1 Form factors and numerics: The analytic GUE form-factor expressions require numerical checks because they rely on large-L and kernel approximations.The appendix compares these approximations with GUE samples across temperatures and system sizes.
D.1 Form factors and numerics: At infinite temperature, numerics reproduce early-time oscillations, the dip, ramp, and plateau, but the predicted ramp slope differs from the numerical result.The discrepancy persists as L increases, so it is not attributed to finite-L sampling effects.
D.1 Form factors and numerics: The modified ramp function ˜r2(t) = 1 − 2t/πL better matches the numerical ramp near the dip.Its error is suppressed as L →∞, although disagreement remains near the transition to the plateau.
D.1 Form factors and numerics: The connected 2-point form factor displays early-time quadratic growth, intermediate linear growth, and a late-time constant plateau.The closed-form approximation mainly captures the linear regime before the plateau.
D.1 Form factors and numerics: The early-time quadratic behavior is not important for the paper’s GUE correlation-function and frame-potential analysis.The passage identifies this behavior as an independent physical question for future study.
D.1 Form factors and numerics: At finite temperature, analytic expressions agree well with numerics at early and late times but deviate around the dip and ramp.The modified ramp improves agreement at small β, while increasing β reveals an incompletely understood β-dependence of the slope.
D.1 Form factors and numerics: Finite-temperature expressions are trusted only in the short-distance approximation and primarily for small β.The appendix also reports numerical checks of the 4-point form-factor expression.
D.2 Frame potentials and numerics: Frame-potential numerics are harder than form-factor numerics because they depend on eigenvectors and require double sums over ensemble elements.For the k-th frame potential, sample sizes greater than L2k are needed, motivating extrapolation in ensemble size.