Quantum chaos and the complexity of spread of states

I. INTRODUCTION

• Spread complexity minimizes wave-function spread over all bases, with the Lanczos-generated Krylov basis uniquely attaining the minimum under the stated evolution conditions.The minimum holds throughout a finite interval for continuous evolution and for all time in discrete evolution.Source: §I · p. 1, §II · p. 2

II. DEFINING SPREAD COMPLEXITY

• The cost assigns positive increasing weights to basis probabilities, so deeper support increases complexity; choosing c_n = n measures average basis depth.Source: §II · p. 2
• The construction avoids the triviality of minimizing separately at each time by minimizing a functional derivative sequence near the initial time.A time-by-time basis choice could simply set its first vector equal to the instantaneous state.Source: §II · p. 2
• Theorem 1 states that the complete Krylov bases minimize the derivative sequence, with equality only for those bases.The proof proceeds inductively by showing that deviations from successive Krylov vectors increase the corresponding derivative.Source: §II · pp. 2–3
• The resulting spread complexity is basis-independent and equals the cost evaluated in the Krylov basis.Source: §II · p. 3

A. Minimization for discrete time evolution

• For discrete unitary evolution, the Krylov basis minimizes the cost at every time step.A non-Krylov basis splits the next Krylov vector across multiple basis elements, increasing cost because the coefficients are strictly increasing.Source: A. Minimization for discrete time… · p. 3
• Discrete-time Krylov minimization applies naturally to quantum circuits, assigning complexity to protocols that transform an input state into a target state.Source: A. Minimization for discrete time… · p. 4

B. Complexity as the exponential of an entropy

• The exponential of the Shannon entropy of basis weights provides an alternative spread complexity whose small-time minimum is also attained in the Krylov basis.This quantity measures the minimum Hilbert-space dimension needed to store the basis-weight distribution.Source: B. Complexity as the exponential of an… · p. 4
• The entropy comparison follows from vanishing early derivatives outside the shared Krylov vectors and strict convexity of the entropy function.The resulting inequality is strict unless the relevant basis element is proportional to the next Krylov vector.Source: B. Complexity as the exponential of an… · p. 4
• The Krylov basis also minimizes the associated participation ratio for small times.Source: B. Complexity as the exponential of an… · p. 4

III. COMPUTING SPREAD COMPLEXITY

• The Lanczos algorithm applies Gram–Schmidt recursively to H^n|ψ(0)⟩, generating the ordered orthonormal Krylov basis.Source: §III · p. 5
• The Lanczos coefficients a_n and b_n define the Hamiltonian’s action in the Krylov basis.Source: §III · p. 5
• In this basis, the Hamiltonian is tridiagonal, called Hessenberg form for finite-dimensional systems.Source: §III · p. 5

A. Krylov basis from the Hessenberg form

• A Hessenberg representation gives a numerically stable route to Lanczos coefficients and Krylov-basis wavefunctions by exponentiating the tridiagonal Hamiltonian.The initial-state coordinates and signs of off-diagonal entries require appropriate basis and phase choices.Source: A. Krylov basis from the Hessenberg form · p. 5

B. Krylov basis from the survival amplitude

• The path-sum algorithm computes moments from weighted transitions, with the bottom node after n steps equal to μ_n = ⟨K0|(iH)^n|K0⟩.Source: Figure 1 · p. 5, B. Krylov basis from the survival… · p. 6
• The recursive survival-amplitude procedure computes Lanczos data from moments but can accumulate rounding errors because it repeatedly divides products of coefficients.The numerical analyses address this by using arbitrary-precision arithmetic.Source: B. Krylov basis from the survival… · p. 6
• The survival amplitude generates Hamiltonian moments, which are converted recursively into Lanczos coefficients using an unwrapped Markov-chain path representation.Odd moments isolate successive a_k coefficients, while even moments isolate b_k coefficients after subtracting previously computable paths.Source: B. Krylov basis from the survival… · pp. 5–6

C. Wave function and Complexity

• Once the Lanczos coefficients and survival amplitude are known, the Krylov amplitudes ψ_n(t) are obtained sequentially from the Lanczos recursion.The survival amplitude is the complex conjugate of ψ_0(t), providing the starting value for the recursion.Source: C. Wave function and Complexity · p. 7
• With c_n = n, spread complexity is the evolving state’s average support depth in the Krylov basis and can be written as the expectation value of a complexity operator.Source: C. Wave function and Complexity · p. 7
• The paper also evaluates an entropic complexity based on the probabilities p_n = |ψ_n|^2.Source: C. Wave function and Complexity · p. 7

D. Survival, TFD and the partition sum

• For TFD evolution, the finite-dimensional Lanczos computation remains within the subspace spanned by {|n,n⟩}.The explored subspace has maximum dimension equal to that of the original Hilbert space.Source: D. Survival, TFD and the partition sum · p. 7
• The TFD survival amplitude is directly related to the partition function, and its survival probability is the spectral form factor.This connection enables Krylov-basis probabilities and links complexity dynamics to spectral diagnostics.Source: D. Survival, TFD and the partition sum · p. 7
• Partition functions and thermal energy moments provide a route connecting spread complexity with geometry, quantum gravity, and spectral classifications of quantum matter.The proposed bridge relies on spread complexity being a functional of the survival amplitude.Source: D. Survival, TFD and the partition sum · p. 8
• For general initial states, complexity growth depends on both the energy spectrum and the structure of energy eigenstates, unlike the spectrum-only TFD case.The additional dependence enters through overlaps between the evolving state and energy eigenstates.Source: D. Survival, TFD and the partition sum · p. 8

IV. RELATION TO KRYLOV COMPLEXITY

• Choosing an inner product maps operator Heisenberg evolution to state Schrödinger evolution in an auxiliary Hilbert space, turning spread complexity into Krylov complexity.Thus any Krylov complexity can be interpreted as quantum state complexity in a suitable auxiliary space.Source: §IV · pp. 8–9
• State spread complexity is more general than conventional Krylov complexity because it uses both Lanczos-coefficient sets, a_n and b_n.Conventional operator growth typically tracks only b_n, so not every state complexity is conventionally Krylov-based.Source: §IV · p. 9
• The approach removes basis ambiguity by minimizing wave-function spread over all orthonormal bases, an idea also applicable to operator complexity.Different operator-complexity notions can be viewed as spread in different bases.Source: §IV · p. 9
• Krylov complexity retains an additional inner-product ambiguity, and the paper does not perform the proposed minimization over that choice.The Wightman inner product is arbitrary, although prior work identifies it with the slowest Krylov-complexity growth.Source: §IV · p. 9

V. ANALYTICAL MODELS

• The identity term in the Lie-algebra Hamiltonian changes only the evolution phase and therefore does not affect quantum complexity.It can nevertheless be used to set the ground-state energy.Source: §V · p. 9
• When the initial state is highest weight, the Krylov basis furnishes a representation of the Hamiltonian’s symmetry group.The tridiagonal Hamiltonian action and fixed ladder-operator algebra allow the Lanczos coefficients to be read off from symmetry.Source: §V · p. 9
• Lie-algebra Hamiltonians generate generalized coherent states, whose Krylov amplitudes are obtained by expanding group-displacement states in an orthonormal basis.This also provides a geometric interpretation of spread complexity.Source: §V · p. 10
• Different diagonal coefficients a_n can produce qualitatively different complexity growth even when off-diagonal coefficients b_n grow linearly.The examples include quadratic, periodic, and exponential growth patterns.Source: §V · p. 10

A. A particle moving in SL(2,R)

• The TFD state of the harmonic oscillator is realized as evolution governed by SL(2,R) symmetry.The oscillator’s Lanczos coefficients match the SL(2,R) representation with h = 1/2.Source: A. A particle moving in SL(2,R) · p. 10
• The time-evolved oscillator TFD is a generalized coherent state whose amplitudes are obtained in the Krylov basis through group-displacement evolution.The parameters α and γ control the growth rates of b_n and a_n, respectively.Source: A. A particle moving in SL(2,R) · pp. 10–11
• Complexity is periodic for the standard oscillator but grows exponentially for the inverted oscillator.The inverted case is treated through analytic continuation from the stable oscillator.Source: A. A particle moving in SL(2,R) · pp. 10–11
• The transition at γ = 2α separates periodic complexity from exponential growth, while the transition-point complexity grows quadratically in time.The free-limit interpretation requires analytic continuation because the partition function is not well defined at ω = 0.Source: A. A particle moving in SL(2,R) · p. 11
• A continuum approximation explains quadratic growth from ballistic propagation on an effective free-particle line when b_n ∼ n.The mapping gives ⟨√n⟩ ∼ t and therefore ⟨n⟩ ∼ t^2.Source: A. A particle moving in SL(2,R) · p. 11

1. Entropic complexity and variance:

• At late times, entropic complexity grows linearly for generic γ and α but only logarithmically when γ = 2α.The free-limit logarithmic entropy implies quadratic growth for the entropic complexity definition.Source: §1 · p. 12
• The entropic complexity has parametrically smaller growth for large scaling dimension h.This comparison concerns the alternative entropic measure rather than the original spread definition.Source: §1 · p. 12
• Variance depends on representation h and indicates that heavy-state distributions are sharply localized around their mean.Near γ = 2α, convergence to the large-time variance limit is t^-2 rather than generically exponential.Source: §1 · p. 12

• The 2j + 1 Krylov basis vectors are identified with the spin-j states |j, −j + n⟩.Source: B. A particle moving in SU(2) · p. 12
• The time-evolving state is constructed using the BCH-decomposed evolution operator and its Krylov-basis coefficients.Source: B. A particle moving in SU(2) · p. 12
• The Krylov-basis support oscillates in time, making the complexity periodic.Source: B. A particle moving in SU(2) · p. 12
• Because SU(2) representations are finite-dimensional, complexity is bounded by the representation dimension and does not exhibit a chaotic plateau.Source: B. A particle moving in SU(2) · p. 12

C. A particle moving in the Heisenberg-Weyl group

• The Heisenberg-Weyl construction identifies the algebra, Lanczos coefficients, evolving state, Krylov coefficients, and survival amplitude used to compute complexity.Source: C. A particle moving in the… · p. 13
• For ω ≠ 0, the excitation potential makes complexity oscillate despite the infinite-dimensional Hilbert space; as ω → 0, complexity grows quadratically without bound.Source: C. A particle moving in the… · p. 13

VI. COMPUTATIONAL MODELS

• These four complexity regimes share the physics of the spectral form factor’s slope, dip, ramp, and plateau, with the linear growth and plateau matching chaotic expectations.Source: §VI · p. 13
• Chaotic TFD states show a linear complexity ramp to a peak of height a e^S, followed by a spectral-rigidity-controlled slope down to a plateau b e^S.Source: §VI · p. 13

• The Schwarzian moments and survival probability are obtained from the partition function and analytically continued thermal quantities.Source: A. The Schwarzian theory · p. 14
• The Schwarzian TFD complexity grows quadratically at both small and large times, as confirmed numerically.Source: A. The Schwarzian theory · pp. 14–15, Figure 5 · p. 15
• The Schwarzian theory’s Lanczos coefficients display a square-root regime at small n and a linear regime at large n, implying corresponding changes in complexity growth.Source: A. The Schwarzian theory · pp. 14–15
• At large n, an − 2bn approaches a constant, matching the free-potential limit of the harmonic oscillator.Source: Figure 4 · p. 15, A. The Schwarzian theory · p. 15
• The large-n Lanczos relation matches the harmonic oscillator’s free limit, so late-time Schwarzian TFD evolution is approximated by motion in SL(2,R).Source: A. The Schwarzian theory · p. 15
• In the infinite-c/β limit, the SL(2,R) description becomes accurate at all times and the Hamiltonian and complexity operator close into a complexity algebra.Source: A. The Schwarzian theory · p. 15
• The quadratic-growth coefficient satisfies a ∝ c and a ∝ σ^2_E, linking the complexity growth rate to the energy variance.Source: Figure 5 · p. 15, A. The Schwarzian theory · p. 15

Lanczos coefficients

• The Lanczos coefficients show a linear small-n regime followed by an O(1)-scale transition to a≈0 and nearly constant b_n.In the linear regime, a_n+d=2b_n∼n; at larger n, b_n slowly decays to zero only near the finite Krylov-chain edge.Source: B. Random Matrices · pp. 16–17, Figure 8 · p. 16, Figure 9 · p. 16

Mechanism

• The linear complexity regime follows from effectively free propagation on the Krylov chain when a_n≈0 and b_n is locally constant.The resulting wave equation produces plane waves moving at constant speed, so the mean Krylov position grows linearly.Source: B. Random Matrices · p. 17

Random-matrix complexity dynamics

• For GUE ensembles, spread complexity grows quadratically at short times, crosses to linear growth, then peaks and descends to saturation at an exponentially large plateau, with peak and plateau scales O(N).Source: B. Random Matrices · p. 17

Relation to spectral form factor

• Complexity’s ramp–peak–slope–plateau parallels the spectral form factor’s slope–dip–ramp–plateau, although their characteristic times need not coincide.The complexity peak occurs at O(N), whereas the spectral-form-factor dip occurs at O(√N), because complexity sums over all Krylov probabilities.Source: B. Random Matrices · pp. 18–19

Spectral correlations

• Removing eigenvalue correlations while preserving the GUE density of states eliminates both the complexity peak and slope, as well as the spectral-form-factor dip and ramp.This comparison identifies spectral correlations as the likely source of the post-peak complexity slope.Source: Figure 12 · p. 18, Figure 13 · p. 18, B. Random Matrices · p. 19

Wavefunction dynamics

• The Krylov-basis wavefunction propagates outward, reflects from the chain edge, and produces the downward complexity slope after the peak.At infinite temperature, entropic complexity can instead continue spreading after the bounce and lack a peak and downward slope.Source: Figure 14 · p. 18, Figure 15 · p. 18, B. Random Matrices · p. 19

Universality classes

• GUE, GOE, and GSE complexity curves transition to their plateaus sharply, smoothly, and with a kink, respectively.The GSE wavefunction can bounce twice before approximate stationarity, linking ensemble-dependent complexity behavior to spectral rigidity.Source: B. Random Matrices · p. 20, Figure 18 · p. 20, Figure 19 · p. 20

Universality classes

• The SYK universality class depends on N mod 8: N mod 8=2 or 6 gives GUE, 0 gives GOE, and 4 gives GSE.Because complexity depends on the ensemble, the displayed calculations choose N=14,18,22,26, which lie in the GUE class.Source: C. SYK Model · p. 21

Lanczos coefficients

• SYK’s Lanczos coefficients have a linear-growth regime followed by a_n≈0 and nearly constant b_n, with the transition at n∼O(1).The relation a_n+d=2b_n∼n makes the early dynamics resemble the free oscillator analyzed earlier.Source: C. SYK Model · p. 21, Figure 22 · p. 21, Figure 23 · p. 21

Complexity dynamics

• SYK in the GUE universality class shows quadratic growth, a linear ramp to a peak, a downward slope, and equilibration at a plateau.The slope and plateau transition are consistent with GUE spectral statistics.Source: C. SYK Model · pp. 21–22, Figure 24 · p. 22

Interpretation

• The SYK results support using complexity dynamics to diagnose random-matrix eigenvalue statistics in chaotic models.The observed slope and plateau shape signal GUE behavior, consistent with earlier SYK spectral-statistics studies [15] [85].Source: C. SYK Model · pp. 21–22

Comparison

• Compared with computational-basis spread complexity, the minimized-over-bases measure reaches exponential values on exponentially long rather than entropy-order timescales.This comparison highlights the role of minimizing spread over all bases [29].Source: C. SYK Model · p. 22

D. Black holes from collapse

• A microcanonical TFD restricts entanglement to a narrow energy window with approximately equal amplitudes, limiting the energy moments used in the analysis.Source: D. Black holes from collapse · p. 22
• For sufficiently short times, the survival probability gives Weyl-Heisenberg-like dynamics and quadratic Krylov complexity growth.The same short-time regime also appears within the SL(2, R) paradigm.Source: D. Black holes from collapse · p. 22
• The intermediate quadratic-growth regime is not established universally because its behavior depends on the ratio of the Lanczos coefficient sets.The studied models satisfy a relation implying quadratic growth over an intermediate time range, but the paper offers only a conjecture.Source: D. Black holes from collapse · p. 23
• With an upper energy bound, sufficiently high moments force Lanczos coefficients to saturate, implying linear complexity growth at a rate of O(Emax).Source: D. Black holes from collapse · p. 23
• Collapse-like states with Emax of order the thermal energy exhibit universal initial quadratic growth and late linear growth controlled by the black-hole mass.The intermediate model-dependent window is absent, and linear growth lasts for exponentially long times in the entropy.Source: D. Black holes from collapse · p. 23

VII. DISCUSSION

• The minimized complexity is computable through Lanczos dynamics and is linked to survival amplitude, energy spectrum, and spectral rigidity in chaotic systems.Its late-time decay distinguishes Gaussian Unitary, Orthogonal, and Symplectic random-matrix ensembles, while the paper proposes extending the analysis to integrable systems and black-hole interiors.Source: §VII · pp. 23–24