# Spread complexity and quantum chaos for periodically driven spin-chains

Amin A. Nizami∗ Department of Physics, Ashoka University, Rajiv Gandhi
Education City, Rai, NCR, India 131029 Ankit W. Shrestha† Department of
Physics, Ashoka University, Rajiv Gandhi Education City, Rai, NCR, India
131029

###### Abstract

The complexity of quantum states under dynamical evolution can be investigated
by studying the spread with time of the state over a pre-defined basis. It is
known that this complexity is minimised by choosing the Krylov basis, thus
defining the spread complexity. We study the dynamics of spread complexity for
quantum maps using the Arnoldi iterative procedure. The main illustrative
quantum many-body model we use is the periodically kicked Ising spin-chain
with non-integrable deformations, a chaotic system where we look at both local
and non-local interactions. In the various cases we find distinctive behaviour
of the Arnoldi coefficients and spread complexity for regular vs. chaotic
dynamics: suppressed fluctuations in the Arnoldi coefficients as well as
larger saturation value in spread complexity in the chaotic case. We compare
the behaviour of the Krylov measures with that of standard spectral
diagnostics of chaos. We also study the effect of changing the driving
frequency on the complexity saturation.

††<EMAIL_ADDRESS><EMAIL_ADDRESS>
##  1 Introduction

The complexity of physical systems in general, and for quantum dynamics of
many-body systems in particular, is encapsulated quantitatively by a number of
measures. These include circuit complexity, quantum Kolmogorov complexity,
algorithmic randomness, entanglement based measures and entropic measures [1,
2, 3, 4, 5]. The detailed dynamics of complexity can also signal the presence
(or absence) of quantum chaos, and thus these complexity measures can often be
used as chaos diagnostics [6, 7, 8].

A relatively new measure of the complexity of operator growth with bearing on
the integrable/chaotic nature of the dynamics is Krylov complexity
(K-complexity). Operator K-complexity was first defined and studied in [9] for
quantum systems in the thermodynamic limit. Its relation with circuit
complexity was recently studied in [10, 11]. It utilises the Heisenberg
picture of quantum dynamics and measures the growth of operator complexity as
encapsulated by successive nested commutators of the system Hamiltonian with
an (initially localised) observable. For maximally chaotic systems, the
initial growth rate (upto the scrambling time) of K-complexity is exponential
111However, this can happen also for cases involving saddle-dominated
scrambling resulting (semi-classically) from unstable fixed points in phase
space [12]. So this is not a sufficient condition for chaos.. For finite size
systems this growth eventually becomes linear and saturates after the
Heisenberg time [13]. For a comprehensive recent review on Krylov construction
in quantum systems, see [14].

The main focus of this paper is the related notion of K-complexity for quantum
states, also known as state complexity or spread complexity. It was first
introduced in [15] and utilises the Schrödinger picture of state evolution. It
measures the spread with time of an initial state over a pre-defined basis,
namely the Krylov basis that is defined iteratively through the Lanczos
algorithm. The next section reviews the basic definitions and concepts
relating to spread complexity that we will require. Spread complexity has been
investigated for a variety of models starting from the Sachdev-Ye-Kitaev (SYK)
model and Random Matrix Models in [15]. It has been studied for spin chains
[16, 17], the quenched LMG model [18, 19], Kitaev chain [20], free fermion
models [21], many-body scars [22], and others [23, 25, 26, 27, 28, 29, 30, 31,
32, 24]. It also serves as an order parameter which can discern topologically
non-trivial phases of matter [33].

In this paper we will study spread complexity for quantum maps such as those
generated by Floquet dynamics of spin-chains. Precision Floquet engineering in
periodically driven quantum systems opens up many avenues, both experimental
and theoretical, in quantum control using optical lattices and ultra-cold
atoms [34, 35, 36]. It can be used to study novel non-equilibrium phenomenon
and engineer novel phases of matter such as time crystals [37].

We will study chaotic dynamics in a quantum many-body system – the
periodically kicked Ising spin-chain. Such a lattice of interacting spin-half
fermions provides an ideal testing ground for many-body quantum chaos. As this
intrinsically quantum mechanical system has no classical limit, it is useful
in exploring facets of chaotic dynamics beyond the semi-classical regime. A
primary objective in our study is to investigate the transition from regular
to maximally chaotic dynamics in Floquet spin-chain models when the coupling
strength or the non-locality of the interaction is tuned. This is done using a
variety of measures, with a particular focus on spread complexity. The first
model here will be the kicked Ising spin-chain with a tunable coupling
constant that interpolates between the integrable and chaotic regimes. We will
also look at a maximally non-local version of the Kicked Ising spin chain and
study the effect of non-locality on the dynamics. Maximally non-local spin-
chain models with constant all-to-all interactions have been studied earlier
as systems exhibiting fast scrambling [38, 39]. Fast scramblers have a
scrambling time $t_{s}\sim\log N$ where $N$ is the system size and have
maximally chaotic dynamics with the quantum Lyapunov exponent saturating the
chaos bound [40]. The second model (discussed in appendix C) is a bosonic
spin-chain: the kicked Bose-Hubbard dimer model.

This paper is organised as follows. Section 2 introduces and reviews the basic
notions of spread complexity and the Arnoldi iterative algorithm utilised for
Krylov construction for quantum maps. We also review the Krylov chain picture
and use it to give a simple argument for the complexity growth being at most
linear in time. Section 3 introduces the kicked Ising spin chain model, both
the local and non-local versions. Here also we study the distinctive behaviour
in the integrable/chaotic domain, of the Arnoldi coefficients, spread
complexity, as well as other chaos quantifiers such as the spectral parameter
$\eta$ and the delocalisation parameter $\sigma$. Some similar results for the
Bose-Hubbard dimer model are relegated to appendix C. In section 4 we conclude
with a discussion of the main results and observations.

##  2 Arnoldi iteration and spread complexity for quantum maps

In this section we review the basic notions of Krylov construction, spread
complexity, Lanczos and Arnoldi iteration that we will need in our subsequent
analysis. For more details, the reader is referred to the introductory
sections of [15, 9, 41, 42, 43].

###  2.1 Krylov complexity for quantum states

Consider a basis $\mathfrak{B}=\\{\ket{B_{n}}\\}$ using which a time-dependent
quantum state is expanded as $\ket{\psi(t)}=\sum_{n}\psi_{n}(t)\ket{B_{n}}$.
Following [15] we define a complexity measure

$C_{\mathfrak{B}}(t)=\sum_{n}n\,|\psi_{n}(t)|^{2}$ (1)

which quantifies the (time-dependent) spread of the state over the basis
elements. A functional minimization over the bases:
$C(t)=min_{\mathfrak{B}}\,\,C_{\mathfrak{B}}(t)$, leads to a basis-independent
measure of state complexity dubbed as spread complexity (see [15] for more
details). These authors also showed that this minimum is obtained for the
Krylov basis. The Krylov construction for quantum states for autonomous as
well as Floquet systems is outlined below.

###  2.2 Review of Lanczos iteration

We consider the time-evolved state

$\ket{\psi(t)}=\exp(-itH)\ket{\psi_{0}}=\sum_{n=0}^{\infty}\frac{(-it)^{n}}{n!}H^{n}\ket{\psi_{0}},$
(2)

which suggests the natural basis
$\\{\ket{\psi_{0}},H\ket{\psi_{0}},H^{2}\ket{\psi_{0}},\dots\\}$ in which the
state can be expanded. We assume that the initial state is not an eigenstate
of the full Hamiltonian. Krylov construction then proceeds by orthonormalising
this set using the Lanczos algorithm. The algorithm iteratively generates a
set of orthonormal basis vectors starting from $\ket{K_{-1}}\equiv 0$,
$\ket{K_{0}}=\ket{\psi_{0}}$ and

$\displaystyle\ket{K_{n}}=\frac{1}{b_{n}}\ket{A_{n}},\;\;\;\;\;\;\;\;\;\;\ket{A_{n+1}}=(H-a_{n})\ket{K_{n}}-b_{n}\ket{K_{n-1}}$
(3)

for $n\geq 1$. The Lanczos coefficients are defined as:

$\displaystyle
a_{n}=\bra{K_{n}}H\ket{K_{n}},\;\;\;b_{n}=\innerproduct{A_{n}}{A_{n}}^{1/2}$
(4)

This algorithm naturally halts for systems with a finite-dimensional Hilbert
space after generating a set of orthonormal states. The Hamiltonian in the
Krylov Basis takes a tri-diagonal form with $a_{n}$ as the diagonal elements
and $b_{n}$ as the sub-diagonal and super-diagonal elements [15]. The time-
evolved state $\ket{\psi(t)}$ can be expanded in the Krylov Basis as:

$\displaystyle\ket{\psi(t)}=\sum_{n=0}^{D_{K}-1}\psi_{n}(t)\ket{K_{n}}$ (5)

where $D_{K}$ is the dimension of the Krylov space. The total probability
$\sum_{n}|\psi_{n}(t)|^{2}=1$ is conserved for unitary evolution. Spread
complexity is then defined as:

$\displaystyle C(t)=\sum_{n=0}^{D_{K}-1}n|\psi_{n}(t)|^{2}$ (6)

which measures the average spread of the wave-function over the Krylov Basis
[15]. An equivalent form of spread complexity is the expectation value of the
Krylov complexity operator $\hat{K}$:

$\displaystyle
C(t)=\bra{\psi(t)}\hat{K}\ket{\psi(t)},\;\;\;\;\;\hat{K}=\sum_{n=0}^{D_{K}-1}n\ket{K_{n}}\bra{K_{n}}$
(7)

Apart from the average spread, we can also study the Shannon entropy of the
spread using the probability distribution $|\psi_{n}(t)|^{2}$. It is defined
as

$\displaystyle
S(t)=-\sum_{n=0}^{D_{K}-1}|\psi_{n}(t)|^{2}\,\ln|\psi_{n}(t)|^{2}$ (8)

###  2.3 Review of Arnoldi iteration

For periodically driven systems, the Krylov construction proceeds similarly
using the unitary Floquet operator $U_{F}$ and the Arnoldi iterative algorithm
which generalises the Lanczos procedure beyond the Hermitian case. Here one
considers the basis which is naturally constructed by stroboscopically probing
the system at equal time periods
$\\{\ket{\psi_{0}},U_{F}\ket{\psi_{0}},U_{F}^{2}\ket{\psi_{0}},...\\}$, where
the $n^{th}$ element of the basis represents the state after $n$ time steps.
In [15] it was also shown that the basis generated in this manner minimizes
the complexity function defined in eq.(1).

Arnoldi orthonormalization entails the following iterative construction.
Starting with $\ket{K_{0}}=\ket{\psi_{0}}$, we define the next basis element

$\displaystyle\ket{K_{1}}=\frac{1}{h_{1,0}}\biggl{[}U_{F}\ket{K_{0}}-h_{0,0}\ket{K_{0}}\biggr{]}.$
(9)

Subsequent basis vectors are generated as

$\displaystyle\ket{K_{n}}=\frac{1}{h_{n,n-1}}\left[U_{F}\ket{K_{n-1}}-\sum_{j=0}^{n-1}h_{j,n-1}\ket{K_{j}}\right],$
(10)

for $n\geq 2$, where $h_{j,k}=\bra{K_{j}}U_{F}\ket{K_{j}}$ are called Arnoldi
coefficients and are analogous to the Lanczos coefficients in the time-
independent formulation. Of these, the normalization constants $h_{n,n-1}$ are
analogous to the Lanczos $b_{n}$’s. This generalization allows the algorithm
to generate orthonormal vectors using non-Hermitian operators and was first
used for Floquet systems in [43, 44] and for open quantum systems in [45]. The
unitary Floquet operator takes the Hessenberg form in the Krylov Basis with
non-zero elements $h_{j,k}$ on and above the sub-diagonal of the matrix. The
Arnoldi coefficients thus define the matrix representation of the system
Floquet operator in the Krylov basis.

As before, the state after $j$ time-steps can be expanded in the Krylov basis

$\displaystyle\ket{\psi_{j}}=\sum_{n=0}^{D_{K}-1}\psi_{n}^{\,j}\ket{K_{n}}$
(11)

and spread complexity and entropy can be defined for Floquet dynamics

$\displaystyle
C_{j}=\sum_{n=0}^{D_{K}-1}n|\psi_{n}^{\,j}|^{2},\;\;\;\;\;\;\;\;\;S_{j}=-\sum_{n=0}^{D_{K}-1}|\psi_{n}^{\,j}|^{2}\,\ln|\psi_{n}^{\,j}|^{2}$
(12)

Besides kicked systems, quantum maps representing such discrete unitary
evolutions also arise naturally in studying the dynamics of quantum circuits
[46].

###  2.4 Krylov Chain picture

In the Krylov construction, the dynamics of a general quantum system is mapped
to a particle-hopping tight-binding model on a one-dimensional lattice with
the Lanczos coefficients representing the hopping amplitudes. The equations
describing this dynamics were first derived in [9] for time-independent
Hamiltonian systems. For the case of Floquet dynamics, the corresponding
equations were first derived in [43] and will be relevant to us here. We take
the action of the Floquet operator on eq. (11) to get:

$\displaystyle\ket{\psi_{j+1}}=\sum_{n=0}^{D_{K}-1}\psi_{n}^{\,j}\;U_{F}\ket{K_{n}}$
(13)

Using the Krylov basis expansion on the left hand side and action of the
Floquet operator on $\ket{K_{n}}$ from eq. (10), we get a set of linear
difference equations for the probability amplitudes at different time steps

$\displaystyle\psi_{n}^{\,j+1}=\sum_{l=n-1}^{D_{K}-1}h_{n,l}\,\psi_{l}^{\,j}$
(14)

This equation describes a particle hopping in a one-dimensional lattice with
Krylov Basis vectors as lattice sites and the coefficients $h_{n,l}$ as
hopping amplitudes. In this case, the hopping is non-local in one direction
(see fig.1) unlike the case for time-independent Hamiltonians. Solving the
difference equations with the initial condition $\psi_{n}^{0}=\delta_{n0}$
gives an equivalent way of calculating the probability amplitudes to determine
the spread complexity.

This picture can be used to argue that the spread complexity growth in a
Floquet system is at most linear in time, an observation made in [46] for
operator complexity growth in dual-unitary models. We will provide a more
general argument. Intuitively, the idea is that since the particle on the
Krylov chain can hop only one step to the right (fig.1), after $j$ time steps
it can at most be $j$ lattice sites away to the right from its starting point.
This leads to a K-complexity that is at most $j$, and hence the growth can at
most be linear in time.

Figure 1: Schematic diagram of a particle hopping in a one-dimensional lattice
in the Arnoldi approach to Floquet dynamics in Krylov space. The equations
(14) encapsulating the dynamics imply that the particle jumps are local to the
right but can be non-local to the left, as shown in the above figure.

A more quantitative argument is as follows. The growth of spread complexity is
maximised when the set of vectors generated by the repeated action of the
Floquet operator for a given initial state is naturally orthonormal. In this
case, eq. (10) implies that the Arnoldi coefficients $h_{n,n-1}$ are equal to
one and the rest of the $h_{j,k}$ are equal to zero. Eq. (14) then takes the
simple form

$\displaystyle\psi_{n}^{\,j+1}=h_{n,n-1}\,\psi_{n-1}^{\,j}=\psi_{n-1}^{\,j}$
(15)

Using the given initial condition, we can deduce that
$\psi_{n}^{\,j}=\delta_{nj}$ for the case of maximum growth and spread
complexity is given by

$\displaystyle C_{j}^{\,max}=\sum_{n=0}^{D_{K}-1}n|\delta_{nj}|^{2}=j$ (16)

which is the equation for a straight line with slope one. We will see below in
the example of the spin-chains which we study that this bound on the initial
growth rate is obeyed in every case. The spread entropy in such cases is zero
since the particle hops from one site to the other without any dispersion in
wavefunction.

##  3 Kicked Ising spin chain: local and non-local interactions

In this section we will investigate facets of regular and chaotic dynamics of
a quantum many-body system – the kicked one-dimensional Ising spin-chain. This
consists of a lattice of spin-half fermions with interactions that can be
local (nearest-neighbour) or non-local. Besides spread complexity, we will
also compute a number of other quantities and compare and contrast their
behaviour. Some recent studies of complexity and chaos in (time-independent)
spin chains include [47, 17, 48, 49, 50]. Aspects of quantum chaos for kicked
spin-chains have been studied recently in [51] utilising spectral parameters
and entanglement entropy.

We will begin with the local model with nearest neighbour interactions. This
model consists of $N$ locally interacting spin-half systems kicked
periodically with an external tilted magnetic field. The Hamiltonian for the
system is [52]:

$\displaystyle
H=H_{0}+V\,\delta_{T}(t)=\sum_{i=1}^{N}\left(J\sigma_{i}^{z}\sigma_{i+1}^{z}+\bm{b.\sigma_{i}}\delta_{T}(t)\right)\,;\,\,\,\delta_{T}(t)=\sum_{n=-\infty}^{\infty}\delta(t-nT)$
(17)

where $\sigma_{i}^{j}$ is the Pauli matrix for the particle at site $i$ along
the $j$ axis, and $\bm{b}$ is the magnetic field vector. The $z-$axis for the
system can be set such that the magnetic field is along the $(x-z)$ plane and
the spin-chain along the $y-$axis. The tilted magnetic field vector can be
resolved into its components using an angular variable $\phi$ such that
$\bm{b}=(b\sin{\phi},0,b\cos{\phi})$. We will mostly work with the time period
$T$ set equal to one. The Floquet operator for the given Hamiltonian is given
by the quantum map:

$\displaystyle
U_{F}=\exp{-\text{i}\,J\sum_{i=1}^{N}\sigma_{i}^{z}\sigma_{i+1}^{z}}\exp{-\text{i\,}b\sum_{i=1}^{N}(\sigma_{i}^{x}\,\sin{\phi}\,+\sigma_{i}^{z}\,\cos{\phi}\,)}$
(18)

The variable $\phi$ can be used as a tuning parameter interpolating from the
integrable limit to maximal chaos. Given the symmetry of the system, we only
need to study it for the range $\phi\in[0,\pi/2]$. Besides $\phi=0$, the
strongly coupled $\phi=\pi/2$ case is also integrable as it can be solved
exactly by the Jordan-Wigner transformation [53]. We will work with open
boundary conditions, and work in a symmetry restricted system subspace, namely
the positive parity sector.

###  3.1 Spectral Parameter and Level Statistics

Random Matrix Theory (RMT) and spectral statistics of many-body hamiltonians
provide a classic diagnostic of chaos in the quantum domain. As per the Berry-
Tabor conjecture [54], the spectral statistics of classically integrable
hamiltonians gives a Poisson distribution. On the other hand, the BGS
conjecture [55] surmises that for the chaotic case, one of the several Wigner-
Dyson universality classes showing level repulsion is realised. For a
pedagogical reference emphasising the spectral and RMT aspects of quantum
chaos, see [56]. For our purposes, the integrability to chaos transition in
the kicked Ising model can be characterized by the spectral statistics of the
quasi-energies of the Floquet operator. Given the eigenvalue relation for the
Floquet operator:

$\displaystyle U_{F}\ket{\psi_{\varphi}}=e^{i\varphi}\ket{\psi_{\varphi}},$
(19)

we can calculate the spacing between consecutive eigenvalues
$s_{n}=\frac{N}{2\pi}(\varphi_{n}-\varphi_{n-1})$ and define a parameter
$\langle\tilde{r}\rangle$:

$\displaystyle\langle\tilde{r}\rangle=\frac{1}{D}\sum_{n=1}^{D}\frac{\min(s_{n},s_{n-1})}{\max(s_{n},s_{n-1})}.$
(20)

The value of the parameter $\langle\tilde{r}\rangle$ depends on the
distribution of level spacing and encodes information about the integrable or
chaotic regime of the dynamics. From RMT, the analytic values of
$\langle\tilde{r}\rangle$ for the Poisson and GOE case is
$\langle\tilde{r}\rangle_{P}\approx 0.38629$ and
$\langle\tilde{r}\rangle_{GOE}\approx 0.53590$ [57]. Thus, we can define a
normalized parameter $\eta$ as:

$\displaystyle\eta=\frac{\langle\tilde{r}\rangle-\langle\tilde{r}\rangle_{P}}{\langle\tilde{r}\rangle_{GOE}-\langle\tilde{r}\rangle_{P}}$
(21)

The value of $\eta$ is close to zero for integrable dynamics and one for the
chaotic case. For the Floquet spin-chain represented by eq. (17) the variation
of $\eta$ for different values of the coupling $\phi$ is given below. Note the
smaller values of $\eta$ close to the integrable limits $\phi=0,\pi/2$.
Similar computations for $\eta$ for time-independent spin-chains have been
done in [17]. Also plotted below are the spectral statistics curves for the
integrable and chaotic case. This makes clear the regular nature of the
dynamics for the weakly-coupled case, as the level spacings show a Poisson
distribution. For a larger value ($\phi=\pi/3$) of the coupling, the
distribution shows level repulsion characterising chaos, and is of the GOE
universality class.

Figure 2: The value of the spectral parameter $\eta$ as a function of angle
$\phi$ for $N=11$ spins as the dynamics changes from regular to chaotic. Note
the low value at the two ends when the system is integrable

Figure 3: The probability distribution of the level spacing for $N=13$ spins –
Poisson for $\phi=\pi/30$ (left) and GOE for $\phi=\pi/3$ (right).

###  3.2 Dynamics of Arnoldi coefficients and spread complexity

We will now turn to features of the Arnoldi coefficients and the dynamics of
spread complexity for the kicked Ising spin-chain. The various relevant
quantities are defined in subsection 2.3. For the following calculations, the
initial state is taken as an $H_{0}$ eigenstate
$\ket{\psi_{0}}=\ket{\psi_{0}^{eig}}$ in the positive parity sector of the
Hilbert space. Most results, however, are generic with same qualitative
features for other initial states. To facilitate comparison, we will keep
fixed values of the coupling constants $b$ and $\phi$ for the regular as well
as the chaotic case.

Fig. 4 shows the behaviour of the Arnoldi coefficients $h_{n,n-1}$ and the
real part of $h_{n,n}$ (the imaginary part has a similar behaviour). The
fluctuations in the Arnoldi coefficients are noticeably larger in the near-
integrable case. The larger fluctuations signal the presence of localisation
on the Krylov chain, a phenomenon that results in a suppressed value of
K-complexity [58, 59]. This difference in the spread complexity saturation
value in the two limiting cases is also clearly seen in figure 5. In the
integrable limit we also observe that the spread entropy shows larger
fluctuations which are quasi-periodic in nature, the reasons for which it
would be interesting to discern 222Note that although the presented results
are for a fixed value of the coupling constants and a particular initial
state, we have checked that the above observations are generic and continue to
hold for other choices..

Figure 4: Dispersion in Arnoldi coefficients in the near integrable and
chaotic limits. The left plot is for the real sub-diagonal $h_{n,n-1}$ whereas
the right plot is for the real part of the diagonal $h_{n,n}$ – the imaginary
part fluctuates in a similar manner. The calculations are done for $N=10$ in
the positive parity sector.

Figure 5: Spread Complexity (left) and Entropy (right) for different values of
the coupling paramterised by $\phi$.

### Dispersion of Arnoldi Coefficients

We will now investigate the relation between chaotic dynamics and the
dispersion of Arnoldi coefficients. The localisation length ($L_{loc}$) of the
wavefunction is related to the dispersion or disorder:
$L_{loc}\sim\sigma_{h}^{-1}$ [60]. For this, we plot the variation in the
standard deviation of Arnoldi coefficients as a function of the coupling
parameter $\phi$. We have scaled all the parameters such that the maximum
value is set to one and the minimum value is set to zero. This means that a
re-scaled variable is defined as

$\displaystyle X_{s}=\frac{X-X_{min}}{X_{max}-X_{min}}$ (22)

analogous to the definition of the spectral parameter in eq. (21). Here $X$
can be the delocalisation parameter, or as discussed below, the time-averaged
magnetisation or spread complexity saturation value. This facilitates
comparison between all these quantities as the coupling is varied.

The different initial states chosen below include $\ket{\psi_{eig}}$ which is
an $H_{0}$ eigenstate (all spins up). $\ket{\psi_{unif}}$ is the normalised
state with the same constant coefficient as each of the component. In the
third case of random initial states, the calculations were done for five
different random initial states and averaged over for each value of $\phi$. It
is noteworthy that different initial states have essentially overlapping
dispersion plots 333Besides the initial states and Arnoldi coefficients in
fig.6, we have checked this universality in the behaviour for several other
cases as well. For autonomous spin chains, a similar observation for Lanczos
coefficients was made in [17]. In this reference some variants of the
delocalisation length involving dispersion about the moving average and with
the log ratio of Lanczos coefficients were also considered. However, the
simple standard deviation suffices for our purposes to delineate the main
features of the fluctuations in the Arnoldi sequences.. We can infer that this
dispersion or delocalisation length is a robust state-independent measure of
chaotic dynamics.

Figure 6: (rescaled) standard deviation of the Arnoldi coefficients
$h_{n,n-1}$ and $h_{n},n$ for different values of $\phi$. The variation in the
spectral parameter is also shown for comparison.

### Saturation value of spread complexity

Another quantity of interest is the saturation value of spread complexity at
late times 444“Late” in this context means beyond the Heisenberg time scale..
The definition we will use for the late-time spread complexity saturation
value will be the discrete analogue of the one used in [47]:

$\displaystyle\overline{C_{j}}=\sum_{n=0}^{D_{K}-1}n\overline{|\psi_{n}|^{2}}\;\;\;\;\;\;\;\;\text{where,
}\overline{|\psi_{n}|^{2}}=\lim_{m\to\infty}\frac{1}{m}\sum_{j=0}^{m}|\psi_{n}^{(j)}|^{2}$
(23)

This saturation value is comparatively higher for the chaotic system and it is
related to the delocalization on the Krylov chain. It has been considered as
an indicator of chaos in recent literature [47, 49, 43]. However, there are
caveats related to the choice of initial states. Recent work [61, 17] argued
that the late-time saturation values for the choice of certain initial states
does not always agree with the other chaos measures such as r-statistics.

The saturation value, for the same set of initial states as before, is plotted
below as a function of the coupling $\phi$. We notice that when the initial
state is an eigenstate of $H_{0}$, the results match with the integrability to
chaos transition as shown by the $\eta$ parameter in fig. 7. However the same
is not always true for other initial states.

Figure 7: Rescaled saturation value of the spread complexity as a function of
the coupling parameter $\phi$ for different initial states.

###  3.3 Magnetisation

The mean magnetisation is a classic order parameter quantifying the disorder
in the many-body state for a system. We will compute the time-dependent
expectation value of $J_{z}=\sum_{i}\sigma^{z}_{i}$ operator with the initial
state being all spins up, and also the variation of the time-averaged value
$\overline{J_{z}}$ as the parameter $\phi$ is changed. The long-time average
$\overline{J_{z}}$ serves as an order parameter for the sytem and is defined
as:

$\displaystyle\overline{J_{z}}=\lim_{T\to\infty}\frac{1}{T}\sum_{j=0}^{T}\bra{\psi_{j}}J_{z}\ket{\psi_{j}}$
(24)

where the state $\ket{\psi_{j}}$ is defined as
$\ket{\psi_{j}}=U_{F}^{j}\ket{\psi_{0}}$.

As shown in the fig. 8 (left) the mean magnetisation shows a periodic
variation with time and a large average value in the weakly coupled case, as
expected in this ferromagnetic case with a small amount of disorder. However
for the chaotic case, the magnetisation drops to near-zero quickly. For the
integrable but strongly coupled case, there are quasi-periodic fluctuations.
The figure on the right shows the behaviour of the (rescaled) order parameter
$\overline{J_{z}}$ and the time averaged spread complexity $\overline{C_{j}}$
discussed in the previous section. For a (time-independent) spin-chain
described by the LMG model, results for the magnetisation were obtained
recently in [19]. There it was shown analytically that when the initial state
is an eigenstate of the pre-quench hamiltonian ($H_{0}$), then
$\overline{J_{z}}$ and $\overline{C(t)}$ are the same upto an additive
constant. This result essentially followed from the observation that the
Lanczos basis is the same as the pre-quench energy eigenbasis. As shown in
Fig. 8 (right), where the initial state is an $H_{0}$ eigenstate,
$\overline{J_{z}}$ and $\overline{C_{j}}$ have similar behaviour although in
this case the Krylov-Arnoldi basis doesn’t match with the $H_{0}$ eigenbasis.

Figure 8: Evolution of expectation value of the total spin operator $J_{z}$
(left) and its time average (rescaled) for different values of parameter
$\phi$ (right), with the averaged spread complexity also shown for comparison.

###  3.4 Non-local Kicked Ising Chain

We will now turn to a variant of the above kicked Ising spin chain model where
the interactions are maximally non-local. In particular, the Hamiltonian
contains all-to-all spin interactions with uniform coupling.

$\displaystyle
H=H_{\text{local}}+\gamma\sum_{i<j}\sigma_{i}^{z}\sigma_{j}^{z}$ (25)

where, $H_{local}$ is the Hamiltonian given in Eq. (17). The Floquet operator
for non-local interactions is given by:

$\displaystyle
U_{F}=\exp{-\text{i}\,\gamma\sum_{i<j}\sigma_{i}^{z}\sigma_{j}^{z}}U_{F\,\text{(local)}}$
(26)

where, $U_{F\,\text{(local)}}$ is the Floquet operator given in Eq. (18). For
time-independent spin-chains, the effects of non-locality on (operator)
complexity have been recently studied in [62].

Even without the periodic drive, such systems with all-to-all two-body
interactions are known to be fast scramblers [63, 64, 38]. This means that the
exponentially fast spreading of quantum information in such systems involves a
time scale, the scrambling time $t_{s}$, that is logarithmic in the system
size $N$. Other examples of fast scramblers are brownian quantum circuits, SYK
model and black holes.

We will study the effect of non-locality on spread complexity in the non-
trivially integrable regime at $\phi=\pi/2$. This system can be solved via
Jordan-Wigner transformation [53]. At the value of $\phi=\pi/2$, the system
has a parity symmetry as well as Z-reflection symmetry.

The effect of turning on the non-local coupling ($\gamma$) is to decrease the
disorder in the Arnoldi coefficients, as made clear by Fig. Appendix C:
Integrable to chaos transition in kicked Bose-Hubbard dimer. This figure also
clearly shows how the saturation value of the spread complexity and its
initial growth rate increases with $\gamma$. This makes manifest that non-
locality of the interaction induces chaos even at relatively small coupling
constants. Given the close relationship between locality and integrability
[65], this is not unexpected. However, it is interesting that even
perturbatively small non-local couplings can open the way for maximal chaos.
We also find that $\gamma$ has no substantive effect when it is turned on with
the system already in the chaotic regime.

Figure 9: Arnoldi coefficients (top left), state complexity (top right),
entropy (bottom left) for the Ising chain with local and non-local
interactions in the integrable and chaotic regime. The bottom right figure
shows how the (normalised) averaged spread complexity and initial slope grows
as the non-local coupling is increased.

###  3.5 Tuning the driving frequency

In our analysis so far, the time period of driving was set to unity. We will
now investigate the effects of varying the driving frequency on the Arnoldi
coefficients and spread complexity. The graphs below shows the Arnoldi
coefficients and spread complexity for different values of time period $T$ in
the chaotic Kicked Ising Chain. We can see that in the high frequency (small
$T$) limit, Arnoldi coefficients show the signature behaviour of Lanczos
coefficients – initial linear growth followed by a descent [13]. In appendix B
we show how the Arnoldi Krylov construction goes over to the Lanczos one, in
the limit of a high frequency drive. The saturation value of the spread
complexity also decreases as the time-period of kicking decreases. This result
may be explained as follows. Recall that a suppressed complexity value results
from localisation on the Krylov chain in the integrable limit. The integrable
case has extra symmetries characterised by the presence of conserved
quantities. These conserved quantities can also arise in the high frequency
limit of the periodic drive. In this limiting case,

$U_{F}=\exp(-iTH_{0})\exp(-iTV)\rightarrow\exp\big(-iT(H_{0}+V)\big{missing})=\exp(-iTH_{F})$
(27)

and thus operators commuting with the effective Floquet hamiltonian $H_{F}$
are conserved and thus, in the high frequency limit, we have a suppressed
complexity saturation value. Note that this suppressed value is not due to
Krylov localisation, as in the case when the local couplings are tuned to
larger values. In this case, the Arnoldi sequences do not show larger
disorder, but nevertheless the saturation value is suppressed.

Figure 10: Arnoldi coefficients (left) and state complexity (right) for the
kicked Ising spin-chain for different values of kicking period $T$. In the
high frequency limit, note the Lanczos-like behaviour of the Arnoldi
coefficients and the suppression in the spread complexity saturation.

We also zoom into the region of initial linear growth and compute the slope as
a function of the time-period of the drive. In fig.11 the bound on the slope
is represented by the black line. Increasing the time period, the initial
growth rate increases and tends towards the maximal possible as shown in Fig.
11 (right). These observations are in accord with the result of section 2,
where we showed that Krylov chain dynamics implies that the initial growth
late of the spread complexity is at most linear (with slope 1).

Figure 11: Initial growth of Arnoldi coefficients (left) and slope of the best
fit line for different values of kicking period $T$ (right).

##  4 Discussion

We have studied the dynamics of a kicked Ising spin-chain (local as well as
non-local) through a number of chaos quantifiers. The utility of the various
Krylov measures is highlighted by the accord with spectral quantifiers of non-
integrable dynamics.

In section 3 we studied the regular to chaotic transition in the kicked Ising
spin-chain resulting from tuning its coupling strength. A noteworthy result
here was the concurrence between several different diagnostics of non-
integrable dynamics namely the spectral parameter $\eta$, the delocalization
parameter $\sigma_{h}$, the saturation value of spread complexity
$\overline{C_{j}}$ and the average magnetization $\bar{J_{z}}$. As noted in
the literature on autonomous systems [17, 61], we also find that the
saturation value of complexity has dependence on the initial state, although
the match with the spectral parameter is good for an initial state that is
localised in the $H_{0}$ eigenbasis.

Our results in this section also show that maximal chaos with fast scrambling
can arise in kicked spin-chains due to the presence of non-local interactions,
even with weak coupling strengths (besides the usual route to chaos due to
strong local couplings). We also discussed the effect of changing the
frequency of the drive and saw that, in the high frequency limit, we obtain a
suppressed saturation value of the spread complexity and the dynamics of
Arnoldi coefficients reduces to the Lanczos case. We reiterate that this
suppressed saturation value is not due to Krylov localisation.

One can further study the transition from integrability to chaos in
harmonically driven or periodically quenched spin-chains utilising the methods
used in this paper. Also, in all these models it would be interesting to
investigate the existence of dynamical quantum phase transitions in the
thermodynamic limit via non-analyticities in the Loschmidt amplitude and
K-complexity saturation [66, 67, 19].

## Acknowledgements

We thank Arul Lakshminarayan, Adolfo del Campo, Irfan N. Mir and Philip
Cherian for useful discussions. AWS would like to thank the Research and
Development Office, Ashoka University and Axis Bank for financial support. AWS
would also like to thank the participants of the workshop “Quantum Dynamics
and Chaos: Modern Perspectives” (Ashoka University, March 9-11, 2024) where
some of the results of this paper were presented. We would like to acknowledge
use of the physics department server and the QuSpin package [68] for the
numerical calculations.

## Appendix A: Special case

We will look at an interesting special case 555We thank Arul Lakshminarayan
for bringing this case to our attention. when the coupling constants take on
particular values in the kicked Ising chain. We start with a modified form of
the Floquet operator

$\displaystyle
U_{F}=\exp{-ih_{x}\sum_{i=1}^{N}\sigma_{i}^{x}}\exp{-iJ\sum_{i=1}^{N-1}\sigma_{i}^{z}\sigma_{i+1}^{z}-ih_{z}\sum_{i=1}^{N}\sigma_{i}^{z}}$
(28)

and work with the values of parameters: $h_{x}=h_{z}=J=\pi/4$. This is the
particular case of the self-dual kicked Ising spin chain also known as a dual
unitary model [52, 69]. In such cases, $U_{F}^{p}$ becomes the identity
operator for some integer $p$ which depends on the number of spins. A similar
observation was made in the case of Clifford circuits in [46]. Note the linear
growth (with slope 1) and recurrence in the spread complexity. As noted in
section 2.4 the $h_{n,n-1}$ coefficients being unity leads to a maximal
(linear growth) of spread complexity, which is what this example illustrates.
The recurrence also has a period that scales with the system size. This
special case was also discussed in [70] where the operator entanglement was
studied.

Figure 12: Arnoldi Coefficients and Spread complexity in the case of resonance
in KIC.

## Appendix B: High frequency limit for the Floquet operator

In the above calculations, we have set the time-period of delta-function kicks
to unity. However, we can study the behavior of Arnoldi coefficients and
spread complexity by changing the frequency of the kick. The general Floquet
operator is given as:

$\displaystyle
U_{F}=e^{-iH_{0}T}e^{-iVT}=e^{-i(H_{0}+V)T-\frac{1}{2}[H_{0},V]T^{2}+\dots}$
(29)

where the second equality comes from BCH relation. In the limit that $T\to 0$,
we can approximate the Floquet operator upto the first order as:

$\displaystyle U_{F}\approx e^{-i(H_{0}+V)T}=e^{-iH^{\prime}T}$ (30)

If we plug this expression in the Arnoldi iteration:

$\displaystyle\ket{A_{n}}$
$\displaystyle=e^{-iH^{\prime}T}\ket{K_{n-1}}-\sum_{j=0}^{n-1}\bra{K_{j}}e^{-iH^{\prime}T}\ket{K_{n-1}}\ket{K_{j}}$
(31)

and take the terms upto the first order in Taylor expansion of the
exponential:

$\displaystyle\ket{A_{n}}$
$\displaystyle=\ket{K_{n-1}}-iTH^{\prime}\ket{K_{n-1}}-\sum_{j=0}^{n-1}[\innerproduct{K_{j}}{K_{n-1}}-iT\bra{K_{j}}H^{\prime}\ket{K_{n-1}}]\ket{K_{j}}$
$\displaystyle=-iTH^{\prime}\ket{K_{n-1}}+\sum_{j=0}^{n-1}iT\bra{K_{j}}H^{\prime}\ket{K_{n-1}}\ket{K_{j}}$

The action of the Hamiltonian in the Krylov Basis is:

$\displaystyle
H^{\prime}\ket{K_{n-1}}=a_{n-1}\ket{K_{n-1}}+b_{n}\ket{K_{n}}+b_{n-1}\ket{K_{n-2}}$
(32)

Plugging this in the above expression and using the orthogonality relation for
Krylov Basis vectors:

$\displaystyle\ket{A_{n}}$
$\displaystyle=-iTH^{\prime}\ket{K_{n-1}}+\sum_{j=0}^{n-1}iT[a_{n-1}\innerproduct{K_{j}}{K_{n-1}}+b_{n}\innerproduct{K_{j}}{K_{n}}+b_{n-1}\innerproduct{K_{j}}{K_{n-2}}]\ket{K_{j}}$
$\displaystyle=-iT\left[H^{\prime}\ket{K_{n-1}}-a_{n-1}\ket{K_{n-1}}-b_{n-1}\ket{K_{n-2}}\right]$

which, upto a constant factor, is the exact expression for Lanczos algorithm.
Thus, we expect the behavior of the Arnoldi coefficients to show features of
Lanczos coefficients in the high frequency limit $T\to 0$.

## Appendix C: Integrable to chaos transition in kicked Bose-Hubbard dimer

Another interesting many-body system that shows integrable to chaos transition
is the kicked Bose-Hubbard dimer model. Recently, [71] explored the transition
in this model using the spectral statistics, Husimi distribution, OTOC, and
entanglement entropy. This appendix will use spread complexity as another
dynamical measure to understand the integrable or chaotic nature of this
bosonic system.

The two-site model is described by the Hamiltonian [71]:

$\displaystyle
H=\nu(b_{0}^{\dagger}b_{1}+b_{1}^{\dagger}b_{0})+\frac{U}{N}\sum_{i=0}^{1}n_{i}(n_{i}-1)+\frac{\mu}{2}(n_{0}-n_{1})\delta_{T}(t)$
(33)

where, $\nu$ is the hopping amplitudes between the sites, $U$ sets the on-site
interaction strength, and $N$ is the total number of particles. Similarly,
$b_{i}$ and $b_{i}^{\dagger}$ are bosonic annihilation and creation operators
and $n_{i}=b_{i}^{\dagger}b_{i}$ is the number operator at site $i$. They
satisfy the commutation relation:

$\displaystyle[b_{i},b_{j}]=\delta_{ij},\;\;\;\;[b_{i}^{\dagger},b_{j}^{\dagger}]=[b_{i},b_{j}]=0$
(34)

The parameter $\mu$ in the Hamiltonian controls the periodically applied
potential difference between the sites with time-period $T$. The model can be
represented in terms of the angular momentum operators:

$\displaystyle H=2J_{x}+\frac{k}{2J}J_{z}^{2}+\mu J_{z}\,\delta_{T}(t)$ (35)

where the $J$ operators are mapped to bosonic creation and annihilation
operators as:

$\displaystyle
J_{x}=\frac{b_{0}^{\dagger}b_{1}+b_{1}^{\dagger}b_{0}}{2},\;\;J_{y}=\frac{i(b_{0}^{\dagger}b_{1}-b_{1}^{\dagger}b_{0})}{2},\;\;J_{z}=\frac{b_{0}^{\dagger}b_{1}-b_{1}^{\dagger}b_{0}}{2}$
(36)

These operators satisfy the commutation relation
$[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}$ and $J^{2}=\frac{N}{2}(\frac{N}{2}+1)$ is
a conserved quantity. Here, $\nu$ is set to one, and other constants are
related as $J=\frac{N}{2}$ and $k=2U$ [71]. The corresponding Floquet operator
is:

$\displaystyle U_{F}=\exp{-i\mu
J_{z}}\exp{-i\left(2J_{x}+\frac{k}{2J}J_{z}^{2}\right)}$ (37)

As before, we will study the late-time saturation value of spread complexity
and fluctuation in Arnoldi coefficients, and compare it with the spectral
statistic parameter $\eta$. We will take eigenstate of the $J_{x}$ operator as
the initial state and the system size $J=250$ for the calculations.

Figure 13: Late-time saturation value of the spread complexity, standard
deviation of the Arnoldi coefficients, and parameter $\eta$ for different
values of coupling parameters $\mu$ and $k$.

It was shown in [71] that the system shows integrable to chaos transition for
$\mu=3.0$ while remains integrable for $\mu=6.0$. We can see that the measures
from spread complexity follows a similar trend – standard deviation of Arnoldi
coefficients gets suppressed and late-time saturation value of spread
complexity increases during the transition to chaos for $\mu=3.0$. The values
fluctuate around a mean value for $\mu=6.0$ as expected for a system that
remains integrable.

## References

  * [1] Michael A. Nielsen “A geometric approach to quantum circuit loWisnier bounds” In _Quant. Inf. Comput._ 6.3, 2006, pp. 213–262 DOI: 10.26421/QIC6.3-2
  * [2] Michael A. Nielsen, Mark R. Dowling, Mile Gu and Andrew C. Doherty “Quantum Computation as Geometry” In _Science_ 311.5764, 2006, pp. 1133–1135 DOI: 10.1126/science.1121541
  * [3] Mark R. Dowling and Michael A. Nielsen “The geometry of quantum computation” In _Quant. Inf. Comput._ 8.10, 2008, pp. 0861–0899 DOI: 10.26421/QIC8.10-1
  * [4] W.. Zurek “Algorithmic randomness and physical entropy” In _Phys. Rev. A_ 40, 1989, pp. 4731–4751 DOI: 10.1103/PhysRevA.40.4731
  * [5] André Berthiaume, Wim van Dam and Sophie Laplante “Quantum Kolmogorov Complexity” In _Journal of Computer and System Sciences_ 63.2, 2001, pp. 201–221 DOI: https://doi.org/10.1006/jcss.2001.1765
  * [6] Tibra Ali et al. “Chaos and Complexity in Quantum Mechanics” In _Phys. Rev. D_ 101.2, 2020, pp. 026021 DOI: 10.1103/PhysRevD.101.026021
  * [7] Jordan Cotler, Nicholas Hunter-Jones, Junyu Liu and Beni Yoshida “Chaos, Complexity, and Random Matrices” In _JHEP_ 11, 2017, pp. 048 DOI: 10.1007/JHEP11(2017)048
  * [8] Daniel A. Roberts and Beni Yoshida “Chaos and complexity by design” In _JHEP_ 04, 2017, pp. 121 DOI: 10.1007/JHEP04(2017)121
  * [9] Daniel E. Parker et al. “A Universal Operator Growth Hypothesis” In _Phys. Rev. X_ 9.4, 2019, pp. 041017 DOI: 10.1103/PhysRevX.9.041017
  * [10] Ben Craps, Oleg Evnin and Gabriele Pascuzzi “A Relation between Krylov and Nielsen Complexity” In _Physical Review Letters_ 132.16 American Physical Society (APS), 2024 URL: http://dx.doi.org/10.1103/PhysRevLett.132.160402
  * [11] Chenwei Lv, Ren Zhang and Qi Zhou “Building Krylov complexity from circuit complexity”, 2023 arXiv: https://arxiv.org/abs/2303.07343
  * [12] Budhaditya Bhattacharjee, Xiangyu Cao, Pratik Nandy and Tanay Pathak “Krylov complexity in saddle-dominated scrambling” In _JHEP_ 05, 2022, pp. 174 DOI: 10.1007/JHEP05(2022)174
  * [13] J… Barbón, E. Rabinovici, R. Shir and R. Sinha “On The Evolution Of Operator Complexity Beyond Scrambling” In _JHEP_ 10, 2019, pp. 264 DOI: 10.1007/JHEP10(2019)264
  * [14] Pratik Nandy et al. “Quantum Dynamics in Krylov Space: Methods and Applications”, 2024 arXiv: https://arxiv.org/abs/2405.09628
  * [15] Vijay Balasubramanian, Pawel Caputa, Javier M. Magan and Qingyue Wu “Quantum chaos and the complexity of spread of states” In _Phys. Rev. D_ 106.4, 2022, pp. 046007 DOI: 10.1103/PhysRevD.106.046007
  * [16] Ankit Gill, Kunal Pal, Kuntal Pal and Tapobrata Sarkar “Complexity in two-point measurement schemes” In _Phys. Rev. B_ 109.10, 2024, pp. 104303 DOI: 10.1103/PhysRevB.109.104303
  * [17] Gastón F. Scialchi, Augusto J. Roncaglia and Diego A. Wisniacki “Integrability to chaos transition through Krylov approach for state evolution”, 2023 arXiv: https://arxiv.org/pdf/2309.13427
  * [18] Mir Afrasiar et al. “Time evolution of spread complexity in quenched Lipkin-Meshkov-Glick model”, 2022 arXiv: https://arxiv.org/abs/2208.10520
  * [19] Pedro H.. Bento, Adolfo Campo and Lucas C. Céleri “Krylov Complexity and Dynamical Phase Transition in the quenched LMG model”, 2023 arXiv: https://arxiv.org/abs/2312.05321
  * [20] Pawel Caputa et al. “Spread complexity and topological transitions in the Kitaev chain” In _JHEP_ 01, 2023, pp. 120 DOI: 10.1007/JHEP01(2023)120
  * [21] Mamta Gautam, Nitesh Jaiswal and Ankit Gill “Spread complexity in free fermion models” In _Eur. Phys. J. B_ 97.1, 2024, pp. 3 DOI: 10.1140/epjb/s10051-023-00636-6
  * [22] Sourav Nandy, Bhaskar Mukherjee, Arpan Bhattacharyya and Aritra Banerjee “Quantum state complexity meets many-body scars” In _J. Phys. Condens. Matter_ 36.15, 2024, pp. 155601 DOI: 10.1088/1361-648X/ad1a7b
  * [23] Arghya Chattopadhyay, Arpita Mitra and Hendrik J.. Zyl “Spread complexity as classical dilaton solutions” In _Phys. Rev. D_ 108.2, 2023, pp. 025013 DOI: 10.1103/PhysRevD.108.025013
  * [24] Johanna Erdmenger, Shao-Kai Jian and Zhuo-Yu Xian “Universal chaotic dynamics from Krylov space” In _JHEP_ 08, 2023, pp. 176 DOI: 10.1007/JHEP08(2023)176
  * [25] Kuntal Pal, Kunal Pal, Ankit Gill and Tapobrata Sarkar “Time evolution of spread complexity and statistics of work done in quantum quenches” In _Phys. Rev. B_ 108.10, 2023, pp. 104311 DOI: 10.1103/PhysRevB.108.104311
  * [26] Mamta Gautam et al. “Spread complexity evolution in quenched interacting quantum systems” In _Phys. Rev. B_ 109.1, 2024, pp. 014312 DOI: 10.1103/PhysRevB.109.014312
  * [27] Aranya Bhattacharya, Rathindra Nath Das, Bidyut Dey and Johanna Erdmenger “Spread complexity for measurement-induced non-unitary dynamics and Zeno effect” In _JHEP_ 03, 2024, pp. 179 DOI: 10.1007/JHEP03(2024)179
  * [28] Khushboo Dixit, S. Haque and Soebur Razzaque “Quantum spread complexity in neutrino oscillations” In _Eur. Phys. J. C_ 84.3, 2024, pp. 260 DOI: 10.1140/epjc/s10052-024-12620-0
  * [29] Kyoung-Bum Huh, Hyun-Sik Jeong and Juan F. Pedraza “Spread complexity in saddle-dominated scrambling”, 2023 arXiv: https://arxiv.org/abs/2312.12593
  * [30] Eoin Carolan, Anthony Kiely, Steve Campbell and Sebastian Deffner “Operator growth and spread complexity in open quantum systems”, 2024 arXiv: https://arxiv.org/abs/2404.03529
  * [31] Vijay Balasubramanian, Javier M. Magan and Qingyue Wu “Quantum chaos, integrability, and late times in the Krylov basis”, 2023 arXiv: https://arxiv.org/abs/2312.03848
  * [32] Bozhen Zhou and Shu Chen “Spread complexity and dynamical transition in two-mode Bose-Einstein condensations”, 2024 arXiv: https://arxiv.org/abs/2403.15154
  * [33] Pawel Caputa and Sinong Liu “Quantum complexity and topological phases of matter” In _Phys. Rev. B_ 106.19, 2022, pp. 195125 DOI: 10.1103/PhysRevB.106.195125
  * [34] André Eckardt “Colloquium: Atomic quantum gases in periodically driven optical lattices” In _Rev. Mod. Phys._ 89.1, 2017, pp. 011004 DOI: 10.1103/RevModPhys.89.011004
  * [35] Christof Weitenberg and Juliette Simonet “Tailoring quantum gases by Floquet engineering” In _Nature Physics_ 17.12 Springer ScienceBusiness Media LLC, 2021, pp. 1342–1348 DOI: 10.1038/s41567-021-01316-x
  * [36] F. Meinert et al. “Floquet Engineering of Correlated Tunneling in the Bose-Hubbard Model with Ultracold Atoms” In _Phys. Rev. Lett._ 116 American Physical Society, 2016, pp. 205301 DOI: 10.1103/PhysRevLett.116.205301
  * [37] Michael P. Zaletel et al. “Colloquium: Quantum and classical discrete time crystals” In _Rev. Mod. Phys._ 95.3, 2023, pp. 031001 DOI: 10.1103/RevModPhys.95.031001
  * [38] Ron Belyansky et al. “Minimal Model for Fast Scrambling” In _Phys. Rev. Lett._ 125.13, 2020, pp. 130601 DOI: 10.1103/PhysRevLett.125.130601
  * [39] Silvia Pappalardi et al. “Scrambling and entanglement spreading in long-range spin chains” In _Phys. Rev. B_ 98.13, 2018, pp. 134303 DOI: 10.1103/PhysRevB.98.134303
  * [40] Juan Maldacena, Stephen H. Shenker and Douglas Stanford “A bound on chaos” In _JHEP_ 08, 2016, pp. 106 DOI: 10.1007/JHEP08(2016)106
  * [41] Pawel Caputa, Javier M. Magan and Dimitrios Patramanis “Geometry of Krylov complexity” In _Phys. Rev. Res._ 4.1, 2022, pp. 013041 DOI: 10.1103/PhysRevResearch.4.013041
  * [42] E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner “Operator complexity: a journey to the edge of Krylov space” In _JHEP_ 06, 2021, pp. 062 DOI: 10.1007/JHEP06(2021)062
  * [43] Amin A. Nizami and Ankit W. Shrestha “Krylov construction and complexity for driven quantum systems” In _Phys. Rev. E_ 108.5, 2023, pp. 054222 DOI: 10.1103/PhysRevE.108.054222
  * [44] Daniel J. Yates and Aditi Mitra “Strong and almost strong modes of Floquet spin chains in Krylov subspaces” In _Phys. Rev. B_ 104.19, 2021, pp. 195121 DOI: 10.1103/PhysRevB.104.195121
  * [45] Aranya Bhattacharya, Pratik Nandy, Pingal Pratyush Nath and Himanshu Sahu “Operator growth and Krylov construction in dissipative open quantum systems” In _JHEP_ 12, 2022, pp. 081 DOI: 10.1007/JHEP12(2022)081
  * [46] Philippe Suchsland, Roderich Moessner and Pieter W. Claeys “Krylov complexity and Trotter transitions in unitary circuit dynamics”, 2023 arXiv: https://arxiv.org/pdf/2308.03851.pdf
  * [47] E Rabinovici, A Sánchez-Garrido, R Shir and J Sonner “Krylov complexity from integrability to chaos” In _Journal of High Energy Physics_ 2022.7 Springer, 2022 URL: https://link.springer.com/content/pdf/10.1007/jhep07(2022)151.pdf
  * [48] Ben Craps et al. “Lyapunov growth in quantum spin chains” In _Phys. Rev. B_ 101.17, 2020, pp. 174313 DOI: 10.1103/PhysRevB.101.174313
  * [49] Ben Craps, Marine De Clerck, Oleg Evnin and Philip Hacker “Integrability and complexity in quantum spin chains” In _SciPost Phys._ 16.2, 2024, pp. 041 DOI: 10.21468/SciPostPhys.16.2.041
  * [50] Hugo A. Camargo et al. “Spread and Spectral Complexity in Quantum Spin Chains: from Integrability to Chaos”, 2024 arXiv: https://arxiv.org/abs/2405.11254
  * [51] Tabea Herrmann, Maximilian F.. Kieler and Arnd Bäcker “Characterizing quantum chaoticity of kicked spin chains” In _Phys. Rev. E_ 108.4, 2023, pp. 044213 DOI: 10.1103/PhysRevE.108.044213
  * [52] M. Akila, D. Waltner, B. Gutkin and T. Guhr “Particle-Time Duality in the Kicked Ising Chain II: Applications to the Spectrum”, 2016 DOI: 10.1088/1751-8113/49/37/375101
  * [53] Elliott H. Lieb, Theodore Schultz and Daniel Mattis “Two soluble models of an antiferromagnetic chain” In _Annals Phys._ 16, 1961, pp. 407–466 DOI: 10.1016/0003-4916(61)90115-4
  * [54] Michael Victor Berry and Michael Tabor “Level clustering in the regular spectrum” In _Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences_ 356.1686 The Royal Society London, 1977 URL: https://www.jstor.org/stable/pdf/79349.pdf?casa_token=2Hp2lFi6258AAAAA:3242_SkOJieE71mGF1KyxyhVrlzEACZ355ZRHM6r8Nx6vYddrxQRNGq1ynJjuWokojclOfrke7AWBENiSPyGtK-MeesRI4oldh76zxGfiQ9YvyHJ60Y
  * [55] O. Bohigas, M.. Giannoni and C. Schmit “Characterization of chaotic quantum spectra and universality of level fluctuation laws” In _Phys. Rev. Lett._ 52, 1984, pp. 1–4 DOI: 10.1103/PhysRevLett.52.1
  * [56] Fritz Haake “Quantum Signatures of Chaos” 54, Springer Series in Synergetics Berlin, Heidelberg: Springer Berlin Heidelberg, 2010 DOI: 10.1007/978-3-642-05428-0
  * [57] Y.. Atas, E. Bogomolny, O. Giraud and G. Roux “Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles” In _Phys. Rev. Lett._ 110.8, 2013, pp. 084101 DOI: 10.1103/PhysRevLett.110.084101
  * [58] E. Rabinovici, A. Sánchez-Garrido, R. Shir and J. Sonner “Krylov localization and suppression of complexity” In _JHEP_ 03, 2022, pp. 211 DOI: 10.1007/JHEP03(2022)211
  * [59] Anatoly Dymarsky and Alexander Gorsky “Quantum chaos as delocalization in Krylov space” In _Phys. Rev. B_ 102.8, 2020, pp. 085137 DOI: 10.1103/PhysRevB.102.085137
  * [60] L Fleishman and DC Licciardello “Fluctuations and localization in one dimension” In _Journal of Physics C: Solid State Physics_ 10.6 IOP Publishing, 1977, pp. L125 URL: https://iopscience.iop.org/article/10.1088/0022-3719/10/6/003/meta
  * [61] Bernardo L. Español and Diego A. Wisniacki “Assessing the saturation of Krylov complexity as a measure of chaos” In _Phys. Rev. E_ 107.2, 2023, pp. 024217 DOI: 10.1103/PhysRevE.107.024217
  * [62] Aranya Bhattacharya, Pingal Pratyush Nath and Himanshu Sahu “Krylov complexity for nonlocal spin chains” In _Phys. Rev. D_ 109.6, 2024, pp. 066010 DOI: 10.1103/PhysRevD.109.066010
  * [63] Yasuhiro Sekino and Leonard Susskind “Fast Scramblers” In _JHEP_ 10, 2008, pp. 065 DOI: 10.1088/1126-6708/2008/10/065
  * [64] Nima Lashkari et al. “Towards the Fast Scrambling Conjecture” In _JHEP_ 04, 2013, pp. 022 DOI: 10.1007/JHEP04(2013)022
  * [65] Darvin Wanisch, Juan Diego Arias Espinoza and Stephan Fritzsche “Information scrambling and the correspondence of entanglement dynamics and operator dynamics in systems with nonlocal interactions” In _Phys. Rev. B_ 107.20, 2023, pp. 205127 DOI: 10.1103/PhysRevB.107.205127
  * [66] M. Heyl, A. Polkovnikov and S. Kehrein “Dynamical Quantum Phase Transitions in the Transverse-Field Ising Model” In _Phys. Rev. Lett._ 110.13, 2013, pp. 135704 DOI: 10.1103/PhysRevLett.110.135704
  * [67] Markus Heyl “Dynamical quantum phase transitions: a review” In _Rept. Prog. Phys._ 81.5, 2018, pp. 054001 DOI: 10.1088/1361-6633/aaaf9a
  * [68] Phillip Weinberg and Marin Bukov “QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems part I: spin chains” In _SciPost Phys._ 2.1, 2017, pp. 003 DOI: 10.21468/SciPostPhys.2.1.003
  * [69] Bruno Bertini, Pavel Kos and Tomaž Prosen “Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos” In _Phys. Rev. Lett._ 121.26, 2018, pp. 264101 DOI: 10.1103/PhysRevLett.121.264101
  * [70] Rajarshi Pal and Arul Lakshminarayan “Entangling power of time-evolution operators in integrable and nonintegrable many-body systems” In _Phys. Rev. B_ 98.17, 2018, pp. 174304 DOI: 10.1103/PhysRevB.98.174304
  * [71] Chenguang Liang, Yu Zhang and Shu Chen “Statistical and dynamical aspects of quantum chaos in a kicked Bose-Hubbard dimer” In _Phys. Rev. A_ 109.3, 2024, pp. 033316 DOI: 10.1103/PhysRevA.109.033316