aainstitutetext: Racah Institute of Physics, The Hebrew University, Jerusalem
9190401, Israelbbinstitutetext: Department of Theoretical Physics, University
of Geneva, 24 quai Ernest-Ansermet, 1214 Genève 4, Switzerlandccinstitutetext:
Department of Physics and Materials Science, University of Luxembourg, L-1511
Luxembourg

# A bulk manifestation of Krylov complexity

E. Rabinovici b A. Sánchez-Garrido c R. Shir b and J. Sonner
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

There are various definitions of the concept of complexity in Quantum Field
Theory as well as for finite quantum systems. For several of them there are
conjectured holographic bulk duals. In this work we establish an entry in the
AdS/CFT dictionary for one such class of complexity, namely Krylov or
K-complexity. For this purpose we work in the double-scaled SYK model which is
dual in a certain limit to JT gravity, a theory of gravity in AdS2. In
particular, states on the boundary have a clear geometrical definition in the
bulk. We use this result to show that Krylov complexity of the infinite-
temperature thermofield double state on the boundary of AdS2 has a precise
bulk description in JT gravity, namely the length of the two-sided wormhole.
We do this by showing that the Krylov basis elements, which are eigenstates of
the Krylov complexity operator, are mapped to length eigenstates in the bulk
theory by subjecting K-complexity to the bulk-boundary map identifying the
bulk/boundary Hilbert spaces. Our result makes extensive use of chord diagram
techniques and identifies the Krylov basis of the boundary quantum system with
fixed chord number states building the bulk gravitational Hilbert space.

##  1 Introduction

Notions of quantum complexity are of interest not only in the field of quantum
computing but also in the field of quantum gravity, via the framework of the
holographic duality or, more specifically, the AdS/CFT correspondence [1, 2].
Any notion of quantum complexity is expected to have a very specific time
evolution profile and to evolve for very long time scales (exponential in the
number of degrees of freedom). Similar time evolution profiles have been found
for certain diffeomorphism-invariant quantities of gravity in AdS space. In
the spirit of the AdS/CFT correspondence, a conjecture has emerged [3, 4, 5],
namely that quantum complexity is dual (in the sense of the holographic
duality) to diffeomorphism-invariant quantities such as the volume of the
wormhole in two-sided AdS space.

Quantum complexity of a quantum state is defined as the minimal number of
local unitary operations (known as “gates”) a quantum computer would need to
apply to some reference state in order to create that state, up to some chosen
precision set by a tolerance parameter. Analogously, quantum complexity can be
defined for operators themselves. The complexity of a given operator would be
the minimal number of local unitary operators whose composition equals the
given operator to within the chosen tolerance parameter.

In a chaotic system the quantum complexity of a state evolves in the following
manner: up to time scales of order the Hilbert space dimension, complexity
grows, including a long period of linear growth (this was recently proven in
[6]), and then, for a finite Hilbert space, complexity saturates at a value of
order the Hilbert space dimension. After saturation, for a finite Hilbert
space, at even larger time scales the value of complexity is expected to
undergo endless Sisyphean Poincaré recurrences visiting again and again
initial complexity values.

A concrete notion of quantum complexity which requires neither the
introduction of a particular set of local unitary operations available to a
quantum computer nor a choice of tolerance parameter, and under which the time
evolution of complexity (for states evolving under a chaotic Hamiltonian)
resembles the time-dependent profile described above, would be a natural
candidate for holographic complexity. Such a candidate should be naturally
defined on the quantum mechanical boundary as well as in the quantum gravity
bulk, preferably relating the complexity evolution of a state on the boundary
to some invariant observable’s time evolution in the bulk.

Krylov complexity, first defined in [7] for operators evolving in time in the
thermodynamic limit and later shown to satisfy the expected profile for
operator quantum complexity at all time scales in [8, 9] in a finite system,
is naturally defined for any quantum system without external (human) input.
Originally defined for an operator evolving in time under a Hamiltonian in the
Heisenberg picture, it was then defined also for states evolving under a
Hamiltonian according to the Schrödinger equation [10]. By its definition,
Krylov complexity is bounded from above by the state/operator Hilbert space
dimension and indeed can be seen to saturate in [9] in the case of operators,
while for infinite Hilbert spaces it does not saturate in general.

There have been several suggestions for the entry in the AdS/CFT dictionary
for quantum complexity. In this work we find a precise one-to-one
correspondence between Krylov complexity in a 1-dimensional quantum mechanical
boundary and a specific bulk observable in a 2-dimensional quantum gravity
theory. In particular, Krylov complexity of the thermofield double state (at
infinite temperature) in the boundary theory, which is double-scaled SYK,
becomes equivalent to wormhole length in the bulk theory, which is 2-sided
AdS2 Jackiw-Teitelboim gravity [11, 12].

Previous works on the subject include [13] which computed operator growth in
SYK via Krylov complexity and compared it with the complexity=volume
formulation in JT gravity; it was shown that the two have similar qualitative
behaviours in time. The work [14] studied the expected behaviour of
K-complexity in low-dimensional gravity theories via relationships with random
matrix theory.

This paper is organised as follows: in section 2 we review all the needed
background for double-scaled SYK (DSSYK) and for Jackiw-Teitelboim (JT)
gravity, and how they are related. In section 3 we quickly review the Krylov
construction for states, introducing the Lanczos coefficients and the Krylov
basis as well as the definition of state Krylov complexity (K-complexity). We
then identify the Lanczos coefficients in DSSYK and discuss the results for
Krylov complexity. In section 4 we study Krylov complexity in the regime of
DSSYK in which it is dual to JT gravity, and we establish the correspondence
of this quantity with the bulk wormhole length, adequately matching the
parameters on either side of the duality. Finally, in section 5 we summarize
our results and discuss their interpretation and their impact on the
understanding of higher dimensional cases.

##  2 Background

In this section we review the current state of knowledge for the double-scaled
Sachdev-Ye-Kitaev (DSSYK) model and chord diagrams [15, 16], the phase space
of JT gravity [17], and the equivalence between DSSYK (at a specific limit)
and JT gravity [18]. While the material in this section largely reviews
results from the literature, it exposes the main ingredients necessary to
match boundary Krylov complexity to its bulk counterpart.

###  2.1 Double-scaled SYK

The Sachdev-Ye-Kitaev (SYK) model [19, 20] is a strongly interacting, many-
body quantum system described by all-to-all interactions between Majorana
fermions $\psi_{i}$ with $1\leq i\leq N$ which satisfy
$\\{\psi_{i},\psi_{j}\\}=2\delta_{ij}$, grouped into $p$-body interactions
$\psi_{i_{1}}\psi_{i_{2}}\dots\psi_{i_{p}}$ through random couplings
$J_{i_{1}i_{2}\dots i_{p}}$ in the form of the Hamiltonian

$H_{SYK}=i^{p/2}\sum_{1\leq i_{1}<i_{2}<\dots<i_{p}\leq N}\,J_{i_{1}i_{2}\dots
i_{p}}\,\psi_{i_{1}}\psi_{i_{2}}\dots\psi_{i_{p}}~{}.$ (1)

The random couplings $J_{i_{1}i_{2}\dots i_{p}}$ are usually taken from a
Gaussian distribution with zero mean $\langle J_{i_{1}i_{2}\dots
i_{p}}\rangle=0$ and an appropriate non-zero variance. The model (1) has been
extensively studied for fixed $p$ and $N$, and at large $N$, as well as in the
large $p$ and large $N$ limit (see e.g. [21, 22, 23, 24]). We will follow [15]
and take both $N$ and $p$ to infinity while keeping the ratio $2p^{2}/N$
fixed, a limit known as double-scaled SYK (or DSSYK), introducing the ratio
parameter $\lambda$:

$\lambda\equiv\frac{2p^{2}}{N}~{},$ (2)

and we will take the variance $\langle J^{2}_{i_{1}i_{2}\dots i_{p}}\rangle$
to be

$\langle J^{2}_{i_{1}i_{2}\dots
i_{p}}\rangle=\frac{1}{\lambda}\binom{N}{p}^{-1}J^{2}~{}.$ (3)

As discussed in appendix A, double-scaled SYK with this choice of coupling
variance is, for small values of $\lambda$, compatible with the results for
SYK where $N$ and $p$ are large and independent respecting the hierarchy $1\ll
p\ll N$, which was studied in [22].

In [15] it was shown that in the limit where $N$ and $p$ go to infinity while
keeping $\lambda$ fixed, the ensemble averaged moments

$M_{2k}\equiv\langle\,\mathrm{Tr}(H^{2k})\,\rangle$ (4)

of the Hamiltonian (1) are given by a sum over chord diagrams. Chord diagrams
represent pairwise Wick contractions among the couplings $J_{i_{1}...i_{p}}$
of the different Hamiltonians in the product inside the trace (4), which make
the collective indices of the corresponding monomials
$\psi_{I}\equiv\psi_{i_{1}}\psi_{i_{2}}\dots\psi_{i_{p}}$ coincide. For the
$2k$-th moment, there are $2k$ paired monomials in each trace. To find the
value of such a chord diagram, it needs to be “untangled” to a configuration
where all contracted pairs are sitting one beside the other (see figure 1).

Figure 1: Untangling a chord diagram.

Once this is achieved, using the fact that $\psi_{I}^{2}=i^{p}$ for $p$ even
(it will be assumed that $p$ is even), one obtains a total phase of $i^{kp}$
coming from all the monomials squared, which exactly cancels the overall phase
coming from the $i^{p/2}$ factor in the normalization of each Hamiltonian (1).
Additionally, the system size dependence coming from the summation over the
indices is mitigated by the binomial in the variance normalization (3),
yielding bounded results in the double-scaling limit. Therefore, the
contribution of a given Wick contraction (i.e. a given chord diagram) boils
down to the overall sign obtained after commuting the Majorana monomials
through each other inside the trace until those with coincident indices are
consecutive. In the double-scaling limit, it is found [15] that the overall
effect of such a commutation is captured by a multiplicative factor $q\equiv
e^{-\lambda}$. The number of such commutations required to untangle a chord
diagram is exactly the number of intersections of chords, and hence one is
left with a simple rule for evaluating moments via chord diagrams:

$M_{2k}=\frac{J^{2k}}{\lambda^{k}}\sum_{\begin{subarray}{c}\text{chord
diagrams}\\\ \text{ with $k$ chords}\end{subarray}}q^{\text{number of
intersections}}~{}.$ (5)

To evaluate the RHS, it is useful to “cut open” each chord diagram and think
of the process of constructing it as a transition from a state $|0\rangle$
with zero chords back to a state with zero chords, through a process of
creating $k$ chords and eventually annihilating them all. An auxiliary Hilbert
space $\\{|0\rangle,|1\rangle,|2\rangle,\dots,|n\rangle\\}$, which represents
states with $0,1,2,\dots,n$ open chords, can now be introduced. An open chord
is a chord that emanated from a given Hamiltonian insertion but which has not
yet been closed by reaching another Hamiltonian insertion: Figure 2 depicts a
chord diagram of six Hamiltonian insertions interpreted as a transition from
zero open chords back to zero open chords through six intermediary steps
(Hamiltonian insertions); if we look, for instance, at the step $i=4$, we note
that there are two open chords. In the transition from $|0\rangle$ back to
$|0\rangle$, chords can be ‘created’ $|l\rangle\mapsto|l+1\rangle$ and
‘annihilated’ $|l\rangle\mapsto|l-1\rangle$ at each step.

Figure 2: “Cutting open” a chord diagram [15]. The cut-open chord diagram can
be thought of as a process in which one starts with zero open chords and ends
with zero open chords, while chords are created and annihilated in between.

All possible chord diagrams with $i$ steps (i.e. $i$ Hamiltonian insertions)
are represented by a state $|\psi^{(i)}\rangle$ in the Hilbert space. We
define such a state through its coordinates over the chord basis,

$\centering|\psi^{(i)}\rangle=\sum_{l\geq
0}\psi^{(i)}_{l}|l\rangle~{}.\@add@centering$ (6)

The coordinates $\psi^{(i)}_{l}$ are defined to be the sum over all chord
diagrams of $i$ steps with $l$ open chords, where each diagram is weighted by
$q$ to the power of the number of intersections featured in it; that is, all
diagrams with $i$ steps and $l$ open chords are represented by a state in the
Hilbert space which is proportional to $|l\rangle$, with projection
$\psi_{l}^{(i)}$, such that the sum of all diagrams of a fixed number of steps
is represented by the linear combination $|\psi^{(i)}\rangle$ given in (6).
The coordinates $\psi_{l}^{(i)}$ satisfy the following recurrence relation:

$\psi_{l}^{(i+1)}=\frac{J}{\sqrt{\lambda}}\psi_{l-1}^{(i)}+\frac{J}{\sqrt{\lambda}}(1+q+\dots+q^{l})\psi_{l+1}^{(i)}~{},$
(7)

where the first term represents the possibility that a chord was created in
the transition from step $i$ to step $i+1$, and the second term represents the
possibility that a chord was annihilated in that transition. In the case in
which the chord is annihilated at step $i+1$, there must have been $l+1$ open
chords at step $i$ so that we are left with $l$ open chords at step $i+1$, and
therefore the chord, in getting annihilated, can intersect up to $l$ other
open chords, and in each case the diagram would get multiplied by $q$ to the
number of such intersections. The term $1+q+...+q^{l}$ in (7) accounts
therefore for all the possible ways to close a chord in the step $i+1$. This
recursion (7) can be written concisely in a basis-independent way as

$|\psi^{(i+1)}\rangle=T|\psi^{(i)}\rangle~{},$ (8)

where, in coordinates over the $\\{|l\rangle\\}$ basis, the transfer matrix
$T$ takes the form:

$T\overset{*}{=}\frac{J}{\sqrt{\lambda}}\begin{pmatrix}0&\frac{1-q}{1-q}&0&0&\dots\\\
1&0&\frac{1-q^{2}}{1-q}&0&\dots\\\ 0&1&0&\frac{1-q^{3}}{1-q}&\dots\\\
0&0&1&0&\dots\\\ \vdots&\vdots&\vdots&\vdots&\ddots\\\ \end{pmatrix},$ (9)

where the asterisk above the equality sign denotes that this is an expression
in coordinates over a particular basis. At the first step $i=0$ there are no
open chords and therefore $|\psi^{(0)}\rangle=|0\rangle$. Thus,

$\centering|\psi^{(i)}\rangle=T^{i}|0\rangle~{}.\@add@centering$ (10)

Since at each step we either create or annihilate a chord, to make a chord
diagram with exactly $k$ chords we would need to apply the operator $T$ $2k$
times, creating $k$ chords and annihilating $k$ chords. The final state is
$|\psi^{(2k)}\rangle=T^{2k}|0\rangle$, and thus

$M_{2k}=\langle 0|\,T^{2k}\,|0\rangle=\langle
0|\psi^{(2k)}\rangle=\psi^{(2k)}_{0}~{}.$ (11)

In view of this, $T$ can be interpreted as an effective Hamiltonian of DSSYK.
In (9) we give its expression in coordinates over the chord basis
$\\{|l\rangle\\}_{l=0}^{\infty}$, where it is represented by a non-symmetric
matrix. In [16, 15] it was pointed out that there exists a diagonal similarity
transformation that brings $T$ into a tridiagonal, symmetric form.
Furthermore, in [18] this transformation is explained in terms of the chord
inner product defined in that article (and which was necessary to define in
order to have an actual Hilbert space): according to such an inner product,
the states $\\{|l\rangle\\}$ are orthogonal to each other, but they are not
normalized (except for $|0\rangle$). The similarity transformation amounts to
re-normalizing these states so that they form an orthonormal basis. By a
slight abuse of notation, from now on $\\{|l\rangle\\}$ will denote the
normalized fixed chord number states, i.e. $\langle
l|l^{\prime}\rangle=\delta_{l,l^{\prime}}$. In this orthonormal basis, $T$
takes the form [15, 18]:

$T\overset{*}{=}\frac{J}{\sqrt{\lambda}}\begin{pmatrix}0&\sqrt{\frac{1-q}{1-q}}&0&0&\dots\\\
\sqrt{\frac{1-q}{1-q}}&0&\sqrt{\frac{1-q^{2}}{1-q}}&0&\dots\\\
0&\sqrt{\frac{1-q^{2}}{1-q}}&0&\sqrt{\frac{1-q^{3}}{1-q}}&\dots\\\
0&0&\sqrt{\frac{1-q^{3}}{1-q}}&0&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}~{}.$ (12)

Algebraically, $T$ can be written as:

$\centering
T=\frac{J}{\sqrt{\lambda}}\left(\alpha+\alpha^{\dagger}\right)~{},\@add@centering$
(13)

where

$\centering\alpha=\sum_{l\geq 0}\sqrt{[l+1]_{q}}\quad|l\rangle\langle
l+1|~{},\@add@centering$ (14)

and where we have defined the q-number (or basic number) as:

$\centering[l]_{q}\equiv\frac{1-q^{l}}{1-q}=\sum_{k=0}^{l-1}q^{k}~{}.\@add@centering$
(15)

In fact, the ladder operators $\alpha,\,\alpha^{\dagger}$, together with the
chord number operator

$\centering\hat{l}=\sum_{l\geq 0}l|l\rangle\langle l|~{},\@add@centering$ (16)

form the algebra of a q-deformed oscillator [25]:

$\centering\begin{split}&[\alpha,\alpha^{\dagger}]_{q}\equiv\alpha\alpha^{\dagger}-q\alpha^{\dagger}\alpha=1\\\
&[\hat{l},\alpha^{\dagger}]=\alpha^{\dagger}\\\
&[\hat{l},\alpha]=-\alpha\end{split}\@add@centering$ (17)

whose representation over the $|l\rangle$ basis satisfies

$\centering\hat{l}|l\rangle=l|l\rangle,\qquad\alpha^{\dagger}|l\rangle=\sqrt{[l+1]_{q}}|l+1\rangle,\qquad\alpha|l\rangle=\sqrt{[l]_{q}}|l-1\rangle~{}.\@add@centering$
(18)

This algebra reduces to the usual oscillator algebra (the Heisenberg-Weyl
algebra) in the limit $q\to 1$, since:

  * (i)

$[A,B]_{q}\overset{q\to 1}{\longrightarrow}[A,B]$

  * (ii)

$[l]_{q}\overset{q\to 1}{\longrightarrow}l$.

From (12), the eigensystem equation for the components of the eigenvectors of
$T$, $\psi_{l}(E)=\langle l|E\rangle$, is given by:

$E\,\psi_{l}(E)=\frac{J}{\sqrt{\lambda(1-q)}}\left(\sqrt{1-q^{l+1}}\,\psi_{l+1}(E)+\sqrt{1-q^{l}}\,\psi_{l-1}(E)\right)$
(19)

In [15, 16], it was shown that the eigenvalues are a function of a continuous
variable $\theta$:

$E(\theta)=\frac{2J\,\cos\theta}{\sqrt{\lambda(1-q)}},\quad
0\leq\theta\leq\pi~{},$ (20)

and the normalized eigenvectors are given by

$\psi_{l}(\mu)=\sqrt{(q;q)_{\infty}}|(e^{2i\theta};q)_{\infty}|\frac{H_{l}(\mu|q)}{\sqrt{2\pi(q;q)_{l}}},\quad\mu=\cos\theta$
(21)

where $(a;q)_{n}\equiv\prod_{k=0}^{n-1}(1-aq^{k})$ is the q-Pochhammer
symbol111https://en.wikipedia.org/wiki/Q-Pochhammer_symbol. and $H_{l}(\mu|q)$
are the q-Hermite polynomials. Since

$\psi_{0}(\mu)=\sqrt{\frac{(q;q)_{\infty}}{2\pi}}|(e^{2i\theta};q)_{\infty}|~{},$
(22)

$\psi_{l}(\mu)$ can be written as

$\psi_{l}(\mu)=\psi_{0}(\mu)\frac{H_{l}(\mu|q)}{\sqrt{(q;q)_{l}}}~{},\quad\mu=\cos\theta~{}.$
(23)

See appendix C for a detailed review of the derivation of these results.

###  2.2 Phase space of JT gravity

We now turn to review some background on JT gravity and in particular its
phase space, following [17, 26, 27]. This theory is dual to the low-energy
regime of DSSYK, as we shall revise in section 2.3.

JT gravity is a 2-dimensional gravity theory with no propagating bulk degrees
of freedom. Its action involves the metric $g_{\mu\nu}$ and the dilaton field
$\Phi$ and is given by

$S_{JT}=\int_{\mathcal{M}}d^{2}x\sqrt{-g}\Big{[}\Phi_{0}R+\Phi(R+2/l_{AdS}^{2})\Big{]}+2\int_{\partial\mathcal{M}}\sqrt{\gamma}\Big{[}\Phi_{0}K+\Phi(K-1/l_{AdS})\Big{]}$
(24)

where $\gamma_{\mu\nu}$ is the induced metric on the boundary, $K$ is the
extrinsic curvature of the boundary and $l_{AdS}$ is a length scale of the
2-dimensional spacetime which turns out to be 2-dimensional Anti-de Sitter
(AdS2). This length is related to the cosmological constant. The terms
involving the constant $\Phi_{0}$ are topological in nature and will not
affect our discussion at the classical level. The boundary conditions

$\displaystyle ds^{2}\big{|}_{\partial\mathcal{M}}$
$\displaystyle=-\frac{dt_{b}^{2}}{\epsilon^{2}}$ (25)
$\displaystyle\Phi\big{|}_{\partial\mathcal{M}}$
$\displaystyle=\frac{\phi_{b}}{\epsilon}$ (26)

fix the induced metric $\gamma_{\mu\nu}$, with $t_{b}$ the time on the
boundary, and fix the dilaton field $\Phi$ to a positive constant $\phi_{b}$
at the boundary. In the large volume limit we have $\epsilon\to 0$, i.e.
$\epsilon$ plays the role of a boundary regulator.

Variation of the action (24) gives the following equations of motion:

$\displaystyle 0$ $\displaystyle=R+2/l_{AdS}^{2}$ (27) $\displaystyle 0$
$\displaystyle=(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}/l_{AdS}^{2})\Phi~{}.$ (28)

The first equation of motion $R=-2/l_{AdS}^{2}$ tells us that the metric
should have constant negative curvature, or in other words, it is described by
AdS2. A metric for AdS2 can be constructed via an embedding in a 3-dimensional
Minkowski spacetime with two time directions and one space direction:

$ds^{2}=-dT_{1}^{2}-dT_{2}^{2}+dX^{2}.$ (29)

AdS2 is the induced geometry on the hypersurface

$-T_{1}^{2}-T_{2}^{2}+X^{2}=-l_{AdS}^{2},$ (30)

which has boundaries at $X\to\pm\infty$, see figure 3.

Figure 3: Left: The surface $-T_{1}^{2}-T_{2}^{2}+X^{2}=-l_{AdS}^{2}$ embedded
in 3-dimensional space. The wormhole length is represented by the red line,
and the boundaries are represented by blue lines. The yellow dots represent
the anchoring points at the boundary, between which the wormhole length is
computed. The yellow lines represent the Schwarzschild horizon and the areas
colored yellow represent the regions covered by the Schwarzschild coordinates.
Right: Schematic Penrose diagram of the same geometry.

A set of global coordinates which cover the whole space is:

$\displaystyle T_{1}$ $\displaystyle=l_{AdS}\,\sqrt{x^{2}+1}\cos\tau$ (31)
$\displaystyle T_{2}$ $\displaystyle=l_{AdS}\,\sqrt{x^{2}+1}\sin\tau$
$\displaystyle X$ $\displaystyle=l_{AdS}\,x$

with $0\leq\tau<2\pi$ and $-\infty<x<\infty$. Plugging these coordinates into
(29) provides an AdS2 metric on the surface (30):

$ds^{2}/l_{AdS}^{2}=-(1+x^{2})d\tau^{2}+\frac{1}{1+x^{2}}dx^{2}~{}.$ (32)

Allowing $\tau\in\mathbb{R}$ in (32) yields the universal cover of AdS2.
Turning to the second equation of motion, (28), its solution is given in the
embedding coordinates $(T_{1},T_{2},X)$ by $\Phi=AT_{1}+BT_{2}+CX$ [28, 17].
The unique solution, modulo $SO(1,2)$ rotations in embedding space, which
respects the boundary condition (26), namely that $\Phi$ has the same
(positive) value on both boundaries, can be expressed as $\Phi\propto T_{1}$
[17]. That is:

$\Phi(x,\tau)=\Phi_{h}\sqrt{1+x^{2}}\,\cos\tau~{}.$ (33)

Another set of coordinates, with a well-defined time on the boundary, is given
by the Schwarzschild coordinates

$\displaystyle T_{1}$ $\displaystyle=l_{AdS}\,r/r_{s}$ (34) $\displaystyle
T_{2}$
$\displaystyle=l_{AdS}\,\sqrt{(r/r_{s})^{2}-1}\sinh(r_{s}t/l_{AdS}^{2})$
$\displaystyle X$
$\displaystyle=l_{AdS}\,\sqrt{(r/r_{s})^{2}-1}\cosh(r_{s}t/l_{AdS}^{2})~{},$

with $r>r_{s}$ and $-\infty<t<\infty$. These coordinates do not cover the
whole space, as shown in figure 3. The induced metric in Schwarzschild
coordinates is given by

$ds^{2}=-\frac{r^{2}-r_{s}^{2}}{l_{AdS}^{2}}dt^{2}+\frac{l_{AdS}^{2}}{r^{2}-r_{s}^{2}}dr^{2}~{}.$
(35)

The solution to (28) with boundary condition (26) acquires a simple form in
terms of the Schwarzschild coordinates:

$\Phi(r,t)=\phi_{b}\,r/l_{AdS}~{}.$ (36)

Note that when $r=r_{s}$, the two coordinate sets (31, 34) give the
relationship $1=\sqrt{1+x^{2}}\cos\tau$ which from (33) means that at
$r=r_{s}$ we have $\Phi=\Phi_{h}$. On the other hand, from (36) we have at
$r=r_{s}$ that $\Phi=\phi_{b}r_{s}/l_{AdS}$. We thus have the relationship

$r_{s}=l_{AdS}\frac{\Phi_{h}}{\phi_{b}}~{}.$ (37)

At the boundary, $r=l_{AdS}/\epsilon$ and so the metric (35) and dilaton (36)
become

$\displaystyle ds^{2}\big{|}_{\text{boundary}}$
$\displaystyle=-\frac{dt^{2}}{\epsilon^{2}}(1-\epsilon^{2}(r_{s}/l_{AdS})^{2})$
(38) $\displaystyle\Phi\big{|}_{\text{boundary}}$
$\displaystyle=\phi_{b}/\epsilon$ (39)

which from the boundary conditions (25, 26) shows that the Schwarzschild time
$t$ becomes the boundary time when $\epsilon\to 0$. At the boundary, the
relationship between $\tau$ and $t$ can be worked out by considering the
relationship between the two sets of coordinates (31, 34):

$\left(\frac{r}{r_{s}}\frac{1}{\cos\tau}\right)^{2}-1=\Big{[}\left(\frac{r}{r_{s}}\right)^{2}-1\Big{]}\cosh^{2}(r_{s}t/l_{AdS}^{2})~{}.$
(40)

On the boundary $r=\frac{l_{AdS}}{\epsilon}$, and in the limit $\epsilon\to 0$
the relationship between the global time $\tau$ and the boundary time $t_{b}$
is found to be

$\cos\tau=\frac{1}{\cosh(r_{s}t_{b}/l_{AdS}^{2})}~{}.$ (41)

####  2.2.1 The length of the wormhole and its canonical conjugate

In [17] it was shown that the phase space coordinates of JT gravity consist
of: geodesic distance between the two boundaries, at a time slice defined by
boundary time; and its canonical conjugate. The geodesic distance between two
points on the two asymptotic boundaries of AdS2 can be identified as the
length of a (Lorentzian) wormhole, see figure 3. In order to compute it for a
given boundary time $t_{b}$, we need the metric (32) and to set
$\tau=\text{const}$. We also need the value of $x$ at the boundary points
labeled by time $t_{b}$. Again, the relationship between the two sets of
coordinates (31, 34) gives

$x=\sqrt{(r/r_{s})^{2}-1}\,\cosh(r_{s}t/l_{AdS}^{2})~{}.$ (42)

At the boundary $r=l_{AdS}/\epsilon$ so setting $t=t_{b}$ in the above
equation and taking $\epsilon\to 0$ means that

$x_{b}=\frac{l_{AdS}}{\epsilon\,r_{s}}\cosh(r_{s}t_{b}/l_{AdS}^{2})~{}.$ (43)

The wormhole length is then

$\displaystyle l/l_{AdS}$
$\displaystyle=\int_{-x_{b}}^{x_{b}}\frac{dx}{\sqrt{1+x^{2}}}=2\,\textrm{arcsinh}(x_{b})=2\log\left(x_{b}+\sqrt{x_{b}^{2}+1}\right)$
(44)
$\displaystyle=2\log\left(\frac{2l_{AdS}}{\epsilon\,r_{s}}\cosh(r_{s}t_{b}/l_{AdS}^{2})\right)+O(\epsilon)~{}.$
(45)

Using $r_{s}=l_{AdS}\Phi_{h}/\phi_{b}$ from (37), the above result can be
expressed as

$l/l_{AdS}=2\log\left(\frac{2\phi_{b}}{\epsilon}\right)+2\log\left[\frac{1}{\Phi_{h}}\cosh\left(\frac{\Phi_{h}}{l_{AdS}\,\phi_{b}}\,t_{b}\right)\right]+O(\epsilon)~{}.$
(46)

The first term is infinite in the limit $\epsilon\to 0$ (since in the large
volume limit the boundary of AdS is at infinity), and this result needs to be
renormalized. The renormalized length of the wormhole is given by

$\tilde{l}/l_{AdS}=l-2\log\left(\frac{2\phi_{b}}{\epsilon}\right)=2\log\left[\cosh\left(\frac{\Phi_{h}}{l_{AdS}\,\phi_{b}}\,t_{b}\right)\right]-2\log\Phi_{h}~{}.$
(47)

This result can also be written in terms of the value of the Hamiltonian (or
energy) on the boundary [17]:

$H=\frac{2\Phi_{h}^{2}}{l_{AdS}\,\phi_{b}}=E,$ (48)

from which

$\tilde{l}/l_{AdS}=2\log\left[\cosh\left(\sqrt{\frac{E}{2l_{AdS}\,\phi_{b}}}\,t_{b}\right)\right]-\log\left(\frac{l_{AdS}\,E\phi_{b}}{2}\right)~{}.$
(49)

Together with the canonical conjugate of $\tilde{l}$, given by [17]:

$P\,l_{AdS}=\sqrt{2l_{AdS}\,E\phi_{b}}\tanh\left(\sqrt{\frac{E}{2l_{AdS}\,\phi_{b}}}t_{b}\right),$
(50)

the Hamiltonian of JT gravity takes the form [17]:

$H=\frac{1}{l_{AdS}\,\phi_{b}}\left(\frac{l_{AdS}^{2}\,P^{2}}{2}+2e^{-\tilde{l}/l_{AdS}}\right)~{}.$
(51)

This Hamiltonian is written in terms of a two-sided222 We call $\tilde{l}$ a
two-sided length because it joins the two disconnected (and regularized)
boundaries of AdS2. phase space variable $\tilde{l}$ and its conjugate
momentum, and thus the quantum description of the system will consist of a
Hilbert space spanned by the eigenfunctions of this two-sided length. This
Hilbert space is not factorizable as a product of other Hilbert spaces
describing the degrees of freedom of each side separately, as was discussed in
[17].

###  2.3 Bulk Hilbert space

This section reviews how the chord Hilbert space of DSSYK can be understood
equivalently as the bulk Hilbert space of JT gravity. The guiding principle in
[18] is the following: in a given chord diagram, it is possible to identify a
left and a right region by arbitrarily choosing two points on the
circumference that will separate them; such points are identified with
(Euclidean) future and past infinity, and one can define a constant
(Euclidean) time slice by a line whose anchoring points are on different
regions: the state on such a slice is given by the number of open chords
intersecting it (defined in a unique way [18]). This is nothing but a re-
interpretation of the discussion around (6), but the picture is now very
reminiscent of the non-factorizable, two-sided Hilbert space of JT gravity
reviewed in the preceding section. The connection can be made even more
explicit by showing that the effective DSSYK Hamiltonian becomes, in the
suitable limit, the Liouville Hamiltonian of JT gravity [15, 18], where the
length operator is in fact proportional to the chord number operator. We
review this below.

The effective Hamiltonian of the averaged theory for DSSYK, given by (13), may
be written as:

$\centering
T=\frac{J}{\sqrt{\lambda}}\left(\alpha+\alpha^{\dagger}\right)=\frac{J}{\sqrt{\lambda(1-q)}}\left(D\sqrt{1-q^{\hat{n}}}+\sqrt{1-q^{\hat{n}}}D^{\dagger}\right),\@add@centering$
(52)

where $\alpha,\,\alpha^{\dagger}$ are the ladder operators of the $q$-deformed
harmonic oscillator, while $D^{\dagger}$ is a non-normalized version of the
creation operator, which therefore acts as a unit-displacement operator on the
semi-infinite Krylov (or chord) chain:

$\centering D^{\dagger}|n\rangle=|n+1\rangle,\qquad
D|n\rangle=|n-1\rangle,\qquad D|0\rangle=0.\@add@centering$ (53)

On this chain, $\hat{n}$ plays the role of a position operator. Despite the
inherent discreteness of the semi-infinite lattice at hand, it is possible to
define a conjugate canonical momentum $p$ for $\hat{n}$ as the generator of
translations333If the discrete lattice had a finite length $K$, its Hilbert
space would be finite and therefore it would be impossible to fulfill the
canonical commutation relation $[\hat{n},p]=i$, but we would still call $p$ a
canonical momentum in the sense that it satisfies (54). Taking the suitable
limit $K\to\infty$ one recovers the semi-infinite and discrete Krylov lattice
and, restricted to a physically relevant domain, the commutator $[n,p]$ does
tend to $i$, as explained in detail in [29, 30].:

$\centering D^{\dagger}\equiv e^{-ip}.\@add@centering$ (54)

Observing (54) we note that the discreteness of the lattice manifests itself
in the fact that only displacements of unit length in $n$-space are allowed. A
continuum limit would be achieved by allowing displacements of arbitrary
length $\Delta n$, of the form $e^{-i\,\Delta n\,p}$. In this limit we would
have that $p=-i\partial_{n}$.

Before proceeding further, it is convenient to redefine the position variable
so that it acquires dimensions of length and also so that it is well-behaved
in the small-$\lambda$ limit that we shall shortly take. Introducing a
fundamental length scale $l_{f}$ about which we don’t need to be specific444It
will eventually become the AdS length. It is not a parameter of the DSSYK
theory, so one may equivalently say that chord number is related to bulk
length normalized by AdS units., we define a new length variable $l$ as:

$\centering l=l_{f}\lambda n,\@add@centering$ (55)

whose conjugate canonical momentum is therefore $k=\frac{p}{l_{f}\lambda}$,
becoming $k=-i\partial_{l}$ in the continuum limit. With this all, the
Hamiltonian becomes:

$\centering T=\frac{J}{\sqrt{\lambda(1-q)}}\left(e^{i\lambda
l_{f}k}\sqrt{1-e^{-\frac{l}{l_{f}}}}+\sqrt{1-e^{-\frac{l}{l_{f}}}}e^{-i\lambda
l_{f}k}\right).\@add@centering$ (56)

The spectrum of this Hamiltonian is bounded (for $\lambda>0$) and symmetric
about zero, as can be seen in (20), so we shall do the harmless replacement
$T\to-T$ that ensures that the ground state corresponds to the minimal
momentum $k$ [18]. This choice is arbitrary at this point but will be
important later on, as it will ensure that the triple-scaled Hamiltonian is
bounded from below. Let us write the explicit form of the Hamiltonian after
the replacement $T\mapsto-T\equiv\tilde{T}$:

$\centering\tilde{T}=-\frac{J}{\sqrt{\lambda(1-q)}}\left(e^{i\lambda
l_{f}k}\sqrt{1-e^{-\frac{l}{l_{f}}}}+\sqrt{1-e^{-\frac{l}{l_{f}}}}e^{-i\lambda
l_{f}k}\right).\@add@centering$ (57)

We can now take the remaining limit that brings us to the so-called triple-
scaling limit. It is a small-$\lambda$ limit in which $l$ is taken to be
accordingly large, as follows555Our definition of the triple-scaling limit
differs by a factor of $2$ from that in [18]. See appendix B for a discussion
on this choice, which eventually does not imply any fundamental modification
of the bulk theory.:

$\centering\lambda\to 0,\qquad
l\to\infty,\qquad\frac{e^{-\frac{l}{l_{f}}}}{(2\lambda)^{2}}\equiv
e^{-\frac{\tilde{l}}{l_{f}}}\;\text{fixed.}\@add@centering$ (58)

The last condition in (58) can be rewritten as:

$\centering\frac{\tilde{l}}{l_{f}}=\frac{l}{l_{f}}-2\log\left(\frac{1}{2\lambda}\right).\@add@centering$
(59)

We shall call $\tilde{l}$ the renormalized length666It will become the actual
renormalized bulk length eventually.. In this triple-scaled limit, the
Hamiltonian (57) becomes777This can be shown in the continuum limit, using
$k=-i\partial_{l}$, it still holds for the discrete, semi-infinite chain using
the commutation relation $[l,k]=i$, which is satisfied within the relevant
physical domain as argued in [29, 30].:

$\centering\tilde{T}-E_{0}=2\lambda
J\left(\frac{l_{f}^{2}k^{2}}{2}+2e^{-\frac{\tilde{l}}{l_{f}}}\right)\;+\mathit{O}\left(\lambda^{2}\right),\@add@centering$
(60)

where $E_{0}=\frac{-2J}{\lambda}$ is a constant energy shift. The moral of
this triple-scaling is now clear: we are zooming near the ground state $E_{0}$
of the Hamiltonian (57), and in this regime it takes the form of the
Hamiltonian of Liouville quantum mechanics, whose spectral density is
proportional to $\sinh(2\pi\sqrt{E})$ [31], which is in turn the Hamiltonian
describing the dynamics of the single pair of phase space variables of JT
gravity, as seen in (51). This establishes the correspondence between triple-
scaled SYK and JT gravity: their Hilbert spaces are identified and the
Hamiltonian generating dynamics is the same. Incidentally, we note that
instances of DSSYK with different values of $\lambda$ and $J$ may collapse, in
the triple-scaling limit, onto the same Liouville Hamiltonian if their product
$\lambda J$ is equal, since this is the only parameter controlling the
Liouville Hamiltonian (60), and hence the gravity dual. In other words, the
following parameter identification connects Hamiltonians (51) and (60):

$l_{f}=l_{AdS},\qquad\qquad\qquad 2\lambda J=\frac{1}{l_{AdS}\phi_{b}}.$ (61)

We will come back to these identifications in section 4, where boundary
K-complexity is matched to the corresponding bulk length computation.

##  3 Krylov complexity and chords

In this section we quickly review the definition of Krylov complexity and then
provide exact results for it in DSSYK. Krylov complexity measures the
spreading of states or operators in a quantum system under Hamiltonian time
evolution, over a special ordered basis (sometimes called the Krylov chain)
adapted to the state/operator’s time evolution. Here, we will focus on the
definition of K-complexity for states. Given an initial state $|0\rangle$ at
$t=0$, its time evolution generated by the Hamiltonian $H$ in the Schrödinger
picture is given by

$|\phi(t)\rangle=e^{-iHt}|0\rangle=\sum_{n=0}^{\infty}\frac{(-it)^{k}}{k!}H^{k}|0\rangle$
(62)

and is written as a linear combination over the basis
$\\{|0\rangle,H|0\rangle,H^{2}|0\rangle,\dots\\}$. The survival amplitude,
which is the overlap of the time-evolving state with the state at $t=0$ (also
known as fidelity), is given by

$\langle 0|\phi(t)\rangle=\langle
0|e^{-iHt}|0\rangle=\sum_{n=0}^{\infty}\frac{(-it)^{k}}{k!}\langle
0|H^{k}|0\rangle\equiv\sum_{n=0}^{\infty}\frac{(-it)^{k}}{k!}M_{k}$ (63)

where we defined the moments $M_{k}\equiv\langle 0|H^{k}|0\rangle$. If the
survival probability $\langle 0|\phi(t)\rangle$ is an even function of $t$
then all odd moments $M_{2k+1}$ are zero. This, in particular, is the case for
the effective Hamiltonian of DSSYK with the initial state being the zero-chord
state $|0\rangle$.

The Krylov basis is obtained via the Lanczos algorithm which is essentially a
Gram-Schmidt orthonormalization procedure over the basis
$\\{|0\rangle,H|0\rangle,H^{2}|0\rangle,\dots\\}$. For an even survival
probability function (with odd moments being zero) the Lanczos algorithm takes
the form888In the general case, where odd moments are not necessarily zero,
the Lanczos algorithm is a little more involved. See e.g.
https://en.wikipedia.org/wiki/Lanczos_algorithm. :

  1. 1.

$|A_{1}\rangle=H|0\rangle$, compute $b_{1}=|A_{1}|$, if $b_{1}=0$ stop.
Otherwise define $|1\rangle=|A_{1}\rangle/b_{1}$.

  2. 2.

For $n\geq 1$: $|A_{n+1}\rangle=H|n\rangle-b_{n}|n-1\rangle$, compute
$b_{n+1}=|A_{n+1}|$, if $b_{n+1}=0$ stop. Otherwise define
$|n+1\rangle=|A_{n+1}\rangle/b_{n+1}$.

Here it was assumed that the initial state $|0\rangle$ is normalized. In this
way the ordered Krylov basis (or Krylov chain) $\\{|n\rangle\\}_{n=0}^{K-1}$
is constructed and the Lanczos coefficients $\\{b_{n}\\}_{n=1}^{K-1}$ are
defined, where $K$ is the dimension of the Krylov space. $K$ can be infinite
if the original Hilbert space has infinite dimension, and must be finite
otherwise. By construction, the Krylov basis vectors form an orthonormal
basis, satisfying $\langle m|n\rangle=\delta_{mn}$. In the Krylov basis, the
Hamiltonian acquires the following tridiagonal form determined by the Lanczos
coefficients:

$H|n\rangle=b_{n+1}|n+1\rangle+b_{n}|n-1\rangle~{}.$ (64)

The Lanczos coefficients can be determined from the moments of the survival
amplitude (63), via the iterative procedure [32]:

$\displaystyle M_{2k}^{(n)}$
$\displaystyle=\frac{M_{2k}^{(n-1)}}{b_{n-1}^{2}}-\frac{M_{2k-2}^{(n-2)}}{b_{n-2}^{2}},\quad
n=1,2,\dots,L\quad k=n,n+1,\dots,L$ $\displaystyle b_{n}^{2}$
$\displaystyle=M_{2n}^{(n)}$ (65)

where $M_{2k}^{(0)}\equiv M_{2k}$ as well as $M_{2k}^{(-1)}=0$ and
$b_{-1}=1=b_{0}$. Using this algorithm, the first $L$ Lanczos coefficients can
be determined from the moments $M_{2k}$ with $k=0,\dots,L$.

The time-evolving state can now be expanded over the Krylov basis:

$|\phi(t)\rangle=\sum_{n=0}^{K-1}\phi_{n}(t)|n\rangle$ (66)

where $\phi_{n}(t)\equiv\langle n|\phi(t)\rangle$ can be thought of as a
“wavefunction” spreading over the ordered Krylov basis with initial condition
$\phi_{n}(t=0)=\langle n|0\rangle=\delta_{n0}$. From unitarity of the time
evolution, the norm is preserved, i.e. $\sum_{n=0}^{K-1}|\phi_{n}(t)|^{2}=1$
for all $t$. Note that $\phi_{0}(t)$ is the survival probability defined in
(63).

A ‘position’ operator, $\hat{n}$, over the ordered Krylov basis can be defined
as

$\displaystyle\hat{n}=\sum_{n=0}^{K-1}n\,|n\rangle\langle n|,$ (67)

and Krylov complexity is then defined as the expectation value of $\hat{n}$ as
a function of time, or equivalently, the average position of the wavefunction
$|\phi(t)\rangle$ over the ordered Krylov basis:

$C_{K}(t)=\langle\hat{n}(t)\rangle=\langle\phi(t)|\hat{n}|\phi(t)\rangle=\sum_{n=0}^{K-1}n\,|\phi_{n}(t)|^{2}~{}.$
(68)

From (64) we can obtain a recurrence relation for the eigenvectors of $H$,
$H|E\rangle=E|E\rangle$, since the components of each eigenvector $|E\rangle$
over the Krylov basis, defined as $\psi_{n}(E)\equiv\langle n|E\rangle$,
satisfy:

$E\,\psi_{n}(E)=b_{n+1}\psi_{n+1}(E)+b_{n}\psi_{n-1}(E)~{}.$ (69)

The components of the eigenvectors of $H$ over the Krylov basis are useful in
the determination of the wavefunctions $\phi_{n}(t)$, and if known, together
with the eigenvalues of $H$, they provide the following closed-form expression
for the wavefunction as a function of time:

$\phi_{n}(t)\equiv\langle n|\phi(t)\rangle=\langle
n|e^{-iHt}|0\rangle=\sum_{E}e^{-iEt}\langle n|E\rangle\langle
E|0\rangle=\sum_{E}e^{-iEt}\,\psi_{n}(E)\,\psi_{0}^{*}(E)~{}.$ (70)

###  3.1 Lanczos coefficients in DSSYK

Observing the steps leading to the expression of the effective Hamiltonian in
coordinates over the basis of fixed chord number states, given in matrix form
in (12), the connection to Lanczos coefficients is argued as follows. The
fixed chord number states $\\{|l\rangle\\}$ are an orthonormal set of linear
combinations of states with up to $l$ Hamiltonian insertions $T^{l}|0\rangle$;
and using them as a basis, they bring the transfer matrix (or effective
Hamiltonian) to a tridiagonal form. They thus turn out to be Krylov basis
elements, with the matrix elements of the symmetric, tridiagonal $T$ giving
the Lanczos coefficients:

$\centering
b_{n}=J\sqrt{\frac{[n]_{q}}{\lambda}}=J\sqrt{\frac{1-q^{n}}{\lambda(1-q)}}~{}.\@add@centering$
(71)

Consequently, the chord number operator $\hat{l}$, which gives the position in
the chord number basis, gives equivalently the position in the Krylov basis,
so it is the K-complexity operator.

There is certainly more than one orthonormal basis that can bring the
effective Hamiltonian $T$ to tridiagonal form. However, given a seed state
$|0\rangle$, there is a unique (up to global phases) orthonormal basis
$\\{|n\rangle\\}$ such that the state $|n\rangle$ is a linear combination of
$\\{T^{k}|0\rangle\\}_{k=0}^{n}$, as can be shown inductively. Such a basis
can therefore be identified, up to global phases, with the Krylov basis
adapted to the state $|0\rangle$ and the Hamiltonian $T$, which can be built
efficiently through the Lanczos algorithm and brings the Hamiltonian to
tridiagonal form with real entries. The fact that the matrix expression (12)
for the effective Hamiltonian in the fixed chord number basis has positive
entries indicates that such a basis coincides exactly with the Krylov basis
(i.e. they do not differ even by a global phase). This argument proves that
the fixed chord number states are Krylov elements and therefore that the
entries of the tridiagonal version of $T$ in (12) are the Lanczos
coefficients. In addition, we can perform a non-trivial check of the fact that
the sequence (71) is actually the Lanczos coefficients sequence by taking the
moments $M_{2k}$ computed via chord-diagram combinatorics and showing that the
usual transformation (65) will indeed yield a sequence of Lanczos coefficients
in agreement with expression (71). Using chord diagrams, we compute explicitly
the first three even moments:

$\centering\begin{split}&M_{2}=\frac{J^{2}}{\lambda}\\\
&M_{4}=\frac{J^{4}}{\lambda^{2}}(2+q)\\\
&M_{6}=\frac{J^{6}}{\lambda^{3}}(5+6q+3q^{2}+q^{3})\end{split}\@add@centering$
(72)

where $M_{6}$ involves evaluating the 15 chord diagrams shown in figure 4.

Figure 4: The 15 chord diagrams contributing to $M_{6}$: the top row shows
chord diagrams with zero intersections (5 diagrams), the second row shows
chord diagrams with 1 intersection (6 diagrams), the third row shows diagrams
with 2 intersections (3 diagrams) and the fourth row shows the only chord
diagram with 3 intersections. These numbers correspond to the calculation of
$M_{6}$ in (72).

With this, we can apply the recursion method (65):

$\centering\begin{split}&b_{1}^{2}=M_{2}=\frac{J^{2}}{\lambda}=\frac{J^{2}}{\lambda}[1]_{q}\\\
&b_{2}^{2}=\frac{M_{4}}{M_{2}}-M_{2}=\frac{\frac{J^{4}}{\lambda^{2}}(2+q)}{\frac{J^{2}}{\lambda}}-\frac{J^{2}}{\lambda}=\frac{J^{2}}{\lambda}(1+q)=\frac{J^{2}}{\lambda}[2]_{q}\\\
&b_{3}^{2}=\frac{\frac{M_{6}}{M_{2}}-M_{4}}{\frac{M_{4}}{M_{2}}-M_{2}}-\frac{M_{4}}{M_{2}}=(...)=\frac{J^{2}}{\lambda}(1+q+q^{2})=\frac{J^{2}}{\lambda}[3]_{q}~{}.\end{split}\@add@centering$
(73)

Equations (73) illustrate that the Lanczos coefficients (71) are consistent
with the Hamiltonian moments (72).

####  3.1.1 State dependence of the Lanczos coefficients

An important discussion is in order: what initial state are these Lanczos
coefficients being computed for? They are the Lanczos coefficients associated
to the evolution under the effective Hamiltonian $T$ of the initial state
$|0\rangle$, given by the state

$\centering|\phi(t)\rangle=e^{-itT}|0\rangle,\@add@centering$ (74)

as the moments to which the coefficients (71) are in one-to-one correspondence
are the coefficients of the Taylor series of the survival amplitude of (74):

$\centering\langle
0|e^{-itT}|0\rangle=\sum_{k=0}^{+\infty}\frac{(-it)^{2k}}{(2k)!}M_{2k},\@add@centering$
(75)

where the moments $M_{2k}$ are given in (11).

Furthermore, the state $|0\rangle$ that seeds the evolution can be thought of
as an effective version of the infinite-temperature thermofield “double” state
in the ensemble-averaged theory, and the survival amplitude (75) is in fact
the ensemble-averaged analytic continuation of the partition function, which
is itself known to give the survival amplitude of the thermofield double state
[33, 10]. We shall now elaborate on this. The ensemble-averaged analytic
continuation of the partition function, evaluated at infinite temperature,
reads:

$\centering\left.\left\langle
Z(\beta+it)\right\rangle\right|_{\beta=0}=\left\langle\text{Tr}\left[e^{-itH}\right]\right\rangle.\@add@centering$
(76)

Now, one can check that the trace in (76) can be rewritten as the expectation
value of the evolution operator in a certain state $|\Omega\rangle$, as
follows:

$\centering\text{Tr}\left[e^{-itH}\right]=\langle\Omega|e^{-itH}|\Omega\rangle,\qquad\qquad\qquad|\Omega\rangle\equiv\frac{1}{\sqrt{\mathcal{N}}}\sum_{E}|E\rangle,\@add@centering$
(77)

where $\mathcal{N}\to\infty$ denotes the Hilbert space dimension, and the
factor $1/\sqrt{\mathcal{N}}$ is required for consistency with the convention
$\text{Tr}[\mathbb{1}]=1$, ensuring normalization of the state
$|\Omega\rangle$. By inspection, we note that this state is an
“unconventional” version of the infinite-temperature thermofield “double”
state, in the sense that it does not belong to the tensor product of two
identical Hilbert spaces but rather to the only Hilbert space available in the
problem. This feature will be consistent with the bulk interpretation of this
Hilbert space as the non-factorizable two-sided Hilbert space of JT gravity
[17], as discussed in [18].

Finally, taking the ensemble average amounts to replacing the non-averaged
Hamiltonian $H$ by the effective Hamiltonian $T$, and the state
$|\Omega\rangle$ by the zero-chord state $|0\rangle$ :

$\centering\left\langle\text{Tr}[e^{-itH}]\right\rangle=\Big{\langle}\langle\Omega|e^{-itH}|\Omega\rangle\Big{\rangle}=\langle
0|e^{-itT}|0\rangle,\@add@centering$ (78)

which connects, as promised, the averaged analytic continuation of the
partition function (76) to the survival amplitude (75) and provides the
justification for considering the zero-chord state $|0\rangle$ as an effective
version of the infinte-temperature thermofield “double” state in the averaged
theory.

####  3.1.2 Regimes of the Lanczos sequence

In order to better understand the different regimes of K-complexity as a
function of time, it will be useful to analyze the regimes that can be
identified in the Lanczos sequence (71). At this point, we recall that
$q=e^{-\lambda}=e^{-\frac{2p^{2}}{N}}$. For $\lambda>0$ we have $0<q<1$ and
the Lanczos coefficients $b_{n}$ in (71) are bounded. They have a horizontal
asymptote at $b_{\infty}=\frac{J}{\sqrt{\lambda(1-q)}}>0$. In the limit
$\lambda\to+\infty$ (i.e. $q\to 0$) the Lanczos sequence is constant
$b_{n}\sim\frac{J}{\sqrt{\lambda}}\to 0$, while in the limit $\lambda\to 0$ we
have $q\to 1$ and the Lanczos coefficients grow indefinitely as $b_{n}\sim
J\sqrt{\frac{n}{\lambda}}$. The limit $\lambda\to 0$ is of particular
importance since in [18] the connection to gravity is done in a triple scaling
limit in which $\lambda\ll 1$ (i.e. $q\to 1$).

In order to identify regimes in the Lanczos sequence, the above analysis needs
to be performed more carefully. The arguments above have considered limits of
$\lambda$ keeping $n$ fixed, whereas below we shall study the behavior of the
$b_{n}$ sequence given a fixed parameter $\lambda>0$ for $n$ either
sufficiently big or small compared to $\lambda$.

Inspecting (71) we note that, given $0<q<1$, for sufficiently large $n$, the
factor $q^{n}$ will be small and therefore we can use the approximation:

$\centering
b_{n}\approx\frac{J}{\sqrt{\lambda(1-q)}}\left(1-\frac{q^{n}}{2}\right),\qquad\text{for
large }n.\@add@centering$ (79)

Conversely, if $|n\log(q)|$ is sufficiently small, we can approximate
$1-q^{n}=1-e^{-n\log(1/q)}\approx n\log(1/q)=\lambda n>0$ (for $n>0$), so
that:

$\centering b_{n}\approx J\sqrt{\frac{n}{1-q}},\qquad\text{for small
}n.\@add@centering$ (80)

We can estimate the critical value $n_{*}(q)$ that marks the transition
between the regimes (80) and (79) by taking the value of $n$ for which
$n\log(1/q)$ becomes of order $1$, which yields:

$\centering
n_{*}(q)\approx\frac{1}{\log\left(\frac{1}{q}\right)}=\frac{1}{\lambda}.\@add@centering$
(81)

Summarizing, we have found that the Lanczos coefficients start growing as
$\propto\sqrt{n}$ before $n_{*}(q)$, after which they saturate at a horizontal
asymptote following $\propto 1-\frac{q^{n}}{2}$:

$\centering
b_{n}=\left\\{\begin{split}&J\sqrt{\frac{n}{1-q}},\qquad\qquad\qquad\quad\;\,n\lesssim\frac{1}{\lambda}\\\
&\frac{J}{\sqrt{\lambda(1-q)}}\left(1-\frac{q^{n}}{2}\right),\qquad
n\gtrsim\frac{1}{\lambda}\end{split}\right.\@add@centering$ (82)

Note that $\lim_{q\to 1^{-}}n_{*}(q)=\lim_{\lambda\to
0^{+}}n_{*}(q(\lambda))=+\infty$, consistent with the fact that when $q\to 1$
we have $b_{n}\propto\sqrt{n}$, i.e. only the square-root behavior is
featured. Additionally, inspecting (81) we realize that for $\lambda>1$ (or,
equivalently, $q<e^{-1}\approx 0.37$) the transition value of $n$ is
$n_{*}<1$, which implies that the first regime in (82) will not be visible at
all in the actual Lanczos sequence, since strictly speaking $n\in\mathbb{N}$.
In the cases of interest for us we will need to consider both regimes, as the
connection to gravity occurs in a limit where $\lambda$ is small. Figure 5
depicts several Lanczos sequences for different values of $q$, and figure 6
shows the different regimes in the Lanczos sequence for a particular value of
$q$ close to $1$, together with the $q$-dependence of the transition value
$n_{*}(q)$.

Figure 5: Lanczos coefficients in DSSYK for different values of $q$, denoted
$b_{n}\equiv b(n,q)$. Left: Linear scale along both axes, focusing at small
values of $n$. Right: Log-log plot. In this scale, the initial linear shape is
compatible with a square-root behavior of $b_{n}$; figure 6 illustrates this
in more detail.

Figure 6: Left: Lanczos coefficients of DSSYK for $\lambda=0.01$ (i.e.
$q\approx 0.99005$), together with the limiting regimes for small and large
$n$. Right: Transition value $n_{*}(q)$ as a function of $q$ (in blue). The
constant $n_{*}=1$ has been marked in orange, for reference. Note the
logarithmic scale along the vertical axis. In particular, we note that
$n_{*}(q)$ has a vertical asymptote at $q=1$, implying that in the limit $q\to
1$, which is of interest in this paper, there will be a clear separation of
scales in the Lanczos sequence.

For reference, we can rewrite (82) for small $\lambda$ (i.e. $q$ close to
$1$):

$\centering
b_{n}\approx\left\\{\begin{split}&J\sqrt{\frac{n}{\lambda}},\qquad\qquad\qquad
n\lesssim\frac{1}{\lambda}\\\
&\frac{J}{\lambda}\left(1-\frac{e^{-n\lambda}}{2}\right),\qquad
n\gtrsim\frac{1}{\lambda}\end{split}\right.\@add@centering$ (83)

where the corrections are subleading in an expansion in powers of
$\sqrt{\lambda}$.

###  3.2 K-complexity regimes in DSSYK

The regimes of the Lanczos coefficients (82) allow us to distinguish different
regimes in the growth of K-complexity. The initial condition of the wave
packet is $\phi_{n}(0)=\delta_{n0}$, where $\phi_{n}(t)$ are the coordinates
of the state $|\phi(t)\rangle$, defined in (74), over the Krylov basis (i.e.
the basis of fixed-chord-number states); see (70) for the precise definition.
Therefore, at sufficiently early times, the state wave packet only probes the
first regime of (82), where the Lanczos coefficients exhibit a square-root
behavior. Following [34, 10] we can recycle the results for the Heisenberg-
Weyl algebra. Namely, the Hamiltonian is given by

$\centering T=\gamma\left(a+a^{\dagger}\right),\@add@centering$ (84)

where $\gamma=\frac{J}{\sqrt{1-q}}=\frac{J}{\sqrt{1-e^{-\lambda}}}$ and
$a,\,a^{\dagger}$ act over the Krylov basis as ladder operators of a simple
bosonic harmonic oscillator, satisfying the usual algebra

$\centering[a,a^{\dagger}]=1,\@add@centering$ (85)

i.e. in this regime the q-deformation of the algebra (17) is not important as
long as we restrict to the subspace of states $\\{|n\rangle\;/\;n\lesssim
n_{*}(q)\\}$. Using the Baker-Campbell-Hausdorff formula and the simple
commutation relation (85), the exact wave function $\phi_{n}(t)$ is computed
in [34], as this is nothing but the evolution along a one-parameter family of
coherent states:

$\centering\phi_{n}(t)=e^{-\frac{\gamma^{2}t^{2}}{2}}\frac{(-i\gamma
t)^{n}}{\sqrt{n!}}~{}.\@add@centering$ (86)

The result (86) for the Krylov space wave function can equivalently be
obtained using the spectral decomposition of the tridiagonal Hamiltonian.
Using this method, in section 3.4 we shall derive a formal expression for the
wave functions $\phi_{n}(t)$ at arbitrary $q$ and show that it reduces to (86)
in the $q\to 1$ limit in section 3.4.1. Likewise, in appendix E we present an
independent diagonalization of the tridiagonal Hamiltonian at $q=1$.

From this wave function K-complexity is given by:

$\centering
C_{K}(t)=\sum_{n=0}^{+\infty}n|\phi_{n}(t)|^{2}=\gamma^{2}\,t^{2}=\frac{(tJ)^{2}}{1-q}=\frac{(tJ)^{2}}{1-e^{-\lambda}}\,.\@add@centering$
(87)

For small $\lambda$, we can approximate (87) as
$C_{K}(t)\approx\frac{(tJ)^{2}}{\lambda}$, up to subleading corrections in a
$\lambda$-expansion.

We can use $C_{K}(t)$ as an estimate for the position of the peak of the
coherent packet $\phi_{n}(t)$. Thus, the packet will start to probe the second
region of (82), where the Lanczos coefficients approach a horizonal asymptote,
when $C_{K}(t)$ becomes of the order of $n_{*}(q)$. We therefore define a
transition time scale $t_{*}(q)$ by the relation

$\centering C_{K}\Big{(}t_{*}(q)\Big{)}=n_{*}(q),\@add@centering$ (88)

giving

$\centering
t_{*}(q)=\frac{1}{J}\sqrt{\frac{1-q}{\log\left(\frac{1}{q}\right)}}=\frac{1}{J}\sqrt{\frac{1-e^{-\lambda}}{\lambda}},\@add@centering$
(89)

which behaves as $t_{*}\approx J^{-1}$ for small $\lambda$. We note that, as
$\lambda$ goes to zero, the transition value for $n$ goes to infinity, since
$n_{*}=\frac{1}{\lambda}$. On the other hand, K-complexity at early times,
$C_{K}(t)\sim\frac{1}{\lambda}(tJ)^{2}$, has an accordingly increasing growth
rate, so that the $\lambda$-dependence cancels out in the expression of the
transition time $t_{*}$, which is only controlled by $J^{-1}$.

Well after $t_{*}(q)$ the wave packet probes a region where the Lanczos
coefficients are effectively constant, $b_{n}\approx
b_{\infty}\equiv\frac{J}{\sqrt{\lambda(1-q)}}$. According to [8], the wave
functions $\phi_{n}(t)$ in this case are given by Bessel functions, the
position of whose front-most peak evolves in time as $\sim 2b_{\infty}\,t$.
Using the peak position as an estimate for K-complexity, we obtain in this
case:

$\centering
C_{K}(t)\approx\frac{2\,tJ}{\sqrt{\lambda(1-q)}}\,.\@add@centering$ (90)

In summary:

$\centering
C_{K}(t)=\left\\{\begin{split}&\frac{(tJ)^{2}}{1-q}\,,\qquad\qquad\qquad\quad\;\,t\lesssim
t_{*}(q)\\\ &\frac{2\,tJ}{\sqrt{\lambda(1-q)}}\,+c(\lambda)\,,\qquad t\gg
t_{*}(q)\end{split}\right.\@add@centering$ (91)

where $c(\lambda)$ is some $\lambda$-dependent constant that should, strictly
speaking, be there in order to ensure the matching of the two regimes. For
small $\lambda$ this becomes:

$\centering
C_{K}(t)\approx\left\\{\begin{split}&\frac{(tJ)^{2}}{\lambda}\,,\qquad\qquad\;\,t\lesssim\frac{1}{J}\\\
&\frac{2\,tJ}{\lambda}+c(\lambda)\,,\qquad
t\gg\frac{1}{J}.\end{split}\right.\@add@centering$ (92)

Figure 7 depicts the different regimes of K-complexity, compared with the
numerical result, for a value of $q$ close to $1$.

Figure 7: K-complexity regimes in DSSYK. The plots compare the exact
$C_{K}(t)$ computed numerically (in blue) with the early (green) and late
(red) time regimes described in (91). The numerical computation imposed an
artificial cutoff $K=5000$ in the length of the Lanczos sequence, but the plot
only depicts a time range during which the wave packet doesn’t yet probe the
right edge of the artificially finite Krylov chain, hence not featuring
spurious finite-size effects. Left: Linear scale on both axes. Note that the
curve for the late-time approximation is parallel to the exact result but it
doesn’t exactly overlap with it; this is because of the lack of knowledge of
the constant $c(\lambda)$ in (91), which in the case of this plot was chosen
to make the early- and late-time regimes match at $t=t_{*}(q)$. Right:
Logarithmic scale along the horizontal axis. This allows us to note the good
agreement at early times between the exact result and the corresponding
approximation. Additionally, the logarithmic scale washes away the effect of
the constant $c$, and the red and blue curves do overlap at late times,
confirming the expected linear growth of $C_{K}(t)$ at times well after
$t_{*}(q)$.

###  3.3 Continuum approximation to K-complexity in DSSYK

In this section we obtain an analytic expression for the K-complexity of the
infinite-temperature thermofield “double” state in DSSYK (i.e. the zero-chord-
number state) making use of a continuum approximation of Krylov space.
Generically, the approximation developed in [8], reviewed in appendix D,
consists of promoting $n$ to a continuous variable in a way such that the
recursion relation describing the evolution of the wave function $\phi_{n}(t)$
becomes a differential equation which is, at leading order, a chiral wave
equation with a velocity field given by $v(n)\equiv 2b_{n}$. As a result of
this equation, the wave packet propagation is ballistic and one can estimate
the value of K-complexity by the position of the peak of the wave function,
which behaves as a point particle propagating through the above-mentioned
velocity field $v(n)$. As reviewed in appendix D, for a finite Krylov space of
dimension $K$, this continuum approximation is controlled by the small
parameter $\varepsilon=\frac{1}{K-1}$, but it turns out that, due to some
divisions by $\varepsilon$ in intermediate steps of the discussion, the
applicability of this approximation in the strict thermodynamic limit
$K\to\infty$ is not well under control, even if strictly speaking
$\varepsilon=0$ in that limit.

In the particular model at hand in this article, however, the parameter
$\lambda$ in (2) can be used to control the continuum approximation even in
the $K\to\infty$ limit, as we shall argue here. Inspecting the expression of
the Lanczos coefficients (71) we note that we can define a dimensionless
position variable as $x\equiv\lambda n$ in such a way that its spacing goes to
zero when $\lambda\to 0$, becoming a continuous variable. More precisely, a
continuum limit for the Krylov space of DSSYK can be defined as the limit in
which $\lambda\to 0$, $n\to\infty$ with $x=\lambda n$ fixed. In this limit,
the Lanczos coefficients adopt the limiting form of a continuous function of
$x$:

$\centering b_{n}=J\sqrt{\frac{1-e^{-\lambda
n}}{\lambda(1-q)}}\longrightarrow\frac{J}{\lambda}\sqrt{1-e^{-x}}+\mathit{O}(\lambda^{0})\equiv
b(x).\@add@centering$ (93)

As in the argument reviewed in appendix D, the starting point for studying
this continuum limit is the recursion relation satisfied by the wave function.
Defining a real $\varphi_{n}(t)$ such that $\phi_{n}(t)=i^{n}\varphi_{n}(t)$,
we have that (64) and (66) imply the recursion:

$\centering\dot{\varphi}_{n}(t)=b_{n}\varphi_{n-1}(t)-b_{n+1}\varphi_{n+1}(t).\@add@centering$
(94)

Since in the above-mentioned $\lambda\to 0$ limit the Lanczos coefficients
$b_{n}$ are given by a continuous function loosely denoted by $b(x)$, we can
assume that the wave functions $\varphi_{n}(t)$ are also described by a
continuous function $f(t,x)$ such that $\varphi_{n}(t)=f(t,n\lambda)$. With
this, the recursion (94) reads:

$\centering\partial_{t}f(t,x)=b(x)f(t,x-\lambda)-b(x+\lambda)f(t,x+\lambda).\@add@centering$
(95)

Expanding in powers of $\lambda$ we find:

$\partial_{t}f(t,x)=-v(x)\partial_{x}f(t,x)-\frac{v^{\prime}(x)}{2}f(t,x)+\mathit{O}(\lambda),$
(96)

which at leading order becomes a first-order wave equation for $f(t,x)$ with
velocity field $v(x)\equiv 2\lambda b(x)$. Thanks to the fact that $b(x)$
scales like $\frac{1}{\lambda}$ in (93), the velocity field $v(x)$ is of order
$\lambda^{0}$:

$v(x)=2\lambda b(x)=2J\sqrt{1-e^{-x}}+\mathit{O}(\lambda)\overset{\lambda\to
0}{\longrightarrow}2J\sqrt{1-e^{-x}}.$ (97)

In order to manipulate (96) further, we change the position variable to $y$
such that $x=0\Rightarrow y=0$ and $dy=\frac{dx}{v(x)}$. This, together with
the wave function redefinition

$\centering
g(t,y)\equiv\sqrt{v\left(x(y)\right)}f\left(t,x(y)\right),\@add@centering$
(98)

yields the equation

$\left(\partial_{t}+\partial_{y}\right)g(t,y)=0+\mathit{O}(\lambda).$ (99)

Equation (99) becomes exact for $\lambda=0$. This reasoning is analogous to
the analysis in appendix D, where the small parameter is
$\varepsilon=\frac{1}{K-1}$ instead of $\lambda$; however, in that case
redefinitions like (98) and $dy=\frac{dx}{v(x)}$ turn out to be problematic
because they imply multiplication or division by zero when $\varepsilon=0$,
which does not occur in the present case because $v(x)$ remains finite in the
$\lambda\to 0$ limit.

The initial condition $\phi_{n}(t)=\delta_{n0}$ on the Krylov chain
translates, in the continuum limit, into $f(0,x)=\delta(x)$, and therefore
$g(0,y)$ is also proportional to $\delta(y)$. We note that, given an initial
condition $g(0,y)\equiv g_{0}(y)$, the solution of (99) simply propagates the
initial condition as $g(t,y)=g_{0}(y-t)$, and hence in our case we have
$g(t,y)\propto\delta(y-t)$. Consequently, the position expectation value of
the wave packet, which gives K-complexity (up to the factor $\lambda$ used to
define $x=\lambda n$ in the continuum limit) is simply given by the position
of the peak $x_{p}(t)$, which behaves as a point particle evolving in time
following the velocity vield $v(x)$:

$\centering
t=\int_{0}^{y_{p}(t)}dy=\int_{0}^{x_{p}(t)}\frac{dx}{v(x)}~{}.\@add@centering$
(100)

This approximation can therefore be thought of as a classical approximation
where the evolution of the wave packet is replaced by the propagation in
$x$-space of a point particle. In fact, it is possible to show that the
relation $\dot{x}=v(x)=2J\sqrt{1-e^{-x}}$ is a classical solution of the
equation of motion generated by a Hamiltonian with an exponential potential:

$\centering H^{\prime}\equiv E_{0}+2\lambda
J\left(\frac{l_{f}^{2}k^{2}}{2}+\frac{2}{(2\lambda)^{2}}e^{-\frac{l}{l_{f}}}\right),\@add@centering$
(101)

where we have written $x$ as $\frac{l}{l_{f}}$. $H^{\prime}$ is a Liouville
Hamiltonian, but it is not quite equal to the effective Hamiltonian
$\tilde{T}$ of DSSYK given in (57) at small $\lambda$. This is a manifestation
of the fact that these two Hamiltonians are only classically equivalent. The
classical limit999As discussed in [18], the classical limit is defined as the
limit in which $\lambda\to 0$ and $\lambda k$ is held fixed. In this way,
$[l,k]=i$ but $[l,\lambda k]=i\lambda\to 0$. of (57) is:

$\centering\tilde{T}_{\text{class}}=-\frac{2J}{\lambda}\cos(\lambda
l_{f}k)\sqrt{1-e^{-\frac{l}{l_{f}}}}\qquad+\qquad\text{subleading}~{},\@add@centering$
(102)

and one can verify that Hamiltonians (101) and (102) yield the same Euler-
Lagrange equation of motion $\ddot{x}=2J^{2}e^{-x}$, which produces the above-
mentioned trajectory $v(x)$ upon choosing the initial conditions $x(0)=0$ and
$\dot{x}(0)=0$. However, the two Hamiltonians differ at the quantum level. In
order to retrieve a (quantum) Liouville Hamiltonian from DSSYK it is
necessary, as described in section 2.3, to take a triple-scaling limit in
which $\lambda$ is taken to be small but $\frac{l}{l_{f}}=\lambda n$ is taken
to be sufficiently large such that
$\frac{e^{-l/l_{f}}}{(2\lambda)^{2}}=e^{-\tilde{l}/l_{f}}$ is fixed, in order
to remain close to the ground state of the system. We will revisit this in
section 4, where we will compute K-complexity in the regime in which DSSYK is
dual to JT gravity.

We now resume the computation of K-complexity in the continuous $\lambda\to 0$
limit of DSSYK. Since formally $x=\lambda n$ and $v(x)=2\lambda b(x)$, we can
use (100) and give an approximation for K-complexity in DSSYK where we just
promote $n$ to be a continuous variable and keep $\lambda$ fixed:

$\centering\int_{0}^{n_{p}(t)}\frac{dn}{v(n)}=t,\qquad v(n)\equiv
2b_{n}=2J\sqrt{\frac{1-q^{n}}{\lambda(1-q)}}.\@add@centering$ (103)

The integral (103) can be performed analytically, yielding an implicit
equation for $n_{p}(t)$:

$\centering
tJ=\sqrt{\frac{1-q}{\lambda}}\text{arctanh}\sqrt{1-q^{n_{p}(t)}},\@add@centering$
(104)

which can be solved for $n_{p}(t)$. We reach:

$\centering C_{K}(t)\approx
n_{p}(t)=\frac{2}{\lambda}\log\left\\{\cosh\left[tJ\sqrt{\frac{\lambda}{1-q}}\right]\right\\}.\@add@centering$
(105)

This approximation to K-complexity, which results from promoting $n$ (instead
of $x=\lambda n$) to a continuous variable, is expected to be good at small
$\lambda$, which is when the continuum limit (for the variable $x$) applies.
The numerical analysis presented below confirms this expectation.

In section 4 we will recover this functional behavior of K-complexity from a
gravity computation, being more specific about the bulk-boundary matching.
Furthermore, expression (105) recovers exactly the early- and late-time
regimes identified in (91):

  * •

At early times, using $\cosh(x)=1+\frac{x^{2}}{2}+\mathit{O}(x^{4})$ and
$\log(1+x)=x+\mathit{O}(x^{2})$, we find

$\centering
C_{K}(t)\approx\frac{2}{\lambda}\log\left\\{1+(tJ)^{2}\frac{\lambda}{2(1-q)}\right\\}\approx\frac{(tJ)^{2}}{1-q}\,,\@add@centering$
(106)

which matches perfectly the first line in (91).

  * •

At late times we just make use of the asymptotic behavior of
$\cosh{x}\sim\frac{e^{x}}{2}$, giving

$\centering
C_{K}(t)\approx\frac{2}{\lambda}\log\left\\{\frac{1}{2}\exp\left[tJ\sqrt{\frac{\lambda}{1-q}}\right]\right\\}=\frac{2\,tJ}{\sqrt{\lambda(1-q)}}\,-\frac{2\log{2}}{\lambda},\@add@centering$
(107)

which also agrees with the second line of (91) and even provides an estimate
for the constant $c(\lambda)$.

Finally, the transition time between (106) and (107) can be estimated as the
value of $t$ for which the argument of the $\cosh$ in (105) becomes of order
1. This yields

$\centering
t_{*}(q)\approx\frac{1}{J}\sqrt{\frac{1-q}{\lambda}},\@add@centering$ (108)

which agrees with our previous estimate (89), and therefore it also behaves as
$t_{*}\sim J^{-1}$ for small $\lambda$.

The fact that the regimes (106) and (107) match (91) is a non-trivial check,
since the analysis leading to (91) did not assume the continuum approximation:
it was achieved by analysing separately the different sectors of the Lanczos
sequence and using known solutions for the discrete recurrence relation for
such cases. We elaborate on this below:

  * •

At early times $t<t_{*}(q)$, both the continuum approximation and the exact
discrete result derived using the Heisenberg-Weyl algebra yield
$C_{K}(t)\approx\frac{(tJ)^{2}}{1-q}$. We may intuitively understand this
agreement by noting that the continuum approximation can be seen as the
classical propagation of a point particle in Krylov space, while the discrete
result consists of the propagation of coherent states, which are known to
behave semiclassically.

  * •

At late times $t\gg t_{*}(q)$ the exact, discrete solution for $\phi_{n}(t)$
consists of Bessel functions, see appendix E.1, whose front-most peak
propagates in $n$-space at a constant velocity equal to
$2b_{\infty}=\frac{2J}{\sqrt{\lambda(1-q)}}$. Approximating K-complexity by
the position of this peak yields the estimate
$C_{K}(t)\approx\frac{2\,tJ}{\sqrt{\lambda(1-q)}}$, which agrees with the
point-particle propagation from the continuum approximation at late times.
However, such ballistic approximation does not always accurately apply to the
discrete, exact solution, which features a wave packet that develops a tail as
it propagates through Krylov space.

Numerics show that the continuum (and therefore ballistic) approximation is
better the closer $q$ is to $1$, as illustrated in figures 8, 9 and 10: we
studied the values $q=0.99005,\,0.74082,\,0.3$, for which the continuum
approximation induces an error of around $0.1\%$, $3\%$ and $10\%$,
respectively. In fact, the exact solution for the discrete problem with
constant Lanczos coefficients $b_{n}=b$ yields
$C_{K}(t)=\frac{16}{3\pi}bt+o(t)\approx 1.7\,bt+o(t)$, as shown in (195). An
approximation estimating $C_{K}(t)$ from the position of the front-most peak
of the wave packet would give, even in the discrete case, $C_{K}(t)\sim 2bt$,
resulting in a relative deviation (at late times) of
$\frac{2-\frac{16}{3\pi}}{\frac{16}{3\pi}}\approx 0.18$. In figure 11 we
observe that the relative deviation between the continuum approximation and
the numerical result for K-complexity at $q=0.01$ tends precisely to this
value, confirming that the error is (mainly) due to the ballistic
approximation, which misses the development of a tail behind the wavefront
that the exact discrete solution features. In all numerical computations we
artificially truncated the Krylov chain, giving it a finite dimension
$K=5000$: time-dependent results are therefore only reliable during the time
interval in which the wave-packet does not yet probe this artificial edge of
Krylov space, and this is the time range to which the above-mentioned plots
are restricted.

Figure 8: Comparison between the exact $C_{K}(t)$ computed numerically and the
continuum approximation (105) for DSSYK with $\lambda=0.01$ ($q\approx
0.99005$), i.e. the same value as in figure 7. The agreement is excellent. Top
left: Linear scale along both axes. Top right: Logarithmic scale along the
horizontal axis. Bottom: Normalized deviation as a function of time.

Figure 9: Continuum approximation vs exact (numerical) $C_{K}(t)$ for
$\lambda=0.3$ (i.e. $q\approx 0.74082$).

Figure 10: Continuum approximation vs exact (numerical) $C_{K}(t)$ for
$\lambda\approx 1.20397$ (i.e. $q=0.3$).

Figure 11: Left: Exact $C_{K}(t)$ computed numerically for $\lambda\approx
4.61$ (i.e. $q=0.01$), when the Lanczos sequence (71) becomes very close to a
constant $b_{n}\approx\frac{J}{\sqrt{\lambda}}$, vs the result of the
continuum approximation. Right: Logarithmic scale along the horizontal axis.
Bottom: Relative deviation. Its long-time average is $\sim 0.177$, very close
to the value expected for the relative deviation of the ballistic estimate
with respect to the exact result in the discrete analysis.

For reference, we can rewrite (105) for small $\lambda$ (i.e. $q$ close to
$1$):

$\centering
C_{K}(t)\approx\frac{2}{\lambda}\log\left\\{\cosh\left(tJ\right)\right\\},\@add@centering$
(109)

which in turn recovers the early- and late-time regimes listed in (92).

Understood as a classical description, the results of this continuum
approximation of Krylov space can be useful for the gravity interpretation:
[18] discusses that, since chord number (which we now understand as
K-complexity) is a discrete quantity, it provides a discretized, or quantized,
version of the bulk length. At this point, we propose to make the converse
argument and claim that this continuum approximation of Krylov space should
match gravity results obtained from a classical, geometric description. We
will provide further evidence in favor of this point of view in section 4. A
further argument in favor of interpreting the continuum approximation (109) as
a classical description of Krylov space dynamics consists in noting, as we
have explained, that it coincides with the classical trajectory generated by
the Hamiltonian (101).

###  3.4 Exact formal results for K-complexity in DSSYK

In this section we will present the formal result for Krylov complexity in
DSSYK as a function of $q$, following the prescription for the wavefunctions
in (70). We will use the results for the eigenvectors and eigenvalues of the
effective Hamiltonian $T$ (shown at the end of section 2.1) directly in (70).
The sum over energies will be replaced by an integral over $\theta$, as
discussed in [15, 16] and reviewed after equation (155) in appendix C. Thus:

$\displaystyle\phi_{n}(t)$
$\displaystyle=\int_{0}^{\pi}d\theta\,e^{-itE(\cos\theta)}\,\psi_{n}(\cos\theta)\,\psi_{0}(\cos\theta)$
(110)
$\displaystyle=\int_{0}^{\pi}\frac{d\theta}{2\pi}\,e^{-2iJt\frac{\cos\theta}{\sqrt{\lambda(1-q)}}}\,\frac{(q;q)_{\infty}}{\sqrt{(q;q)_{n}}}\,|(e^{2i\theta};q)_{\infty}|^{2}\,H_{n}(\cos\theta|q).$
(111)

In the second line we used the expressions (20), (22) and (23) for the
eigenvalues and eigenvectors of $T$. The small $\lambda$ limit, $q\to 1$, of
this expression is discussed in section 3.4.1. Exact results for the
wavefunctions for $q=0$ and for $q=1$ are given in appendix E.

It can be checked that the wavefunction is normalized, i.e.
$\sum_{n=0}^{\infty}|\phi_{n}(t)|^{2}=1$, as follows:

$\displaystyle\sum_{n=0}^{\infty}|\phi_{n}(t)|^{2}$
$\displaystyle=[(q;q)_{\infty}]^{2}\int_{0}^{\pi}\frac{d\theta}{2\pi}\int_{0}^{\pi}\frac{d\phi}{2\pi}e^{-\frac{2iJt}{\sqrt{\lambda(1-q)}}(\cos\theta-\cos\phi)}|(e^{2i\theta};q)_{\infty}|^{2}|(e^{2i\phi};q)_{\infty}|^{2}$
$\displaystyle\times\sum_{n=0}^{\infty}\frac{1}{(q;q)_{n}}H_{n}(\cos\theta|q)H_{n}(\cos\phi|q)=\frac{(q;q)_{\infty}}{2\pi}\int_{0}^{\pi}d\theta\,|(e^{2i\theta};q)_{\infty}|^{2}=1$

where (170) was used in the second equality and (173) with $n=m=0$ was used in
the final equality.

We are now ready to write down a formal expression for K-complexity in double-
scaled SYK as a function of $q=e^{-\lambda}$:

$\displaystyle C_{K}(t)=\sum_{n=0}^{\infty}n|\phi_{n}(t)|^{2}=$
$\displaystyle[(q;q)_{\infty}]^{2}\int_{0}^{\pi}\frac{d\theta}{2\pi}\int_{0}^{\pi}\frac{d\phi}{2\pi}e^{-\frac{2iJt}{\sqrt{\lambda(1-q)}}(\cos\theta-\cos\phi)}|(e^{2i\theta};q)_{\infty}|^{2}|(e^{2i\phi};q)_{\infty}|^{2}$
$\displaystyle\times\sum_{n=0}^{\infty}\frac{n}{(q;q)_{n}}H_{n}(\cos\theta|q)H_{n}(\cos\phi|q)~{}.$
(112)

####  3.4.1 The $q\to 1$ limit

In the $q\to 1$ limit we can reproduce the result (86) by carefully taking the
$\lambda\to 0$ limit in the general expression for the wavefunction (111). We
begin by finding expressions101010See
https://mathworld.wolfram.com/q-PochhammerSymbol.html. for $(x;q)_{\infty}$
and $(q;q)_{\infty}$ for $q=e^{-\lambda}$ in the limit $\lambda\to 0$, as well
as for $H(\cos\theta|q)$ and $(q;q)_{n}$:

$\displaystyle(x;q)_{\infty}$
$\displaystyle\approx\exp\Big{[}-\frac{1}{\lambda}\mathrm{Li}_{2}(x)+\frac{1}{2}\log(1-x)+O(\lambda)\Big{]}$
(113) $\displaystyle(q;q)_{\infty}$
$\displaystyle\approx\sqrt{\frac{2\pi}{\lambda}}\exp\Big{[}-\frac{\pi^{2}}{6\lambda}+\frac{\lambda}{24}+O(\lambda^{2})\Big{]}$
(114) $\displaystyle(q;q)_{n}$ $\displaystyle\approx n!\lambda^{n}$ (115)
$\displaystyle H(\cos\theta;q)$ $\displaystyle\approx
2^{n}\cos^{n}(\theta)~{}.$ (116)

From the first identity above we find that for $q=e^{-\lambda}$ in the limit
$\lambda\to 0$,

$\displaystyle|(e^{2i\theta};q)_{\infty}|^{2}$
$\displaystyle=(e^{2i\theta};q)_{\infty}(e^{-2i\theta};q)_{\infty}\approx\sqrt{1-e^{2i\theta}}\sqrt{1-e^{-2i\theta}}\exp\Big{[}\mathrm{Li}_{2}(e^{2i\theta})+\mathrm{Li}_{2}(e^{-2i\theta})\Big{]}$
$\displaystyle=2|\sin\theta|\exp\Big{[}\mathrm{Li}_{2}(e^{2i\theta})+\mathrm{Li}_{2}(e^{-2i\theta})\Big{]}.$
(117)

Plugging these expansions into (111) while keeping only first order terms in
$\lambda$ and expanding $\sqrt{1-q}\approx\sqrt{\lambda}$, we have:

$\displaystyle\phi_{n}(t)$ $\displaystyle\approx
2\sqrt{\frac{2\pi}{\lambda}}\frac{2^{n}}{\sqrt{n!\lambda^{n}}}\int_{0}^{\pi}\frac{d\theta}{2\pi}e^{-\frac{1}{\lambda}\left[\frac{\pi^{2}}{6}+\mathrm{Li}_{2}(e^{2i\theta})+\mathrm{Li}_{2}(e^{-2i\theta})+2itJ\cos\theta\right]}|\sin\theta|\cos^{n}(\theta).$
(118)

To proceed we will use the identity111111See for example
https://dlmf.nist.gov/25.12.

$\mathrm{Li}_{2}(e^{2i\theta})+\mathrm{Li}_{2}(e^{-2i\theta})=\frac{\pi^{2}}{3}-2\pi\theta+2\theta^{2}$
(119)

which gives us
$-\frac{1}{\lambda}[\pi^{2}/2-2\pi\theta+2\theta^{2}]=-\frac{2}{\lambda}(\theta-\pi/2)^{2}$
in the exponential term. We change variables to $x=\theta-\pi/2$ to get

$\displaystyle\phi_{n}(t)\approx
2\sqrt{\frac{1}{2\pi\lambda}}\frac{2^{n}}{\sqrt{n!\lambda^{n}}}\int_{-\pi/2}^{\pi/2}dx\,e^{-\frac{2}{\lambda}(x^{2}-itJ\sin
x)}|\cos x|(-\sin x)^{n}~{}.$ (120)

For $\lambda\to 0$, and $tJ\ll 1$, the main contribution to this integral is
for $|x|\ll 1$, hence we expand $\sin(x)\approx x$ and $\cos x\approx 1$:

$\displaystyle\phi_{n}(t)\approx\sqrt{\frac{2}{\pi\lambda}}\frac{2^{n}}{\sqrt{n!\lambda^{n}}}\int_{-\pi/2}^{\pi/2}dx\,e^{-\frac{2}{\lambda}(x^{2}-2itJx)}(-x)^{n}~{}.$
(121)

Completing the square, we find another Gaussian integral:

$\displaystyle\phi_{n}(t)\approx\sqrt{\frac{2}{\pi\lambda}}\frac{(-2)^{n}}{\sqrt{n!\lambda^{n}}}e^{-\frac{(tJ)^{2}}{2\lambda}}\int_{-\pi/2}^{\pi/2}dx\,e^{-\frac{2}{\lambda}\left(x-\frac{itJ}{2}\right)^{2}}x^{n}~{}.$
(122)

Using a saddle-point approximation for $\lambda\to 0$, we set
$x\approx\frac{itJ}{2}$ and perform the Gaussian integral:

$\phi_{n}(t)\approx\sqrt{\frac{2}{\pi\lambda}}\frac{(-2)^{n}}{\sqrt{n!\lambda^{n}}}e^{-\frac{(tJ)^{2}}{2\lambda}}\sqrt{\frac{\pi\lambda}{2}}\left(\frac{itJ}{2}\right)^{n}=e^{-\frac{(tJ)^{2}}{2\lambda}}\frac{(-itJ/\sqrt{\lambda})^{n}}{\sqrt{n!}}~{},$
(123)

arriving at the result (86) which is indeed expected for $\lambda\to 0$ and
$tJ\ll 1$. Interestingly, this analysis agrees with our previous estimate
(108) and tells us that in the limit $q\to 1$ the transition time remains
finite, $\lim_{q\to 1^{-}}t_{*}(q)=J^{-1}$, and hence the Heisenberg-Weyl
result still only applies at early times.

##  4 Bulk wormhole length and Krylov complexity

We have now assembled all the ingredients necessary to carry out the main task
we set ourselves in this work: giving a precise match of bulk and boundary
Krylov complexity in the context of DSSYK. Let’s recall the two main building
blocks for the argument:

  * •

In section 2.3 we reviewed the identification between the bulk JT Hilbert
space and the boundary Hilbert space of double-scaled SYK, whose dynamics in
the triple-scaling limit are generated by the Liouville Hamiltonian. From this
identification one concludes that fixed chord number states are bulk length
eigenstates.

  * •

In section 3 we showed that fixed chord-number states are equal to the Krylov
basis elements associated to the zero chord-number state (or the infinite-
temperature thermofield “double” state).

Putting these two items together it follows that the Krylov basis elements are
bulk length eigenstates. Therefore, the position expectation value on the
Krylov chain (i.e., K-complexity, as expressed in (68) via the position
operator (67)) gives the length expectation value. This establishes the
correspondence between K-complexity and two-sided bulk length.

In order to be more specific, we need to go to the regime of DSSYK in which
its Hamiltonian becomes that of JT gravity. We argued in section 3.3 that only
taking the continuous $\lambda\to 0$ limit is not sufficient for this, because
in that case the DSSYK Hamiltonian is only classically equivalent to a
Liouville Hamiltonian. As reviewed in section 2.3, in order to find a quantum
Liouville Hamiltonian as some limiting form of the full DSSYK Hamiltonian, it
is necessary to perform a triple-scaling limit (58) that effectively zooms in
near the ground state of the spectrum. In terms of the regularized length
$\tilde{l}$ defined in this limit, the resulting Hamiltonian is (60), which we
restate here:

$\centering\tilde{T}=E_{0}+2\lambda
J\left(\frac{l_{f}^{2}k^{2}}{2}+2e^{-\frac{\tilde{l}}{l_{f}}}\right)\;+\mathit{O}\left(\lambda^{2}\right).\@add@centering$
(124)

Incidentally, we note that (124) can be reached as the result of applying the
triple-scaling limit to (101). In this procedure, the regularized length
introduced through the triple-scaling limit becomes necessary in order to get
a well-behaved limiting Hamiltonian in which kinetic and potential terms
appear at the same order. For the sake of clarity, figure 12 depicts a diagram
relating the various limiting forms of the Hamiltonian discussed in this
paper.

Figure 12: Diagram relating the various limiting forms of the Hamiltonian
studied in this paper. The top left expression gives the effective Hamiltonian
for the averaged theory of DSSYK, written in (57). Its classical limit is the
top right equation, stated in (102). As explained in section 3.3, this
Hamiltonian yields the same Euler-Lagrange equation of motion as the Liouville
Hamiltonian depicted in the bottom right block and presented in (101). As
indicated by the yellow arrow, this Hamiltonian is relevant because the
continuous analysis of the DSSYK Hamiltonian presented in 3.3 yields a
position expectation value that follows a classical trajectory generated by
this bottom-right Liouville Hamiltonian. Finally, the bottom left box contains
the triple-scaled Hamiltonian (60): a Liouville Hamiltonian to which the full
DSSYK Hamiltonian reduces in the triple-scaling limit. This limit has the
effect of zooming near the ground state, and therefore this is an effective
low-energy Hamiltonian. This is the regime where DSSYK is dual to JT gravity.
More technically, the introduction of the regularized position variable is
required in order to obtain potential and kinetic terms of the same order in a
$\lambda$-expansion, as outlined by the horizontal arrow at the bottom of the
diagram. For simplicity and despite the notational abuse, in this figure $k$
denotes the (dimensionless) conjugate momentum of $x$.

In short, in this section we will perform a K-complexity calculation analogous
to that in section 3.3, specialized to the regime of DSSYK in which it becomes
dual to JT gravity. For this, let’s first argue that it is still possible to
relate the evolution of regularized length to K-complexity in the triple-
scaled system. Eigenstates of $\tilde{l}$ can be regarded as the continuum
version of position eigenstates on a lattice with labels $\tilde{n}$ such that
$\frac{\tilde{l}}{l_{f}}=\lambda\tilde{n}$. We note that $\tilde{n}$ differs
from $n$ in a $\lambda$-dependent shift, necessary to zoom in near the ground
state of the original $n$-lattice. Via an argument analogous to that in
section 3.1.1 relating the infinte-temperature thermofield “double” state of
the full Hamiltonian to the $|n=0\rangle$ state in the averaged theory, we can
associate the infinite-temperature thermofield “double” state of the low-
energy Hamiltonian to the $|\tilde{n}=0\rangle$ state in the averaged theory.
Finally, from the fact that
$\frac{\widehat{\tilde{l}}}{l_{f}}=\lambda\widehat{\tilde{n}}$ holds as an
operator identity, taking expectation value on both sides yields the relation
$\frac{\tilde{l}(t)}{l_{f}}=\lambda\widetilde{C_{K}}(t)$, between bulk
regularized length in JT and the K-complexity $\widetilde{C_{K}}(t)$ of the
infinite-temperature thermofield double state of triple-scaled SYK in the
averaged theory.

We can elaborate further on K-complexity from the point of view of the
$\tilde{n}$-lattice. Its Lanczos coefficients are obtained by performing the
triple-scaling protocol on (71):

$\centering
b_{n}=\frac{J}{\lambda}\sqrt{1-(2\lambda)^{2}q^{\tilde{n}}}+\mathit{O}(\lambda^{0})=b-2\lambda
Jq^{\tilde{n}}+\mathit{O}(\lambda^{2}),\@add@centering$ (125)

where the constant $b=\frac{J}{\lambda}+\mathit{O}(\lambda^{0})$ is related to
the ground-state energy. We note that $q^{\tilde{n}}=e^{-\lambda\tilde{n}}$ is
fixed in the triple-scaling limit in such a way that
$\lambda\tilde{n}\equiv\frac{\tilde{l}}{l_{f}}$ does not scale with $\lambda$.
In this limit, the variable $\tilde{l}$ becomes continuous; however, the
analysis of the recurrence equation in Krylov space in this continuum limit is
slightly different from the case in section 3.3: here, one does not reach the
conclusion that $2\lambda b(\tilde{x})$ (where
$\tilde{x}\equiv\frac{\tilde{l}}{l_{f}}$) plays the role of a velocity,
because the orders of the $\lambda$-expansion of $b(\tilde{x})$ get mixed with
those coming from the displacement operator. Instead, it can be checked that
the differential equation that this continuous analysis yields is nothing but
the Schrödinger equation dictated by the triple-scaled Hamiltonian (124),
consistently. Furthermore, the small-$\lambda$ prefactor of this Hamiltonian
allows for a classical approximation in which the expectation value of
$\tilde{l}$ is given by the solution to the equation of motion dictated by the
Liouville Hamiltonian (124) derived on the boundary. We perform such a
classical evaluation using the initial conditions
$\frac{\widetilde{l}(0)}{l_{AdS}}=\widetilde{x}_{0}$,
$\dot{\widetilde{l}}(0)=0$ and, setting $l_{f}=l_{AdS}$ as announced in (61),
we reach:

$\centering\lambda\widetilde{C_{K}}(t)=\frac{\widetilde{l}(t)}{l_{AdS}}=\widetilde{x}_{0}+2\log\left\\{\cosh\left(2\lambda
Je^{-\widetilde{x}_{0}/2}\,t\right)\right\\},\@add@centering$ (126)

which is a solution with energy121212For the sake of notational simplicity, we
define the energy $E$ as the difference between the total energy and the
ground-state energy, i.e. $E\equiv\langle\tilde{T}\rangle-E_{0}$, where
$\tilde{T}$ is given in (124).

$\centering E=4\lambda Je^{-\tilde{x}_{0}}.\@add@centering$ (127)

For fixed $\widetilde{x}_{0}$, this is consistent with the fact that the
Hamiltonian (124) describes configurations that constitute excitations of
energies of order $\lambda$, i.e. close to the ground state. This allows us to
rewrite (126) as:

$\centering\lambda\widetilde{C_{K}}(t)=\frac{\widetilde{l}(t)}{l_{AdS}}=2\log\left\\{\cosh\left(t\,\sqrt{E\lambda
J}\right)\right\\}-\log\left(\frac{E}{4\lambda J}\right).\@add@centering$
(128)

Recalling that the identification between the bulk JT Hamiltonian (51) and the
triple-scaled boundary Hamiltonian (124) reviewed in section 2.3 implies
$2\lambda J=\frac{1}{l_{AdS}\phi_{b}}$, as stated in (61), we have that (128)
reads:

$\centering\lambda\widetilde{C_{K}}(t)=\frac{\widetilde{l}(t)}{l_{AdS}}=2\log\left\\{\cosh\left(t\,\sqrt{\frac{E}{2l_{AdS}\phi_{b}}}\right)\right\\}-\log\left(\frac{El_{AdS}\phi_{b}}{2}\right),\@add@centering$
(129)

which coincides exactly with the gravity computation of the regularized length
(49).

A final identification is required for consistency. Comparing the expression
for the configuration energy (127) with that in the gravity computation (48),
we find that:

$\centering\widetilde{x}_{0}=-2\log\Phi_{h}~{}.\@add@centering$ (130)

That is: from the bulk point of view, the choice of $\Phi_{h}$ amounts to to
the choice of a coordinate patch adapted to an accelerated (Rindler) observer
in the boundary [35, 28] that sees the bulk as a black hole whose horizon lies
at the locus where the dilaton takes the value $\Phi_{h}$. From the
perspective of the triple-scaled Hamiltonian derived from the boundary theory,
this is equivalent to choosing the initial condition for the solution of the
equation of motion. The relation between such an initial condition and the
dilaton field at the horizon is precisely (130). In particular, the
K-complexity of the infinite-temperature thermofield “double” state starts at
zero when $t=0$, corresponding to the choice $\Phi_{h}=1$.

To summarize, in this section we have revisited the need to perform a triple-
scaling limit in order to find a (quantum) Liouville Hamiltonian near the
ground state of DSSYK, hence retrieving the regime in which the bulk theory is
described by JT gravity. This limit defines a notion of regularized length in
terms of which the Krylov problem can be posed in a manner analogous to that
in section 3, where we studied DSSYK away from the triple-scaling limit.
Making use of the bulk/boundary Hilbert space identification reviewed in
section 2.3, we were able to identify the K-complexity of the infinite-
temperature thermofield “double” state of the triple-scaled Hamiltonian with
the expectation value of bulk regularized length, which we could evaluate
classically thanks to the smallness of $\lambda$, which plays the role of an
$\hbar$ parameter controlling the semiclassical expansion. This correspondence
is summarized in (128), where the excitation energy is given by

$\centering E=\frac{2\Phi_{h}^{2}}{l_{AdS}\phi_{b}}=4\lambda
Je^{-\tilde{x}_{0}}.\@add@centering$ (131)

For completeness, we may now elaborate on the features of the emergent bulk
described by the triple-scaled SYK Hamiltonian, together with the accelerated
observer patch specified by the initial condition $\tilde{x}_{0}$. This
discussion is based on the parameter identifications (61) and (130). The bulk
consists of a black hole with a horizon radius given by131313Since
$l_{f}=l_{AdS}$, we shall avoid notational cluttering by simply denoting this
length scale by $L$. (37):

$\centering r_{s}=\frac{\Phi_{h}}{\phi_{b}}L=2\lambda
JL^{2}e^{-\tilde{x}_{0}/2}~{}.\@add@centering$ (132)

Similarly, the black hole temperature can be computed through [27]:

$\centering T_{\text{BH}}=\frac{r_{s}}{2\pi L^{2}}=\frac{\lambda
J}{\pi}e^{-\tilde{x}_{0}/2}~{}.\@add@centering$ (133)

For the particular case of the infinite-temperature thermofield “double” state
of triple-scaled SYK, we have $\tilde{x}_{0}=0$ and therefore
$T_{BH}=\frac{\lambda J}{\pi}$, that is, a temperature of order $\lambda$.
This is consistent with the fact that it is the infinite temperature
thermofield “double” state of the low-energy Hamiltonian, rather than that of
the full DSSYK Hamiltonian, with respect to which it is actually a low-
temperature state because it zooms in near the ground state of the system.

As we have stressed, neither $r_{s}$ nor $T_{\text{BH}}$ are a property of
global AdS2, but of the coordinates adapted to an observer that sees a horizon
in the same way that a Rindler observer in flat space would perceive a horizon
due to its acceleration141414They may be understood as an actual black hole
radius and temperature whenever the 2-dimensional gravity setup comes from the
near-horizon limit of a higher dimensional near-extremal black hole [27, 36].
In such a case, $\Phi_{h}$ can also be understood as a measure of the black
hole entropy.. Accordingly, the bulk theory (60) is solely controlled by the
parameters $L$ and $2\lambda J$, while the choice of a patch (specified by
$\Phi_{h}$) is equivalent to the choice of the initial condition
$\widetilde{x}_{0}$ for the equation of motion of (124).

##  5 Summary and discussion

In this work we have studied K-complexity in the double-scaled SYK and its
bulk dual, JT gravity in AdS2. We began by establishing that the fixed chord-
number states, in terms of which the Hilbert space of DSSYK can be
constructed, are nothing but the Krylov basis elements for the effective
Hamiltonian of the averaged theory and for the infinite-temperature
thermofield “double” state, understood as the linear combination with equal
weights of all the eigenstates of the theory. The associated sequence of
Lanczos coefficients was read off the matrix elements of the effective
Hamiltonian expressed in coordinates over the chord basis, and they were used
to compute the K-complexity profile as a function of time following various
procedures involving increasingly higher levels of sophistication and detail,
which we summarize below.

In a first approach, we used the different regimes (in $n$-space) of the
Lanczos coefficients in order to estimate the K-complexity profile in its
different time regimes. These regimes are defined as the intervals of time
during which the wave packet defined over Krylov space remains mostly
contained within the corresponding region where the Lanczos coefficients take
a particular limiting form. We found that the $b_{n}$ sequence of the
infinite-temperature thermofield “double” state transitions from a square-root
growth, $b_{n}\sim\sqrt{n}$, to saturation at a plateau in $n$-space, which
implied a K-complexity profile that transitions from quadratic growth at early
times to linear growth at times greater than $J^{-1}$, the inverse of the
DSSYK coupling strength. We did not obtain a saturation of K-complexity at
late times as, due to the fact that we are studying the double-scaled SYK, the
system is already in the thermodynamic limit, where the Hilbert space has
infinite-dimension: K-complexity saturation is a non-perturbative effect that
appears at finite late times that are exponentially large in the number of
degrees of freedom [8, 9], and therefore this time scale gets pushed away to
infinity in the limit under consideration. We also note that in this limit the
spectrum of the effective Hamiltonian is continuous. Even for finite systems
with a continuous spectrum non-perturbative behaviors require a case-by-case
consideration: in some cases the behavior associated with the discreteness of
the spectrum survives (e.g. in averaged but finite SYK the plateau behavior
persists in the spectral form factor), while calculating a two-point function
in a black hole background [37, 38] yields a result that tends asymptotically
to zero as a function of time.

A further refinement consists in using a continuum approximation to Krylov
space dynamics in order to obtain a smooth function $C_{K}(t)$ for the
K-complexity profile. As a result of this approximation, the evolution of the
wave packet in Krylov space was reduced to the propagation of a point particle
driven by a space-dependent velocity field whose profile is given by the
(continuous limit of the) Lanczos coefficients. We found that $C_{K}(t)$ grows
in time following a log-cosh functional form that recovers the quadratic- and
linear-growth behaviors in the corresponding regimes, consistent with the
previous analysis. The accuracy of this continuum approximation is determined
by $\lambda$, the “ ’t Hooft coupling” in DSSYK, which was found to play the
role of an $\hbar$-like parameter controlling the semiclassical expansion.

Last, we provided a formal expression of K-complexity obtained by introducing
a resolution of the identity in the definition of the Krylov space wave packet
in terms of the spectral decomposition of the effective Hamiltonian of DSSYK,
which was previously worked out analytically in [15, 16]. The resulting
expression involves integrals and sums of q-deformed special functions. We
analyzed the exact analytical complexity profile for specific limits of the ’t
Hooft coupling of DSSYK, $\lambda\to\infty$ and $\lambda\to 0$, verifying that
it reduces to the expected K-complexity profiles for a flat Lanczos sequence
and for a sequence dictated by the Heisenberg-Weyl algebra, respectively.

As argued in [18], taking a controlled small-$\lambda$ limit in double-scaled
SYK (the so-called triple-scaling limit), one zooms in near the ground state
of the system. In this limit, where one expects dynamics to be governed by the
Schwarzian action [22], one finds, consistently, that the Hamiltonian takes
the form of a Liouville Hamiltonian, which is the ADM Hamiltonian of JT
gravity [17]. [18] exploited this further and noted that the Hilbert spaces of
JT and DSSYK can be identified in such a way that fixed chord-number states
are bulk length eigenstates. We combined this with our result on chord states
being Krylov basis elements associated to the infinite-temperature thermofield
double (TFD) state, and concluded that Krylov elements are, in turn, bulk
length eigenstates. Therefore, K-complexity agrees quantitatively with bulk
length through the relation (129), which matches the K-complexity of the TFD
state of the triple-scaled Hamiltonian to the time evolution of bulk
regularized length, previously computed in [17]. This matching was established
upon a consistent identification of parameters in the bulk and boundary
theories given in (61) and (130). The log-cosh behavior previously obtained
for $C_{K}(t)$ is responsible for the well-known log-cosh growth of bulk
length in the unperturbed geometry, which we matched including numerical pre-
factors. From the bulk point of view, it is natural to argue that the lack of
late-time saturation of the length as a function of time is due to the absence
of (doubly) non-perturbative contributions of higher-genus, which lead to the
plateau behaviour of the spectral form factor [39, 40] and correlation
functions [41], as we are only considering the disk topology; this is
consistent with the fact that the boundary theory is considered in the
thermodynamic limit151515See for example [42] for a bulk computation of a
notion of bulk volume associated to complexity which does saturate at late
times due to contributions of non-trivial topology.. It is worth noting, at
this point, that the growth of the bulk length in a back-reacted geometry in
the presence of a matter shock wave may be studied by computing the
K-complexity of an operator insertion in DSSYK161616We thank E. Witten for
this suggestion. Work in progress..

The agreement between the DSSYK Hamiltonian and JT gravity occurs in the
triple-scaling limit (which implies a small value of $\lambda$) because this
is the regime in which the SYK Hamiltonian takes a Liouville form. In this
regime, the theories are dual to each other, both classically and quantum-
mechanically. In this work we were able to explore this duality by a
semiclassical computation of both K-complexity on the boundary and the
geometric description in the bulk. We expect that quantum corrections will
still match by construction, since both the Hilbert spaces and the
Hamiltonians are identified [15, 18]. On top of this, going away from the
small-$\lambda$ limit in DSSYK would amplify the corrections at higher orders
in $\lambda$ in the effective Hamiltonian (60). Hence, the gravity theory
gradually deviates from JT towards the arbitrary-$\lambda$ Hamiltonian (57).
It would be interesting to explore this in future work.

The calculations in this work provide an explicit matching between
K-complexity and bulk length in an instance of lower-dimensional holography.
It is legitimate to wonder how generic this is, and what lessons may be
extracted for higher dimensions. We advocate for an optimistic point of view
and claim that this low-dimensional model may be reductionist for some aspects
but not for the ones of interest: we investigated the relation between
K-complexity and bulk length, using a model of holography where the gravity
theory possesses a phase space consisting exclusively of the two-sided length
variable (and its conjugate momentum); in this sense, the model is the
simplest it can be while still preserving the ingredients relevant for the
question at hand. Furthermore, some higher-dimensional scenarios involving
near-extremal black holes may be reduced to a two-dimensional gravity problem
when analysed in the near-horizon regime (see e.g. [36] and references
therein), which would make direct contact with JT gravity and therefore with
our results. It would be interesting to work this out in detail. Another
avenue to explore K-complexity in higher dimensional scenarios may consist of
studying black hole microstates in microcanonical windows at fixed energy,
whose Hilbert space dimension will be finite if the theory is defined on a
compact manifold; the finiteness of such a Hilbert subspace should allow us to
probe non-perturbative effects such as the saturation of complexity [14].

## Acknowledgements

It is a pleasure to thank Vijay Balasubramanian, José Barbón, Micha Berkooz,
Damián Galante, Vladimir Navrolansky and Edward Witten for insightful
discussions. This work has been supported in part by the Fonds National Suisse
de la Recherche Scientifique (Schweizerischer Nationalfonds zur Förderung der
wissenschaftlichen Forschung) through Project Grant 200020182513, and the NCCR
51NF40-141869, The Mathematics of Physics (SwissMAP). ER would like to thank
the special fund for high energy physics of the PBC for partial support of
this work. RS would like to thank Vladimir Narovlansky for a discussion about
chord diagrams some years ago. We would like to thank José Barbón for some
insightful correspondence on the continuum limit of Krylov space.

##  Appendix A Coupling variance choices

For reference, this appendix discusses the relation between the various
choices of SYK coupling variance in the literature. Different choices are
suitable depending on the specific limit in which the system is studied, as
they will produce bounded results for observables such as expectation values
of operators and correlation functions.

In [22], the SYK model is studied analytically at large $N$, first at fixed
$p$ and later on at also large $p$. The Hamiltonian in that article is:

$H^{(1)}=i^{p/2}\sum_{1\leq i_{1}<...<i_{p}\leq
N}J^{(1)}_{i_{1}...i_{p}}\chi_{i_{1}}...\chi_{i_{p}},$ (134)

where the Majoranas are normalized such that they square to
$\frac{1}{2}\mathbb{1}$, that is:

$\left\\{\chi_{i},\chi_{j}\right\\}=\delta_{ij}.$ (135)

For the sake of studying the model in the large-$N$ limit at fixed $p$, the
coupling variance used in [22] is

$\centering\left\langle\left(J_{i_{1}...i_{p}}^{(1)}\right)^{2}\right\rangle=\frac{(p-1)!J^{2}}{N^{p-1}}.\@add@centering$
(136)

When [22] turns to studying SYK at large $N$ and large $p$, the Hamiltonian
keeps the same structure,

$H^{(2)}=i^{p/2}\sum_{1\leq i_{1}<...<i_{p}\leq
N}J^{(2)}_{i_{1}...i_{p}}\chi_{i_{1}}...\chi_{i_{p}},$ (137)

but the coupling variance is chosen to have an extra $p$-dependent scaling to
get a more uniform limit:

$\left\langle\left(J_{i_{1}...i_{p}}^{(2)}\right)^{2}\right\rangle=\frac{2^{p-1}(p-1)!}{pN^{p-1}}J^{2}.$
(138)

The origin of the $2^{p-1}$ factor is related to the normalization of the
Majoranas (135): Since they square to $\frac{1}{2}\mathbb{1}$, they will give
an additional suppression of $2^{-p}$ every time a Wick contraction pairs two
Hamiltonians inside of a trace, which needs to be compensated by the coupling
variance in order to obtain finite results in the large $q$ limit. More
explicitly, redefining the Majoranas as

$\chi_{i}=\frac{1}{\sqrt{2}}\psi_{i},$ (139)

we obtain operators $\psi_{i}$ that are normalized in a way such that they
square to one:

$\left\\{\psi_{i},\psi_{j}\right\\}=2\delta_{ij}.$ (140)

In terms of these operators, the Hamiltonian (137) reads:

$H^{(2)}=i^{p/2}\sum_{1\leq i_{1}<...<i_{p}\leq
N}J^{(2)}_{i_{1}...i_{p}}2^{-\frac{p}{2}}\psi_{i_{1}}...\psi_{i_{p}}\equiv
i^{p/2}\sum_{1\leq i_{1}<...<i_{p}\leq
N}\widetilde{J^{(2)}}_{i_{1}...i_{p}}\psi_{i_{1}}...\psi_{i_{p}},$ (141)

so that the effective coupling strength variance gets modified to

$\left\langle\left(\widetilde{J^{(2)}}_{i_{1}...i_{p}}\right)^{2}\right\rangle=\left\langle\left(J^{(2)}_{i_{1}...i_{p}}\right)^{2}\right\rangle
2^{-p}=\frac{(p-1)!}{2pN^{p-1}}J^{2}.$ (142)

When turning to double-scaled SYK, extensively studied in [15], it is more
enlightening to work with normalized Majoranas $\psi_{i}$ satisfying (140)
since, as mentioned in section 2.1, the monomials $\psi_{I}$, where $I$
denotes a collective index $i_{1}...i_{p}$, square to a phase, and therefore
the combinatorial factor coming from the sum in the Hamiltonian can be
directly compensated by the normalization of the coupling strength in order to
yield bounded moments of the Hamiltonian. The Hamiltonian in [15] is:

$H^{(3)}=i^{p/2}\sum_{1\leq i_{1}<...<i_{p}\leq
N}J^{(3)}_{i_{1}...i_{p}}\psi_{i_{1}}...\psi_{i_{p}},$ (143)

and the coupling strength variance is given by

$\left\langle\left(J^{(3)}_{i_{1}...i_{p}}\right)^{2}\right\rangle={\binom{N}{p}}^{-1}J^{2}.$
(144)

As an example, this normalization ensures that
$\left\langle\text{Tr}\left[\left(H^{(3)}\right)^{2}\right]\right\rangle=J^{2}$,
even at finite and independent $N$ and $p$.

In the current article, we are interested in double-scaled SYK but we don’t
use the coupling strength variance (144). Instead, we follow [18, 43] and use
a variance which, when taken in the double-scaling limit, has an asymptotic
behavior that makes contact with (142) in a way about which we shall be
precise below. Our Hamiltonian is:

$H=i^{p/2}\sum_{1\leq i_{1}<...<i_{p}\leq
N}J_{i_{1}...i_{p}}\psi_{i_{1}}...\psi_{i_{p}}.$ (145)

We study this system in the double-scaling limit, taking $N$ and $p$ to
infinity keeping the ratio

$\lambda=\frac{2p^{2}}{N}$ (146)

fixed. Given $\lambda$, the coupling strength variance is chosen to be:

$\left\langle
J_{i_{1}...i_{p}}^{2}\right\rangle=\frac{1}{\lambda}{\binom{N}{p}}^{-1}\,J^{2}.$
(147)

To compare with (142), we can perform an asymptotic analysis already within
the double-scaling limit. That is, we take (147) and implicitly assume
everywhere that $N\equiv N(p)=\frac{2p^{2}}{\lambda}$. Then, for large $p$ we
find the asymptotic behavior:

$\centering\left\langle
J_{i_{1}...i_{p}}^{2}\right\rangle=\frac{1}{\lambda}{\binom{N(p)}{p}}^{-1}\,J^{2}\sim\frac{e^{\lambda/4}}{2p}\frac{(p-1)!}{{N(p)}^{p-1}}J^{2}=\frac{(p-1)!}{2p{N(p)}^{p-1}}J^{2}+\mathit{O}(\lambda).\@add@centering$
(148)

Where $f(p)\sim g(p)\leftrightarrow\lim_{p\to+\infty}\frac{f(p)}{g(p)}=1$.
This means that, in the double-scaling limit, the variance (147) is equivalent
to the choice (142) upon the additional assumption of small $\lambda$, which
is precisely the regime on which papers like [18, 43] focus, since it zooms
near the ground state of SYK and yields the Liouville Hamiltonian which makes
contact with Schwarzian dynamics and hence with an AdS bulk dual.

In view of this, the small-$\lambda$ regime of DSSYK can be seen as a
controlled way to approach large-$N$ and large-$p$ SYK with the hierarchy
$1\ll p\ll N$, as also discussed in [44].

##  Appendix B A note on the triple-scaling limit

A priori, there is a whole family of triple scaling limits that one can take,
which applied to the Hamiltonian (57) yield Liouville-like Hamiltonians with a
different relative weight between the kinetic term and the potential term. The
family of limits is parametrized by some $a\in\mathbb{R}$ as follows:

$\centering\lambda\to 0,\qquad
l\to\infty,\qquad(a\lambda)^{-2}e^{-\frac{l}{l_{f}}}=e^{-\frac{\tilde{l}}{l_{f}}}\,\text{fixed}.\@add@centering$
(149)

Applied to (57), this limit yields:

$\centering\tilde{T}-E_{0}=2\lambda
J\left(\frac{l_{f}^{2}k^{2}}{2}+\frac{a^{2}}{2}e^{-\frac{\tilde{l}}{l_{f}}}\right)\;+\mathit{O}\left(\lambda^{2}\right).\@add@centering$
(150)

However, all the Hamiltonians in the family (150) are equivalent in the sense
that they only differ by a finite translation of the length variable. Their
density of states is therefore the same and the eigenfunctions of one
Hamiltonian in the family can be obtained from those of any other by just a
shift. In our work, we choose $a=2$ (instead of the $a=1$ convention in [18])
because in this case the resulting triple-scaled Hamiltonian makes contact
with the form of the Liouville Hamiltonian for JT gravity written explicitly
in [17], restated in (51). Nevertheless, we stress that, since all the
Hamiltonians in the family (150) are equivalent in the sense that we have just
explained, changing $a$ in the definition of the triple-scaling limit (149)
will not modify the gravity theory fundamentally. This ambiguity in the
definition of $\tilde{l}$ may be seen as related to the choice of
regularization scheme for the boundary divergence from the bulk perspective.

##  Appendix C Eigenvalues and eigenvectors of the effective Hamiltonian

The discussion in this appendix closely follows [15, 16] and is presented for
the sake of completeness.

The symmetric version of $T$ given in (12) satisfies, in the chord basis,

$T\,|l\rangle=\frac{J}{\sqrt{\lambda(1-q)}}\left(\sqrt{1-q^{l+1}}\,|l+1\rangle+\sqrt{1-q^{l}}\,|l-1\rangle\right)~{}.$
(151)

To find the eigenvalues of $T$ we write down the eigensystem equation for the
components of the eigenvectors of $T$, $\psi_{l}(E)=\langle l|E\rangle$:

$E\,\psi_{l}(E)=\frac{J}{\sqrt{\lambda(1-q)}}\left(\sqrt{1-q^{l+1}}\,\psi_{l+1}(E)+\sqrt{1-q^{l}}\,\psi_{l-1}(E)\right)$
(152)

We note that $E$ is required to be independent of the position label $l$ as
(152) is an eigenvalue problem; therefore, assuming there are no bound states,
we can take the large $l$ limit of the above relation where $T$ becomes a
tridiagonal Toeplitz171717A Toeplitz matrix is a matrix with equal entries
along the diagonals. For the tridiagonal case its eigenvalues are well known,
see e.g. https://de.wikipedia.org/wiki/Tridiagonal-Toeplitz-Matrix. matrix
with eigensystem equation

$E\,\psi_{l}(E)=\frac{J}{\sqrt{\lambda(1-q)}}\left(\psi_{l+1}(E)+\psi_{l-1}(E)\right)~{}.$
(153)

The eigenvalues of such an $L$-dimensional tridiagonal Toeplitz matrix are
given by:

$E_{k}=\frac{2J}{\sqrt{\lambda(1-q)}}\cos\left(\frac{k\pi}{L+1}\right),\quad
k=1,\dots,L~{}.$ (154)

When $L$ is large the argument of the cosine, $\theta=\frac{k\pi}{L+1}$,
becomes a continuous variable between $0$ and $\pi$ and the spectrum has
values between $E_{min}=-\frac{2J}{\lambda\sqrt{1-q}}$ and
$E_{max}=\frac{2J}{\lambda\sqrt{1-q}}$. The eigenvalues are then a function of
the continuous variable $\theta$181818When $l$ becomes continuous, $\theta$
can be interpreted as momentum.:

$E(\theta)=\frac{2J\,\cos\theta}{\sqrt{\lambda(1-q)}},\quad
0\leq\theta\leq\pi~{}.$ (155)

Note that as $L\to\infty$, the variable $\theta$ becomes continuous in a
uniform way and thus summing over eigenvalues $\sum_{k=1}^{L}$ becomes
$\int_{0}^{\pi}d\theta$ with uniform density $\rho(\theta)=1$. This fact will
be important for later results.

To find the eigenvectors, we go back to (152) using the result (155) and
defining for convenience $\mu=\cos\theta$ (we will use $v_{l}$ instead of
$\psi_{l}$, reserving the latter to the final normalized result):

$2\mu\,v_{l}(\mu)=\sqrt{1-q^{l+1}}\,v_{l+1}(\mu)+\sqrt{1-q^{l}}\,v_{l-1}(\mu)~{}.$
(156)

This equation can be identified once $v_{l}(\mu)$ is redefined as follows

$v_{l}(\mu)=\frac{\tilde{v}_{l}(\mu)}{\sqrt{(q;q)_{l}}}$ (157)

where $(a;q)_{n}\equiv\prod_{k=0}^{n-1}(1-aq^{k})$ is the $q$-Pochhammer
symbol191919https://en.wikipedia.org/wiki/Q-Pochhammer_symbol.. Plugging this
redefinition into (156) and multiplying both sides of the equation by
$\sqrt{(q;q)_{l}}$ gives:

$2\mu\tilde{v}_{l}(\mu)=(1-q^{l})\tilde{v}_{l-1}(\mu)+\tilde{v}_{l+1}(\mu)$
(158)

which now can be identified as the recurrence relation for the q-Hermite
polynomials (see appendix C.1), hence the solution to (156) is

$v_{l}(\mu)=\frac{H_{l}(\mu|q)}{\sqrt{(q;q)_{l}}},\quad\mu=\cos(\theta)~{}.$
(159)

Using the identity (170) and the fact that $0\leq\theta\leq\pi$, it can be
shown that the norm squared of these eigenvectors is given by

$\sum_{l=0}^{\infty}|v_{l}(\mu)|^{2}=\sum_{l=0}^{\infty}\frac{H_{l}(\mu|q)H_{l}(\mu|q)}{(q;q)^{l}}=\frac{2\pi}{|(e^{2i\theta};q)_{\infty}|^{2}(q;q)_{\infty}}$
(160)

and the normalized eigenvector is given by

$\psi_{l}(\mu)=\sqrt{(q;q)_{\infty}}|(e^{2i\theta};q)_{\infty}|\frac{H_{l}(\mu|q)}{\sqrt{2\pi(q;q)_{l}}},\quad\mu=\cos\theta~{}.$
(161)

We note that

$\psi_{0}(\mu)=\sqrt{(q;q)_{\infty}}|(e^{2i\theta};q)_{\infty}|\frac{H_{0}(\mu|q)}{\sqrt{2\pi(q;q)_{0}}}=\sqrt{\frac{(q;q)_{\infty}}{2\pi}}|(e^{2i\theta};q)_{\infty}|$
(162)

where we used $H_{0}(x|q)=1$ and $(q;q)_{0}=1$. We thus can write

$\psi_{l}(\mu)=\psi_{0}(\mu)\frac{H_{l}(\mu|q)}{\sqrt{(q;q)_{l}}},\quad\mu=\cos\theta.$
(163)

The orthogonality relations for these eigenvectors are shown in appendix C.3.

###  C.1 q-Hermite Polynomials

The q-Hermite polynomial is defined
as202020https://en.wikipedia.org/wiki/Continuous_q-Hermite_polynomials.

$H_{n}(\cos\theta|q)=e^{in\theta}{}_{2}\phi_{0}\Big{[}\begin{matrix}q^{-n},0\\\
-\end{matrix};\;q,q^{n}e^{-2i\theta}\Big{]}$ (164)

where ${}_{2}\phi_{0}$ is a q-hypergeometric function, defined
by212121https://en.wikipedia.org/wiki/Basic_hypergeometric_series.

${}_{j}\phi_{k}\Big{[}\begin{matrix}a_{1},a_{2},\dots,a_{j}\\\
b_{1},b_{2},\dots,b_{k}\end{matrix};\;q,z\Big{]}=\sum_{n=0}^{\infty}\frac{(a_{1};q)_{n}\,(a_{2};q)_{n}\,\dots(a_{j};q)_{n}}{(b_{1};q)_{n}\,(b_{2};q)_{n}\,\dots(b_{k};q)_{n}}\Big{[}(-1)^{n}q^{n(n-1)/2}\Big{]}^{1+k-j}\frac{z^{n}}{(q;q)_{n}}~{}.$
(165)

Using this definition we have

$H_{n}(\cos\theta|q)=\sum_{k=0}^{\infty}\frac{(q^{-n};q)_{k}}{(q;q)_{k}}(-1)^{k}q^{nk-\frac{k(k-1)}{2}}e^{i(n-2k)\theta}$
(166)

Now note that,
$(q^{-n};q)_{k}=(1-q^{-n})(1-q^{-n+1})(1-q^{-n+2})\dots(1-q^{-n+k-1})$ so for
$k=n+1$ we have
$(q^{-n};q)_{n+1}=(1-q^{-n})(1-q^{-n+1})(1-q^{-n+2})\dots(1-1)=0$. We thus
find that for $(q^{-n};q)_{k>n+2}=0$ and the sum becomes truncated at $k=n$:

$H_{n}(\cos\theta|q)=\sum_{k=0}^{n}\frac{(q^{-n};q)_{k}}{(q;q)_{k}}(-1)^{k}q^{nk-\frac{k(k-1)}{2}}e^{i(n-2k)\theta}~{}.$
(167)

From this expression we can get another useful expression for
$H_{n}(\cos\theta|q)$, as follows:

$\displaystyle(q^{-n};q)_{k}$
$\displaystyle=(1-q^{-n})(1-q^{-n+1})(1-q^{-n+2})\dots(1-q^{-n+k-1})$
$\displaystyle=(-1)^{k}(q^{-n}-1)(q^{-n+1}-1)(q^{-n+2}-1)\dots(q^{-n+k-1}-1)$
$\displaystyle=(-1)^{k}\,q^{-n}(1-q^{n})\,q^{-n+1}(1-q^{n+1}))\dots
q^{-n+k-1}(1-q^{n-k+1})$
$\displaystyle=(-1)^{k}\,q^{-nk+\frac{k(k-1)}{2}}\frac{(q;q)_{n}}{(q;q)_{n-k}}$

and thus, another expression for the q-Hermite polynomial is

$H_{n}(\cos\theta|q)=\sum_{k=0}^{n}\frac{(q;q)_{n}}{(q;q)_{k}\,(q;q)_{n-k}}e^{i(n-2k)\theta}~{}.$
(168)

###  C.2 Orthogonality relations

A useful identity for q-Hermite polynomials with $x=\cos\theta,y=\cos\phi$
(see for example [45]) is:

$\sum_{l=0}^{\infty}H_{l}(x|q)H_{l}(y|q)\frac{t^{l}}{(q;q)_{l}}=\frac{(t^{2};q)_{\infty}}{(te^{i(\theta+\phi)};q)_{\infty}(te^{i(\theta-\phi)};q)_{\infty}(te^{-i(\theta-\phi)};q)_{\infty}(te^{-i(\theta+\phi)};q)_{\infty}},$
(169)

which for $t=1$ and $x=y$ is exactly the needed identity. Noting that
$(te^{i(\theta+\phi)};q)_{\infty}(te^{-i(\theta+\phi)};q)_{\infty}=\prod_{k=0}^{\infty}(1-te^{i(\theta+\phi)}q^{k})(1-te^{-i(\theta+\phi)}q^{k})=|(te^{i(\theta+\phi)};q)_{\infty}|^{2}$
and similarly for $e^{i(\theta-\phi)}$, we have:

$\frac{(t^{2};q)_{\infty}}{|(te^{i(\theta+\phi)};q)_{\infty}|^{2}|(te^{i(\theta-\phi)};q)_{\infty}|^{2}}=\frac{(1-t^{2})(qt^{2};q)_{\infty}}{|1-te^{i(\theta+\phi)}|^{2}|1-te^{i(\theta-\phi)}|^{2}|(qte^{i(\theta+\phi)};q)_{\infty}|^{2}|(qte^{i(\theta-\phi)};q)_{\infty}|^{2}},$

where it is made clear that for $t=1$ the expression vanishes for all
$\theta,\phi$ except for $\theta=\pm\phi$. Taking $t=1-\epsilon$, in the limit
$\epsilon\to 0$ $\theta\to\phi$ we have

$\displaystyle|1-te^{i(\theta-\phi)}|^{2}$
$\displaystyle=1-t[e^{i(\theta-\phi)}+e^{-i(\theta-\phi)}]+t^{2}$
$\displaystyle=1-t[1+i(\theta-\phi)-\frac{(\theta-\phi)^{2}}{2}+1-i(\theta-\phi)-\frac{(\theta-\phi)^{2}}{2}+\dots]+t^{2}$
$\displaystyle=1-(1-\epsilon)[2-(\theta-\phi)^{2}+O((\theta-\phi)^{4})]+(1-\epsilon)^{2}$
$\displaystyle=\epsilon^{2}+(\theta-\phi)^{2}+O(\epsilon(\theta-\phi)^{2})+\dots$

and thus

$\displaystyle\frac{(1-t^{2})(qt^{2};q)_{\infty}}{|1-te^{i(\theta+\phi)}|^{2}|1-te^{i(\theta-\phi)}|^{2}|(qte^{i(\theta+\phi)};q)_{\infty}|^{2}|(qte^{i(\theta-\phi)};q)_{\infty}|^{2}}$
$\displaystyle\to\frac{2\epsilon}{\epsilon^{2}+(\theta-\phi)^{2}}\frac{(q;q)_{\infty}}{|(e^{2i\theta};q)_{\infty}|^{2}|(q;q)_{\infty}|^{2}}~{}.$

In the limit $\epsilon\to 0$, the first term is recognized as a delta function
$\delta(t)=\lim_{\epsilon\to
0}\frac{1}{\pi}\frac{\epsilon}{\epsilon^{2}+t^{2}}$ and taking into account
also the possibility $\theta\to-\phi$ it is found that

$\sum_{l=0}^{\infty}\frac{H_{l}(x|q)H_{l}(y|q)}{(q;q)_{l}}=\frac{2\pi[\delta(\theta-\phi)+\delta(\theta+\phi)]}{|(e^{2i\theta};q)_{\infty}|^{2}(q;q)_{\infty}}~{}.$
(170)

###  C.3 Orthogonality of the eigenvectors of DSSYK

In this appendix we show that the eigenvectors (163) satisfy orthogonality
relations in energy and in chord number. Firstly we check orthogonality in
energy:

$\sum_{l=0}^{\infty}\psi_{l}(\cos\theta)\psi_{l}(\cos\phi)=\frac{(q;q)_{\infty}}{2\pi}|(e^{2i\theta};q)_{\infty}|^{2}\sum_{l=0}^{\infty}\frac{H_{l}(\cos\theta|q)H_{l}(\cos\phi|q)}{(q;q)_{l}}=\delta(\theta-\phi)~{},$
(171)

where we used the result (170) together with the knowledge that both $\theta$
and $\phi$ get values between $0$ and $\pi$. Secondly, we check orthogonality
in chord number:

$\int_{0}^{\pi}d\theta\,\psi_{n}(\cos\theta)\psi_{m}(\cos\theta)=\frac{(q;q)_{\infty}}{\sqrt{(q;q)_{n}(q;q)_{m}}}\int_{0}^{\pi}\frac{d\theta}{2\pi}|(e^{2i\theta};q)_{\infty}|^{2}H_{n}(\cos\theta|q)H_{m}(\cos\theta|q)~{}.$
(172)

To compute this we need the following identity:

$\int_{0}^{\pi}\frac{d\theta}{2\pi}|(e^{2i\theta};q)_{\infty}|^{2}H_{n}(\cos\theta|q)H_{m}(\cos\theta|q)=\frac{(q;q)_{n}}{(q;q)_{\infty}}\delta_{nm}~{},$
(173)

which gives immediately

$\int_{0}^{\pi}d\theta\,\psi_{n}(\cos\theta)\psi_{m}(\cos\theta)=\delta_{nm}~{}.$
(174)

##  Appendix D Continuum approximation of Krylov space

This section provides a derivation of the continuum approximation of Krylov
space developed in [8] in order to understand better its applicability and its
relation to ballistic propagation.

In the case in which there are no diagonal Lanczos coefficients, $a_{n}=0$, it
is useful to redefine the wavefunctions as

$\centering\phi_{n}(t)=i^{n}\varphi_{n}(t),\@add@centering$ (175)

so that $\varphi_{n}(t)\in\mathbb{R}$, satisfying the recurrence relation:

$\centering\dot{\varphi}_{n}(t)=b_{n}\varphi_{n-1}(t)-b_{n+1}\varphi_{n+1}(t),\@add@centering$
(176)

for all $n=0,...,K-1$ and with initial condition
$\varphi_{n}(0)=\delta_{n,0}$. Note that, for the sake of rigor, we start by
considering a finite Krylov space of dimension $K$; we shall eventually
consider the limit of large $K$. For consistency, $b_{0}\equiv 0$ and
$\varphi_{-1}(t)\equiv 0$.

We will start by performing some exact manipulations that amount to rewriting
(176) in a more suggestive form. Since Krylov space is discrete, position is
labelled by $n=0,...,K-1$ and the lattice spacing is constant and equal to
$1$. Thus, strictly speaking, it does not go to zero in any sense. In order to
make a continuum limit possible, we use a change of variables
$\xi=\frac{n}{K-1}\equiv\varepsilon n$, where we identify the small parameter
$\varepsilon=\frac{1}{K-1}$. As a result, position in Krylov space is denoted
by a rescaled variable $\xi$ which takes discrete values contained in the
interval $[0,1]$:

$\centering\xi\in\left\\{\frac{n}{K-1},\;n=0,1,...,K-1\right\\}=\left\\{\varepsilon
n,\;n=0,1,...,\frac{1}{\varepsilon}\right\\}.\@add@centering$ (177)

The spacing between the allowed values of $\xi$ is
$\varepsilon=\frac{1}{K-1}$, and we note that in the limit $K\to+\infty$ they
accumulate to $\xi\in[0,1]$, hence yielding a continuum limit. But let’s still
keep $K$ fixed.

An important technicality has to be considered before passing to the continuum
limit. We need to assume that the Lanczos coefficients $b_{n}$ are given by
the evaluation at discrete values of the domain of a certain smooth function
that we loosely denote $b(n)$. This has to be assumed, since analytic
continuation cannot be invoked because for it to apply one needs to have a
function defined over at least an interval, not a discrete domain. Our Lanczos
coefficients for DSSYK (71) certainly satisfy this assumption, but there are
examples in the literature of Lanczos sequences that do not fulfill it: The
works [46, 47] have studied systems where the Lanczos sequence shows
staggering, the even and odd coefficients following different profiles. For
such systems, the Lanczos sequence does not have a well-defined continuum
limit.

Upon the previous variable rescaling, we have:

$\centering b_{n}=b(n)=b\big{(}(K-1)\xi\big{)}=:w(\xi),\@add@centering$ (178)

where we have introduced the function $w$, defined over the rescaled compact
domain (177).

Having assumed that the Lanczos coefficients $b_{n}$ are given by evaluating a
smooth function $b(x)$ at discrete values $x=n$, it is now justified to make
the same assumption for $\varphi_{n}(t)=\varphi(t,n)$. Then again, upon the
variable rescaling:

$\centering\varphi_{n}(t)=\varphi(t,n)=\varphi\big{(}t,(K-1)\xi\big{)}=:f(t,\xi),\@add@centering$
(179)

where the domain for the second variable of $f$ is (177). We can now rewrite
(176) as:

$\centering\partial_{t}f(t,\xi)=w(\xi)f(t,\xi-\varepsilon)-w(\xi+\varepsilon)f(t,\xi+\varepsilon).\@add@centering$
(180)

We emphasize that (180) is an exact rewriting of the recurrence (176), with
$\varepsilon=\frac{1}{K-1}$. A solution $f(t,\xi)$ of (180) gives a solution
for (176) by evaluating $\varphi_{n}(t)=f(t,\frac{n}{K-1})$.

Expanding (180), which we can do since we have assumed that $w(\xi)$ is smooth
and we have argued that $f(t,\xi)$ can also be taken to be so, we arrive at

$\centering\partial_{t}f(t,\xi)+v(\xi)\partial_{\xi}f(t,\xi)+\frac{1}{2}v^{\prime}(\xi)f(t,\xi)=\mathit{O}(\varepsilon^{2}),\@add@centering$
(181)

where we have defined $v(\xi):=2\varepsilon w(\xi)$. The leading term of (181)
is a chiral wave equation with velocity field $v(\xi)$ and mass term
$\frac{1}{2}v^{\prime}(\xi)f(t,\xi)$. This equation can be solved using a
change of variables $\xi\mapsto\chi$ such that $\chi=0$ when $\xi=0$ and
$\frac{d\xi}{v(\xi)}=d\chi$. Since $v(\xi)$ is of order $\varepsilon$, this
change of variables unwraps the compact domain of $\xi$ and has again a non-
compact domain, but the discussion about the compact domain was technically
necessary in order to explain in what sense there is a continuum limit, and
why $\varepsilon=\frac{1}{K-1}$ controls the deviation from it. Introducing
additionally the function

$\centering g(t,\chi):=\sqrt{v(\xi(\chi))}f(t,\xi(\chi)),\@add@centering$
(182)

equation (180) simplifies further into:

$\centering\left(\partial_{t}+\partial_{\chi}\right)g(t,\chi)=\mathit{O}(\varepsilon^{5/2}),\@add@centering$
(183)

where we remind that $\varepsilon=\frac{1}{K-1}$. In the limit $K\to\infty$
equation (183) becomes exactly a chiral wave equation with unit velocity for
the wave function $g$ with coordinates $t,\,\chi$. However, this does not
imply that the continuum approximation of Krylov space becomes exact and
indistinguishable from the discrete problem in the large-$K$ limit: The
intermediate steps of rescaling $n$ to a compact domain $\xi=\varepsilon n$
fail strictly speaking when $\varepsilon=0$, and the continuum approximation
need not apply in general. It may be applicable for propagating wave packets
whose typical length scale is larger than the lattice spacing, but in our
cases of interest the initial condition for the discrete wave packet is
$\varphi_{n}(t=0)=\delta_{n0}$, which therefore probes the discreteness of the
Krylov chain at early times.

Interestingly, studying (183) we note that this continuum approximation
behaves as a classical limit, because it results in ballistic propagation, as
we shall explain. Given an initial condition $g(0,\chi)=g_{0}(\chi)$, which in
our case will be proportional to a delta function centered at $\chi=0$, the
generic solution of (183) simply propagates this packet to the right, without
spreading it:

$\centering g(t,\chi)=g_{0}(\chi-t).\@add@centering$ (184)

Since the packet $g(t,\chi)$ is a propagating delta function, K-complexity
coincides with the position of its peak, which propagates at velocity $1$ in
$\chi$-space. Undoing the change of vatiables:

$\centering
t=\int_{0}^{\chi(t)}d\chi=\int_{0}^{\xi(t)}\frac{d\xi}{v(\xi)},\@add@centering$
(185)

and using $v(\xi)=2\varepsilon w(\xi)=2\varepsilon b(\xi/\varepsilon)$,
together with $\xi=\varepsilon n$, we reach the equation222222We thank José
Barbón for pointing out this expression to us.:

$\centering t=\int_{0}^{n(t)}\frac{dn}{2b(n)},\@add@centering$ (186)

which can be used as an estimate for the position of the peak $n(t)$ in
$n$-space, in an approximation in which $n$ is directly promoted to be a
continuous variable.

We remark that the continuum approximation gives ballistic propagation as an
output, rather than it being an additional, independent assumption. In this
sense, this approximation can be thought of as a classical approximation,
because the propagation of the wave packet has turned into the problem of the
propagation of a localized particle with well-defined position that travels
driven by a velocity field $2b(n)$ in $n$-space.

##  Appendix E Wavefunctions and Krylov complexity for specific values of q

In this appendix we evaluate the exact wavefunctions, $\phi_{n}(t)$, for $q=0$
and for $q=1$. For $q=0$ we can use the result (111) by plugging $q=0$
directly into the equation. For $q=1$ we start from the eigensystem equation
with Lanczos coefficients $b_{n}\sim\sqrt{n}$ and find the eigenvectors in
order to compute $\phi_{n}(t)$. The analysis of $q\to 1$ is done in the main
text.

###  E.1 Exact wavefunctions for q=0

For $q\to 0$ the Lanczos coefficients are given by
$b_{n}\sim\frac{J}{\sqrt{\lambda}}\equiv b$ ($\to 0$) for all $n=0,1,2,\dots$.
When the tridiagonal matrix with these Lanczos coefficients above and below
the diagonal is a Liouvillian, the result for $\phi_{n}(t)$ was found in [8]
to be

$\phi_{n}(t)=i^{n}\varphi_{n}(t)=i^{n}\frac{(n+1)}{bt}J_{n+1}(2bt)~{},$ (187)

whereas in our case (dynamics in the Schrödinger picture) we need to replace
$t\to-t$. We can arrive at this result from (111) by setting $q=0$:

$\displaystyle(0;0)_{\infty}$ $\displaystyle=\prod_{k=0}^{\infty}(1-0\cdot
0^{k})=1=(0;0)_{n}$ (188) $\displaystyle(e^{2i\theta};0)_{\infty}$
$\displaystyle=1-e^{2i\theta}\Rightarrow|(e^{2i\theta};0)_{\infty}|^{2}=4\sin^{2}\theta$
(189) $\displaystyle H_{n}(\cos\theta|0)$
$\displaystyle=\sum_{k=0}^{n}\frac{(0;0)_{n}}{(0;0)_{k}(0;0)_{n-k}}e^{i(n-2k)\theta}=\sum_{k=0}^{n}e^{i(n-2k)\theta}=\frac{\sin[(1+n)\theta]}{\sin\theta}~{},$
(190)

where we used (168) to evaluate $H_{n}(\cos\theta|0)$. Plugging all this into
(111) we have

$\displaystyle\phi_{n}(t)$ $\displaystyle=\frac{2}{\pi}\int_{0}^{\pi}d\theta
e^{-2ibt\cos\theta}\sin\theta\sin[(1+n)\theta]$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\pi}d\theta\,e^{-2ibt\cos\theta}\cos(n\,\theta)-\frac{1}{\pi}\int_{0}^{\pi}d\theta\,e^{-2ibt\cos\theta}\cos[(n+2)\,\theta]$
$\displaystyle=i^{n}J_{n}(-2bt)+i^{n+2}J_{n+2}(-2bt)=i^{n}\Big{[}J_{n}(-2bt)+J_{n+2}(-2bt)\Big{]}$
$\displaystyle=i^{n}\frac{(n+1)}{(-bt)}J_{n+1}(-2bt)~{},$ (191)

where in the second line we used the trigonometric identity
$\sin\alpha\sin\beta=\frac{1}{2}[\cos(\alpha-\beta)-\cos(\alpha+\beta)]$, in
the third line we used one of the integral representations of the Bessel
function of the first kind232323https://dlmf.nist.gov/10.9.
$J_{n}(z)=\frac{{i^{-n}}}{\pi}\int_{0}^{\pi}d\theta\,e^{iz\cos\theta}\cos(n\theta)$,
and in the fourth line we used the Bessel function
identity242424https://functions.wolfram.com/Bessel-
TypeFunctions/BesselJ/17/01/01/.
$J_{n}(z)+J_{n+2}(z)=\frac{2(n+1)}{z}J_{n+1}(z)$. We managed to recover the
result (187) with $t\to-t$ as required.

With the wavefunction given by (191) the result for Krylov complexity can be
evaluated exactly as follows:

$\displaystyle C_{K}(t)$