# Geometric Operator Quantum Speed Limit, Wegner Hamiltonian Flow and Operator
Growth

Niklas Hörnedal  Department of Physics and Materials Science, University of
Luxembourg, L-1511 Luxembourg, G. D. Luxembourg  Nicoletta Carabba  Department
of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg,
G. D. Luxembourg  Kazutaka Takahashi  Department of Physics and Materials
Science, University of Luxembourg, L-1511 Luxembourg, G. D. Luxembourg
Department of Physics Engineering, Faculty of Engineering, Mie University, Mie
514–8507, Japan  Adolfo del Campo  Department of Physics and Materials
Science, University of Luxembourg, L-1511 Luxembourg, G. D. Luxembourg
Donostia International Physics Center, E-20018 San Sebastián, Spain

###### Abstract

Quantum speed limits (QSLs) provide lower bounds on the minimum time required
for a process to unfold by using a distance between quantum states and
identifying the speed of evolution or an upper bound to it. We introduce a
generalization of QSL to characterize the evolution of a general operator when
conjugated by a unitary. The resulting operator QSL (OQSL) admits a geometric
interpretation, is shown to be tight, and holds for operator flows induced by
arbitrary unitaries, i.e., with time- or parameter-dependent generators. The
derived OQSL is applied to the Wegner flow equations in Hamiltonian
renormalization group theory and the operator growth quantified by the Krylov
complexity.

In what time scale does a physical process unfold? Time-energy uncertainty
relations have long been used to estimate characteristic time scales in
physical processes, including lifetimes in quantum decay, tunneling times, and
the duration of a quantum jump, among others [1, 2, 3, 4]. In the quantum
domain, Mandelstam and Tamm put the time-energy uncertainty relation on firm
ground in their 1945 work [5]. They provided its rigorous derivation by
combining the Heisenberg equation of motion and the Robertson uncertainty
relation. They went a step further by identifying the minimum time for the
quantum state of a system to evolve into a distinct state, using the energy
dispersion of the initial state as an upper bound to the speed of evolution.
In doing so, they introduced an early example of a quantum speed limit (QSL).

Over the last decades, such an approach has been refined and generalized to a
great extent [6, 7]. Margolus and Levitin found an alternative bound to the
speed of evolution in terms of the mean energy of the system [8], and ensuing
works showed that an infinite family of bounds exist in terms of other moments
of the generator of evolution [9, 10]. QSLs have also been derived for time-
dependent Hamiltonians [11, 12, 13], open systems described by master
equations [14, 15, 16, 17], and with a stochastic evolution under continuous
quantum measurements [18]. They have been further extended to the classical
domain, with applications ranging from Hamiltonian dynamics to stochastic
thermodynamics [19, 20, 21, 22, 23, 24, 25]. QSLs generally involve a notion
of distance between quantum states and an upper bound to the speed of quantum
evolution. While the Bures angle, defined in terms of the Uhlmann fidelity
[26], is often the default choice for the distance between quantum states,
other alternatives can provide tighter QSLs [27, 28, 17, 29]. This freedom is
particularly important in the context of many-body systems given the
orthogonality catastrophe and the growth of the Hilbert space with the system
size [30, 31, 32, 33, 34].

The emphasis on quantum state distinguishability was a key stepping stone in
developing QSLs. Today, QSLs find manifold applications in quantum metrology
and parameter estimation [35, 36, 37, 38], quantum control [39, 40, 41, 42],
and quantum thermodynamics [43, 44], among other fields. However, certain
phenomena are naturally described in terms of operator flows, i.e., the
continuous evolution of an operator according to given equations of motion. A
typical example is the description of quantum evolution in the Heisenberg
picture, but the relevance of operator flows is not restricted to quantum
dynamics in rotating frames. Operator flows naturally arise in the Wegner flow
equations for Hamiltonian renormalization [45, 46, 47, 48, 49], the study of
operator growth and quantum complexity [50, 51, 52, 53, 54], and correlation
functions [55], to name some examples.

Motivated by these applications, we have introduced the notion of QSL for
operator flows in Ref. [56], which we shall term Operator QSL (OQSL)
hereafter. Another approach pursuing QSLs for observables was proposed in
[57]. At variance with conventional QSLs, OQSLs involve a notion of distance
between operators (instead of quantum states) and an upper bound on the
corresponding speed of evolution. OQSLs have proved useful in characterizing
the time evolution of autocorrelation functions, setting bounds on dynamical
susceptibilities arising in linear response theory, and the precision in
parameter estimation with thermal quantum systems [56]. However, in their
current form, OQSLs are restricted to flows associated with a constant
generator, i.e., one that is independent of time or the relevant parameter
characterizing the evolution. In addition, OQSLs lack in their present
formulation an intuitive geometric description, common in other bounds arising
in quantum information geometry, such as the Mandelstam-Tamm QSL.

In this work, we introduce an OQSL valid under unitary dynamics generated by a
time- (or parameter-) dependent generator, that can be a generic operator or
an observable, e.g., a Hamiltonian. The resulting bound admits a geometric
interpretation and is shown to be tight. We illustrate its usefulness in the
study of Wegner flow equations in the theory of Hamiltonian continuous
renormalization group [48, 46, 47]. We characterize the OQSL for the
Hamiltonian flow in detail and illustrate the possibility of saturating it
when the flow of Hamiltonian parameters is described by the Toda equations. We
further apply the new OQSL to the problem of operator growth in unitary
quantum dynamics in Krylov space, i.e., as characterized by the Krylov
complexity [54, 58, 59, 60, 61, 62]. We close with a discussion and an outlook
pointing out directions for further work.

##  1 Motivation

In quantum physics, a broad class of time-correlation functions is defined
through a positive semi-definite inner product. Consider any operator $A$
evolving unitarily according to $\dot{A}_{t}=i\mathds{L}A_{t}$, where
$\mathds{L}=[H,\cdot]$ is the Liouvillian superoperator. One class of time-
correlation functions that have been considered in the literature takes the
form $C(t)=(A|A_{t})$, defined with the help of a Hermitian bilinear form in
the space of operators,

$(A|B)=\Tr(A^{\dagger}\rho_{1}B\rho_{2}).$ (1)

The operators $\rho_{1}$ and $\rho_{2}$ are in general positive semi-definite,
commute with the Hamiltonian, and need not have unit trace. A familiar example
of this type of correlation function is obtained by setting
$\rho_{1}=\rho_{2}=\mathds{1}$. Eq. (1) reduces then to the Hilbert-Schmidt
inner product. Another familiar instance corresponds to the choice of the
identity $\rho_{1}=\mathds{1}$ and the canonical Gibbs state
$\rho_{2}=e^{-\beta H}/Z$ at inverse temperature $\beta$, with partition
function $Z=\Tr e^{-\beta H}$. Note that in general, the bilinear form in (1)
is not positive definite, which means that it does not always define an inner
product; instead, it will define a positive semi-definite inner product.111We
use the convention that an inner product must satisfy positive-definiteness,
be linear in the second argument and conjugate symmetric As a result, there
might exist cases in which $A\neq 0$ and $(A|A)=0$ are simultaneously
fulfilled. Another example of a correlation function defined through a
positive semi-definite inner product, i.e. $C(t)=(A|A_{t})$, is given by the
so-called Kubo inner product [63, 64, 55, 65]

$(A|B)=\frac{1}{\beta}\int_{0}^{\beta}d\lambda\langle e^{\lambda
H}A^{\dagger}e^{-\lambda H}B\rangle_{\beta}-\langle
A^{\dagger}\rangle_{\beta}\langle B\rangle_{\beta}.$ (2)

Here, $\langle\cdot\rangle_{\beta}$ denote the thermal expectation value and
$\beta$ is once again the inverse temperature.

The function $\norm{A}=\sqrt{(A|A)}$ is a seminorm and we note that the
condition $\norm{A_{t}}=\norm{A}$ is satisfied in the above examples for all
$t$.222A seminorm fulfills all the properties of regular norm except that non-
zero vectors can have norm 0. We might then ask, given a unitary flow induced
by a possibly time-dependent Hamiltonian $H$, what is the minimal time $\tau$
for an initial observable $A$ to reach some specific value of $C(t)$ provided
that $\norm{A_{t}}=\norm{A}$ is satisfied during the whole evolution? What
follows is a derivation of a speed limit that lower bounds this minimum time.

##  2 An Operator Quantum Speed Limit

The complex Hilbert space $\mathcal{H}$ used to model the system will be
assumed throughout this paper to be finite-dimensional. We define
$\mathcal{B}$ to be the space of linear operators acting on this space.
Moreover, let $\textrm{End}(\mathcal{B})$ be the space consisting of all
vector space endomorphism of $\mathcal{B}$. This linear space is commonly
referred to as the Liouville space in the literature [66, 67] and its elements
are referred to as superoperators.

Positive semi-definiteness of $(\cdot|\cdot)$ makes it possible for $C(t)$ to
be constant for certain paths $A_{t}$ even though the operator is changing in
time. We will see that there is a way of “carving away” the degrees of freedom
in $\mathcal{H}$ that do not contribute to changes in $C(t)$. In doing so, we
obtain an effective Hilbert subspace $\mathcal{H}_{\mathcal{P}}$, where the
restriction of $(\cdot|\cdot)$ onto this subspace defines a proper inner-
product. We will see that the condition $\norm{A_{t}}=\norm{A}$ implies that
$A_{t}$ will be situated on a sphere centered at the origin in
$\mathcal{H}_{\mathcal{P}}$ with radius $\norm*{A}$. This observation will
then enable us to derive a geometric OQSL.

###  2.1 Construction of the effective Hilbert space

We let $\langle\cdot,\cdot\rangle_{\textsc{h}}$ denote the Hilbert-Schmidt
inner product on $\mathcal{B}$. It can be shown that any positive semi-
definite inner product $(\cdot|\cdot)$ can be expressed as
$(\cdot|\cdot)=\langle\cdot,\mathcal{P}\cdot\rangle_{\textsc{h}}$, where
$\mathcal{P}$ is a positive semi-definite superoperator with respect to the
Hilbert-Schmidt inner product. This superoperator will be unique, provided
that $(\cdot|\cdot)$ has been specified—see Appendix A. As we prove in
Appendix B, a useful relation between $(\cdot|\cdot)$ and $\mathcal{P}$ is
that

$\norm{A}=0\iff\mathcal{P}A=0.$ (3)

Since $\mathcal{P}$ is self-adjoint, it follows from the spectral theorem that
the linear space $\mathcal{B}$ can be expressed as a direct sum of the
eigenspaces of $\mathcal{P}$ [68]. Consequently, the image of $\mathcal{P}$
will be spanned by the eigenvectors corresponding to non-zero eigenvalues, and
we thus have that
$\mathcal{B}=\textrm{im}(\mathcal{P})\oplus\textrm{ker}(\mathcal{P})$, where
$\textrm{im}(\mathcal{P})$ and $\textrm{ker}(\mathcal{P})$ are the image and
kernel of $\mathcal{P}$ respectively. Relation (3) then implies that the
restriction of $(\cdot|\cdot)$ to $\textrm{im}(\mathcal{P})$ will be positive
definite and will thus define an inner product on the Hilbert space defined by
$\mathcal{H}_{\mathcal{P}}=\textrm{im}(\mathcal{P})$. We will express this
inner product with the usual bracket notation $\langle\cdot,\cdot\rangle$.

For any operator $A\in\mathcal{B}$, we will let
$\hat{A}\in\mathcal{H}_{\mathcal{P}}$ denote the orthogonal projection of $A$
onto $\mathcal{H}_{\mathcal{P}}$.333By orthogonal, we mean orthogonality
measured by the Hilbert-Schmidt inner product. More explicitly, given the
spectral decomposition $\mathcal{P}=\sum_{k}p_{k}\Pi_{k}$, where $p_{k}$ are
the non-zero eigenvalues of $\mathcal{P}$ and $\Pi_{k}$ are the corresponding
eigenprojections, we have that $\hat{A}=\sum_{k}\Pi_{k}A$. We show in Appendix
C that

$(A|B)=\innerproduct*{\hat{A}}{\hat{B}}\quad\forall A,B\in\mathcal{B}.$ (4)

An important consequence of (4) is that
$C(t)=\innerproduct*{\hat{A}}{\hat{A}_{t}}$. This means that if we are
interested in how $C(t)$ changes over time, then we only need to consider the
projected dynamics $\hat{A}(t)$ of $A(t)$.

###  2.2 Deriving the speed limit

If $d$ is the complex dimension of the Hilbert space
$\mathcal{H}_{\mathcal{P}}$, then we can view it as a real vector space
isomorphic to $\mathds{R}^{2d}$. We endow this space with the Riemannian
metric given by the real part of the inner product
$\langle\cdot,\cdot\rangle$. The condition that $(A|A)=(A_{t}|A_{t})$ holds
for all $t\in[0,\tau]$ then means that $\hat{A}_{t}$ will be situated on the
$(2d-1)$-dimensional sphere $\mathcal{S}_{\norm*{A}}$ with radius $\norm*{A}$
centered at the origin. The shortest path connecting two points on the sphere
lies on a great circle with a distance given by the angle between the points
times the radius. In other words, if $\hat{A}$ and $\hat{B}$ are two operators
on the sphere with radius $\norm{A}$, then the geodesic distance is given by

$\operatorname{dist}(\hat{A},\hat{B})=\norm{A}\arccos(\frac{\real(A|B)}{\norm{A}^{2}}).$
(5)

The length of any curve $\hat{A}_{t}$ on the sphere must be greater or equal
to the geodesic distance. As a consequence, we obtain an operator quantum
speed limit $\tau_{\textsc{qsl}}$ by noting that

$\tau=\frac{\operatorname{length}([\hat{A}_{t}])}{\frac{1}{\tau}\operatorname{length}([\hat{A}_{t}])}\geq\frac{\operatorname{dist}(\hat{A}_{0},\hat{A}_{\tau})}{\frac{1}{\tau}\operatorname{length}([\hat{A}_{t}])},$
(6)

where
$\operatorname{length}([\hat{A}_{t}])=\int_{0}^{\tau}\norm{\mathds{L}_{t}A_{t}}dt$
is the length of the evolution. More explicitly, the speed limit can be
written in terms of $C(t)$ as

$\tau\geq\tau_{\textsc{qsl}},\quad\tau_{\textsc{qsl}}=\frac{\sqrt{C(0)}\arccos(\real\frac{C(\tau)}{C(0)})}{\mathcal{V}_{\tau}},$
(7)

where
$\mathcal{V}_{\tau}=\frac{1}{\tau}\int_{0}^{\tau}\norm{\mathds{L}_{t}A_{t}}dt$
is the time averaged speed of the evolution. For non-constant speeds,
$\mathcal{V}_{\tau}$ needs in practice be replaced by an upper bound on the
speed of the evolution in order to make $\tau_{\textsc{qsl}}$ independent on
the time $\tau$ one is trying to estimate.

We stress that the speed limit $\tau$ is only guaranteed to hold whenever
$\norm{A_{t}}$ is preserved for the evolving operator. This is always
satisfied in the case when $\mathcal{P}=\mathds{1}$ so that $(\cdot|\cdot)$
becomes the Hilbert-Schmidt inner product. For more general choices of
$\mathcal{P}$ however, norm preservation will not be guaranteed to hold for
all initial operators. In fact, the norm is preserved for all initial
operators if and only if $[\mathds{L},\mathcal{P}]=0$. Assuming
$[\mathds{L},\mathcal{P}]=0$, let $\mathcal{P}_{nm}$ and
$\omega_{\alpha}=E_{n}-E_{m}$ be the eigenvalues of $\mathcal{P}$ and
$\mathds{L}$ with respect to a common eigenbasis. Moreover, let $A_{ij}$ be
the components of $A$ with respect to this basis. Define
$v_{\alpha}=\frac{\mathcal{P}_{nm}\absolutevalue{A_{nm}}^{2}}{C(0)}$, where
$C(0)=\sum_{n,m}\mathcal{P}_{nm}\absolutevalue{A_{nm}}^{2}$. We can then
express (7) more explicitly, as

$\tau_{\textsc{qsl}}=\frac{\arccos(\real\sum_{\alpha}v_{\alpha}e^{i\omega_{\alpha}\tau})}{\sqrt{\sum_{\gamma}v_{\gamma}\omega_{\gamma}^{2}}},$
(8)

where the indices $\alpha$ and $\gamma$ runs from 1 to
$\textrm{dim}(\mathcal{H})^{2}$.

######  Example 1.

In the case when $C(t)=\Tr(A^{\dagger}A_{t}e^{-\beta H}/Z)$ we have that
$\mathcal{P}A=Ae^{-\beta H}/Z$ and a common eigenbasis between $\mathds{L}$
and $\mathcal{P}$ is given by the operators $\outerproduct{E_{n}}{E_{m}}$
where the eigenvalues of $\mathcal{P}$ are given by
$\mathcal{P}_{nm}=e^{-\beta E_{m}}/Z$.

######  Example 2.

For the Kubo inner-product we can view $\mathcal{P}$ as the composition
$\mathcal{P}=\mathcal{A}\circ\mathcal{B}\circ\mathcal{C}$ of the three
superoperators defined by $\mathcal{A}(A)=\frac{Ae^{-\beta H}}{Z}$,
$\mathcal{B}(A)=\frac{1}{\beta}\int^{\beta}_{0}d\lambda e^{-\lambda
H}Ae^{\lambda H}$ and $\mathcal{C}A=A-\Tr(A)\mathds{1}$. The superoperators
$\mathcal{A}$ and $\mathcal{B}$ commute with a common eigenbasis being given
by the eigenvectors $\outerproduct{E_{n}}{E_{m}}$. A speed limit for a time-
evolving operator $A_{t}$ using the Kubo inner-product can then be obtained by
using $\tilde{A}_{t}=\mathcal{C}(A_{t})$ and
$\tilde{\mathcal{P}}=\mathcal{A}\circ\mathcal{B}$ instead of $A_{t}$ and
$\mathcal{P}$ and then use that $\tilde{\mathcal{P}}_{nm}=\frac{e^{-\beta
E_{m}}}{\beta Z}\int_{0}^{\beta}e^{\lambda(E_{m}-E_{n})}d\lambda$ and
$\tilde{A}_{nm}=A_{nm}-\sum_{k}A_{kk}$.

Let us note that the OQSL (7) is saturated by any traceless observable
involving solely the coupling of the ground state and an excited state of a
time-independent Hamiltonian, in the same spirit as the superposition of these
eigenstates saturates the standard QSLs for states [69]. To show this, let us
consider a time-independent Hamiltonian $H$ and let us denote by $\ket{0}$ and
$\ket{E}$ its ground state and the eigenstate with energy $E$: $H\ket{0}=0$
and $H\ket{E}=E\ket{E}$. Then, it is straightforward to see that the operator
$A=\frac{1}{\sqrt{2}}(\outerproduct{0}{E}+\outerproduct{E}{0})$ exactly
saturates the OQSL (7) with respect to the Hilbert-Schmidt inner product
$(A|B)=\Tr(A^{\dagger}B)$. Indeed, the autocorrelation function reduces to
$C(t)=\cos(Et)$, in natural units, while the velocity is constant and takes
the value $\mathcal{V}=\|[H,A]\|=E$. Therefore, being $C(0)=1$ due to the
normalization, the OQSL (7) reduces to an identity at any time.

This intuition, that when the operator dynamics is confined to two time-
independent Hamiltonian eigenspaces the evolution occurs along a geodesic
trajectory, can be made more rigorous and general, as we shall describe below
identifying the general conditions for the saturation of the OQSL (7).

###  2.3 Refined speed Limit

Suppose we can find a decomposition of $\hat{A}_{t}$ such that
$\hat{A}_{t}=S+V_{t}$, where $S$ and $V_{t}$ stays orthogonal with respect to
the metric $\real\langle\cdot,\cdot\rangle$ throughout the evolution. We then
have that the minimal time it takes $V$ to reach the operator $V_{\tau}$ is
smaller or equal to the time it takes $\hat{A}$ to reach $\hat{A}_{\tau}$. We
can thus obtain another speed limit by substituting $C(t)$ with $(V|V_{t})$ in
(7). Using that $S$ and $V_{t}$ are orthogonal, one can write
$\real(V|V_{t})=\real C(t)-\norm{S}^{2}$. Our refined speed limit thus takes
the form

$\tau\geq\tau_{\textsc{ref}},\quad\tau_{\textsc{ref}}=\frac{\sqrt{C(0)-\norm{S}^{2}}\arccos(\real\frac{C(\tau)-\norm{S}^{2}}{C(0)-\norm{S}^{2}})}{\mathcal{V}_{\tau}}.$
(9)

This OQSL is a generalization of (7) since we can always trivially consider
the case when $S=0$. Also, since $\mathds{1}$ stays invariant throughout the
flow generated by the Hamiltonian, we can always consider the choice
$S=(\mathds{1}|A)\mathds{1}$. In the particular case when $(\cdot|\cdot)$ is
the Hilbert-Schmidt inner product, this choice of $S$ reduces (9) to the speed
limit, denoted by $T_{\Theta}$, in [28].

What is worth noting is that the speed limit (9) becomes tighter the larger
the norm of $S$ is. We can understand this from a geometrical perspective. The
speed limit is saturated whenever the traced-out curve of $V_{t}$ follows a
great circle on the sphere $\mathcal{S}_{\norm*{V}}$. This means that the
evolution will be contained in a two-dimensional subspace. If we then let $X$
and $Y$ be a pair of orthonormal vectors spanning this subspace, we must have
that $V_{t}/\norm{V}=\cos{\theta(t)}X+\sin{\theta(t)}Y$, where $\theta(t)$ is
some real-valued function of $t$.444In fact, if we choose $X$ and $Y$ so that
$X=V$, then $\theta$ is the angle between $V$ and $V_{t}$ and is given by
$\theta(t)=\arccos(\real\frac{C(\tau)-\norm{S}^{2}}{C(0)-\norm{S}^{2}})$.
Given that $S$ is non-zero, the corresponding curve of $\hat{A}$ must then be
situated on an effective Bloch sphere spanned by the operators $X$, $Y$ and
$S$.555We emphasize that the operators $S$, $X$ and $Y$ are orthogonal with
respect to $\real\langle\cdot,\cdot\rangle$ and not necessarily the Hilbert-
Schmidt inner product. More explicitly, we have that
$\hat{A}=\norm{V}\cos{\theta(t)}X+\norm{V}\sin{\theta(t)}Y+S$ will trace out a
curve following a circle centered at $S$ (see figure 1).

Figure 1:  As the evolving operator saturates the refined speed limit, it will
move along a circle that is displaced from the origin by $S$. As a
consequence, the length of the traced-out curve will be strictly larger than
the geodesic distance in $\mathcal{S}_{\norm*{A}}$. The numerator in (9) is
exactly the length of this traced-out curve, and we can thus conclude that the
refined speed limit is strictly tighter than the original one, given that
$S\neq 0$.

Since this circle is not centered at the origin, it will not be a great circle
on the sphere $\mathcal{S}_{\norm*{A}}$. A consequence of this is that the
length of this curve must be strictly larger than the geodesic distance on
$\mathcal{S}_{\norm*{A}}$. This length is precisely the numerator in (9) and
we can thus conclude that (9) gives a strictly tighter inequality than (7).
The difference between these inequalities becomes greater the larger
$\norm{S}$ is, which can be seen in figure 2.

Figure 2:  As $\norm{S}$ grows larger, the center of the circle that
$\hat{A}_{t}$ follows will be closer to the poles of the sphere. Consequently,
$\hat{A}_{t}$ moves in a more curved path, as highlighted by the yellow
segments where the right figure shows a top view of the sphere, and have to
travel further in order to reach the same angle $\theta$. The result of this
is that the refined speed limit becomes increasingly tight the larger
$\norm{S}$ is.

Consider the case when $\ker(\mathds{L})\cap\mathcal{H}_{\mathcal{P}}$ is
invariant throughout the evolution, for example, when the Hamiltonian commutes
with itself for any two points in time. We then have that the norm of $S$ is
maximal when $S$ is equal to the orthogonal projection of $\hat{A}$ onto the
subspace $\ker(\mathds{L})\cap\mathcal{H}_{\mathcal{P}}$—see Appendix E.666The
orthogonal projection here is with respect to the inner product
$\langle\cdot,\cdot\rangle$. Calling this component $P_{0}$, we thus have that
the tightest possible refinement, in this case, is given by

$\tau\geq\tau_{\textsc{oref}}\geq\tau_{\textsc{ref}}\geq\tau_{\textsc{qsl}},\quad\tau_{\textsc{oref}}=\frac{\sqrt{C(0)-\norm{P_{0}}^{2}}\arccos(\real\frac{C(\tau)-\norm{P_{0}}^{2}}{C(0)-\norm{P_{0}}^{2}})}{\mathcal{V}_{\tau}}.$
(10)

We will refer to this as the optimal refinement of the OQSL.

##  3 More on the Conditions for Saturation

As discussed in section 2, the part of the evolution that induces a change in
the time-correlation function will be situated on the sphere
$\mathcal{S}_{\norm*{A}}$ in the effective Hilbert space
$\mathcal{H}_{\mathcal{P}}$. The OQSL (7) will then be saturated whenever the
traced-out curve follows a great circle. In the case when we could find a
decomposition $\hat{A}_{t}=S+V_{t}$, where $S$ and $V_{t}$ remain orthogonal,
we could consider the refined speed limit (9), which is saturated if and only
if $V_{t}$ follows a great circle on the sphere $\mathcal{S}_{\norm*{V}}$. We
discussed that for saturation of (9), there exists a pair of orthonormal
vectors $X$ and $Y$ such that
$\hat{A}=\norm{V}\cos{\theta(t)}X+\norm{V}\sin{\theta(t)}Y+S$. If we choose
$X=\hat{A}/\norm{A}$, then the function $\theta(t)$ is the angle between $V$
and $V_{t}$ with respect to the real-valued inner product
$\real\langle\cdot,\cdot\rangle$ and is more explicitly given by
$\theta(t)=\arccos(\real\frac{C(\tau)-\norm{S}^{2}}{C(0)-\norm{S}^{2}})$.

This section will discuss the conditions for saturation for two particular
cases. In the first case, we consider the consequences of the operator
$\hat{A}$ having support in only two of the eigenspaces of the Hamiltonian. In
the second case, we assume $\hat{A}$ to be Hermitian, allowing us to draw
connections between the saturation of the OQSLs and the dimension of an
underlying Krylov space.

###  3.1 Evolution with support in only two eigenspaces of the Hamiltonian

Consider the case when the Hamiltonian commutes with itself at any two points
in time. We will consider the case when $\hat{A}$ has non-zero support in only
two of the eigenspaces of the Hamiltonian.777An operator $A$ is said to have
non-zero support in a subspace $\mathcal{X}\subseteq\mathcal{H}$ if
$\exists\ket{\psi}\in\mathcal{X}$ s.t. $A\ket{\psi}\neq 0$. Let $E$ and
$E^{\prime}$ denote the energies of these two eigenspaces through time and let
$\omega=E-E^{\prime}$ be the energy gap. If $P_{E}$ and $P_{E^{\prime}}$ are
the corresponding eigenspace projectors, then
$\hat{A}=P_{E}\hat{A}P_{E}+P_{E}\hat{A}P_{E^{\prime}}+P_{E^{\prime}}\hat{A}P_{E}+P_{E^{\prime}}\hat{A}P_{E^{\prime}}$
and it is straight forward to check that
$\mathds{L}P_{E}\hat{A}P_{E^{\prime}}=\omega P_{E}\hat{A}P_{E^{\prime}}$,
$\mathds{L}P_{E}\hat{A}P_{E^{\prime}}=-\omega P_{E^{\prime}}\hat{A}P_{E}$ and
$\mathds{L}P_{E}\hat{A}P_{E}=\mathds{L}P_{E^{\prime}}\hat{A}P_{E^{\prime}}=0$.
In other words, $\hat{A}$ is spanned by the eigenspaces of the Liouvillian
with eigenvalues 0, $\omega$, and $-\omega$. Let $P_{0}$, $P_{\omega}$, and
$P_{-\omega}$ be the corresponding projections of $\hat{A}$ onto these three
eigenspaces. More explicitly, we have in this specific case that
$P_{0}=P_{E}\hat{A}P_{E}+P_{E^{\prime}}\hat{A}P_{E^{\prime}}$,
$P_{\omega}=P_{E}\hat{A}P_{E^{\prime}}$ and
$P_{-\omega}=P_{E^{\prime}}\hat{A}P_{E}$ and we can write
$\hat{A}=P_{0}+P_{\omega}+P_{-\omega}$. The evolution of $\hat{A}$ is given by

$\begin{split}\hat{A}_{t}&=P_{0}+e^{i\int_{0}^{t}\omega(t^{\prime})dt^{\prime}}P_{\omega}+e^{-i\int_{0}^{t}\omega(t^{\prime})dt^{\prime}}P_{-\omega}\\\
&=P_{0}+\cos(\int_{0}^{t}\omega(t^{\prime})dt^{\prime})\big{(}P_{\omega}+P_{-\omega}\big{)}+\sin(\int_{0}^{t}\omega(t^{\prime})dt^{\prime})\big{(}iP_{\omega}-iP_{-\omega}\big{)}\\\
&=P_{0}+\cos{\theta(t)}X_{\omega}+\sin{\theta(t)}Y_{\omega}.\end{split}$ (11)

Here, we have introduced the non normalized operators
$X_{\omega}=P_{\omega}+P_{-\omega}$ and $Y_{\omega}=iP_{\omega}-iP_{-\omega}$
and the angle $\theta(t)=\int_{0}^{t}\omega(t^{\prime})dt^{\prime}$ between
$\hat{A}-P_{0}$ and $\hat{A}_{t}-P_{0}$. The requirement that $\norm{A_{t}}$
is constant in the interval $[0,\tau]$ implies that $P_{0}$, $X$ and $Y$ are
orthogonal and that $X$ and $Y$ have the same norm—see Appendix F.888Note that
we never need to assume that $\hat{A}$ is Hermitian in order to conclude this.
We can thus conclude that $\hat{A}_{t}-P_{0}$ moves along a great arc and thus
saturates the optimal refined speed limit (10). One of the consequences of
this is that any qubit system with a commutative Hamiltonian must satisfy the
speed limit (10).

The above result can be extended to non-commuting Hamiltonians that keep the
eigenspaces corresponding to $E$ and $E^{\prime}$ invariant. One might wonder
whether the Hamiltonian must keep these eigenspaces invariant for the evolving
operator $\hat{A}_{t}$ to achieve saturation. The answer is no. To see this,
we can consider the commuting Hamiltonian $H_{t}$ above and add to it any non-
trivial Hamiltonian $\tilde{H}_{t}$ that commutes with $\hat{A}_{t}$ for all
times $t\in[0,\tau]$. The Hamiltonian $H_{t}+\tilde{H}_{t}$ then generates the
same path for $\hat{A}_{t}$ as the Hamiltonian $H_{t}$. The difference is that
$H_{t}+\tilde{H}_{t}$ will not keep the eigenspaces of $E$ and $E^{\prime}$
invariant.

###  3.2 Relation to Krylov dimension for Hermitian operators

When $\hat{A}$ is Hermitian, it can be illuminating to describe the saturation
conditions in terms of the dimension of the Krylov space of the evolving
operator $\hat{A}_{t}$. The Krylov space is defined to be the smallest
subspace containing the evolution. In the Hermitian case, this is a real
vector space. We can then conclude that saturation of (7) happens if and only
if the dimension of the Krylov space is equal to two. Similarly, given that
$S\neq 0$, we have that a necessary condition for saturation of (7) is that
the dimension of the Krylov space is equal to three.

In the case when the Liouvillian is time-independent, the Krylov dimension is
given by the number of eigenspaces of $\mathds{L}$ that $\hat{A}$ has support
in—see Appendix D. If one of these eigenspaces is the kernel of $\mathds{L}$,
then we can say that the bound will be saturated if and only if the Krylov
dimension is smaller or equal to three.

##  4 Hamiltonian Flow Equations in Continuous Renormalization Group

Operator flows are ubiquitous in physics and are not restricted to time
evolution. In this section, we show that the continuous renormalization group
provides an arena in which Hamiltonian flows naturally occur and where OQSLs
are of relevance.

As a preamble, we note that the continuous renormalization group is also
extensively used in the study of the complexity of quantum states. For
instance, the continuous version of the Entanglement Renormalization tensor
networks, cMERA [70, 71], implements a real space renormalization in which the
flow of a quantum state is described as a function of a continuous parameter
characterizing the length scale. A cMERA Hamiltonian generates translations of
the cMERA parameter. The use of conventional QSL has been explored in this
context to investigate the complexity of states in quantum field theory [72],
using the path integral description of tensor networks [73, 74]. These works
result from an effort to characterize the growth of quantum complexity in
quantum field theories, a context in which conventional QSLs had been applied
[75, 76]. Indeed, all these results concern flows of quantum states and can
thus be tackled with conventional QSL.

In this section, we consider a different framework for the continuous
renormalization group as a paradigmatic example and test bed for our result,
the OQSL (7). Specifically, we focus on the Hamiltonian flow formulated by
Wegner [45] and Glazek and Wilson [46, 47], as a method for Hamiltonian block-
diagonalization [48, 49]. Consider the Hamiltonian flow
$H(l)=U(l)H(0)U(l)^{\dagger}$ with respect to some parameter $l$, where $U(l)$
is a unitary operator satisfying $U(0)=\mathds{1}$, and $H(0)$ is the
Hamiltonian of which the block diagonal form is desired. The flowing
Hamiltonian satisfies the differential equation

$\frac{dH(l)}{dl}=[\eta(l),H(l)],$ (12)

where $\eta(l)=\frac{dU(l)}{dl}U(l)^{\dagger}$ is the $l$-dependent generator
of the unitary flow. For specific choices of $\eta(l)$, the initial
Hamiltonian $H(0)$ eventually flows to its (block)-diagonal form, thus
achieving the desired diagonalization. At each point of the flow, let us
define the target Hamiltonian $H_{T}(l)$ as the diagonal part of $H(l)$,

$H_{T}(l)=\sum_{n}\epsilon_{n}(l)\ket{n}\bra{n},$ (13)

where we have defined $\epsilon_{n}(l)\equiv H_{nn}(l)$. One instance of a
generator that achieves this is $\eta=[H_{T},H]$, as originally proposed in
[45, 48]. We shall refer to this specific choice as the Wegner flow, for
simplicity, even when any Hamiltonian flow described by (12) and converging to
$H_{T}(l)$ is generally referred to as a Wegner flow in the literature.
Clearly, the choice $\eta=[H_{T},H]$ is not the only possibility and we shall
consider a different choice in an example below. Let us stress that the
$l$-dependent diagonal entries $\epsilon_{n}(l)$ in Eq. (13) do not correspond
to the Hamiltonian eigenvalues unless we have reached the end of the flow
$l=l_{f}$. In that case, the flowing Hamiltonian has been transformed into its
diagonal form to coincide with the target part, $H(l_{f})=H_{T}(l_{f})$. The
final $l_{f}$ is typically reached as $l\to\infty$, as shown explicitly in the
practical example below.

By applying our general result (7) to the evolution (12), with $t\to l$,
$A_{t}\to H(l)$ and $H(t)\to-i\eta(l)$, we obtain the following OQSL on the
Hamiltonian flow

$l\geq l_{\textsc{qsl}},\quad
l_{\textsc{qsl}}=\frac{\|H\|\arccos{\frac{(H(0)|H(l))}{\norm{H}^{2}}}}{\mathcal{V}_{l}},$
(14)

where the speed, averaged over the interval $[0,l]$, is given by

$\mathcal{V}_{l}=\frac{1}{l}\int_{0}^{l}\norm{[\eta(t),H(t)]}dt.$ (15)

For the rest of this section, we will only consider the case when
$(\cdot|\cdot)$ is the Hilbert-Schmidt inner product. In this case we have
that $\norm{[\eta,H]}=\Tr([\eta,H]^{2})$.

###  4.1 Dephasing-like Wegner flow

Before applying the OQSL (14) in an explicit example, we show that the Wegner
flow, although unitary, features formal similarities with a dephasing
evolution, under which the off-diagonal entries of the density matrix
$\rho(t)$ decay and the fixed point is set by its diagonal part only. The
crucial difference is that in the Wegner flow, the diagonal part evolves as
well, and it does so in such a way that the total evolution is unitary and the
final diagonal form coincides with the diagonalized initial Hamiltonian. Here
we aim to characterize the decay of the off-diagonal elements $H_{nm}(l)$ with
$n\neq m$ and reveal its formal analogy with dephasing. To this end, let us
adopt the approach of vectorization [77] and represent an operator
$A=\sum_{n,m}A_{nm}\ket{n}\bra{m}$ as a normalized vector

$\ket{A}=\frac{1}{\|A\|}\sum_{n,m}A_{nm}\ket{n,m},$ (16)

where $|n,m\rangle=\ket{n}\otimes\ket{m}$ and the normalization factor has
been introduced to enhance the comparison with the evolution of a quantum
state $\rho$. Now, since we are interested in the decay of the off-diagonal
elements, let us introduce the (super)projector $\mathds{Q}$ over the non-
diagonal part of the Hamiltonian,

$\mathds{Q}=\mathds{1}_{\mathcal{B}}-\mathds{P}_{H_{T}},$ (17)

where $\mathds{1}_{\mathcal{B}}$ is the identity on the operator space
$\mathcal{B}$ and $\mathds{P}_{H_{T}}$ is the (super)projector over the target
Hamiltonian

$\mathds{P}_{H_{T}}=\outerproduct{H_{T}}{H_{T}}=\frac{1}{\|H_{T}\|^{2}}\sum_{n,m}\epsilon_{n}\epsilon_{m}\ket{n,n}\bra{m,m}.$
(18)

From this expression, it follows that
$\mathds{P}_{H_{T}}\ket{H_{T}}=\ket{H_{T}}$ and $\mathds{Q}\ket{H_{T}}=0$. To
characterize the dephasing-like decay realized by the Wegner flow, let us
consider the overlap between the flowing Hamiltonian and its non-diagonal part

$\mathcal{A}_{\mathds{Q}}(l)\equiv\bra{H(l)}\mathds{Q}\ket{H(l)},$ (19)

which quantifies how far $H(l)$ is from being diagonal and vanishes as $l\to
l_{f}$, that is as $H(l)\to H_{T}(l)$. By substituting Eqs. (17) and (18) we
obtain

$\mathcal{A}_{\mathds{Q}}(l)=1-\frac{\|H_{T}\|^{2}}{\|H\|^{2}},$ (20)

where we stress that $\|H_{T}\|^{2}=\sum_{n}\epsilon^{2}_{n}(l)$ depends on
$l$. The quantity $\mathcal{A}_{\mathds{Q}}$ identically vanishes at the end
of the flow, when the Hamiltonian is diagonalized and $H(l)=H_{T}(l)$.
Remarkably, if we choose the Wegner generator to be $\eta=[H_{T},H]$ [48],
then $\mathcal{A}_{\mathds{Q}}$ decays monotonically. Indeed, its decay rate
reads as

$\frac{d}{dl}\mathcal{A}_{\mathds{Q}}(l)=-\frac{1}{\|H\|^{2}}\frac{d}{dl}\sum_{n}\epsilon^{2}_{n}=\frac{1}{\|H\|^{2}}\frac{d}{dl}\sum_{n\neq
m}|H_{nm}|^{2},$ (21)

where the last inequality follows from the conservation of the total norm,
$\frac{d}{dl}\|H\|=0$. If $\eta=[H_{T},H]$ [48], it is straightforward to
compute that

$\frac{dH_{nm}}{dl}=\sum_{k}(\epsilon_{n}+\epsilon_{m}-2\epsilon_{k})H_{nk}H_{km}$
(22)

and therefore

$\frac{d}{dl}\mathcal{A}_{\mathds{Q}}(l)=-\frac{2}{\|H\|^{2}}\sum_{n,m}(\epsilon_{n}-\epsilon_{m})^{2}|H_{nm}|^{2}<0.$
(23)

As a result, the overlap (19), quantifying how far we are from the target, is
monotonically decreasing during the flow. This feature suggests an analogy
with the well-known model of dephasing, where the purity is found to decrease
monotonically [78], in a similar manner as in Eq. (23). To show this, let us
consider the case of pure dephasing

$\frac{d\rho}{dt}=-[X,[X,\rho]],$ (24)

where $X$ is a time-independent Hermitian operator satisfying
$X\ket{n}=x_{n}\ket{n}$. Then it can be shown [78] that the purity
$\Tr\rho^{2}$ decreases monotonically with time as

$\frac{d\Tr\rho^{2}}{dt}=-2\sum_{nm}(x_{n}-x_{m})^{2}|\rho_{mn}|^{2}.$ (25)

Now, Eq. (23) can be also expressed as a “purity”-decay of the off-diagonal
Hamiltonian $H_{\text{off-diag}}(l)=H(l)-H_{T}(l)$ and it reads as

$\frac{d\Tr H_{\text{off-diag}}^{2}}{dl}=\frac{d}{dl}\sum_{i\neq
j}|H_{ij}|^{2}=-2\sum_{i,j}(\epsilon_{i}-\epsilon_{j})^{2}|({H_{\text{off-
diag}}})_{ij}|^{2},$ (26)

which formally corresponds to Eq. (25) with $\rho\to H_{\text{off-diag}}$ and
$X\to H_{T}$. We conclude that the Wegner flow (12) generated by
$\eta=[H_{T},H]$,

$\frac{dH}{dl}=[[H_{T},H],H],$ (27)

which is unitary, suppresses the off-diagonal part of the Hamiltonian as if it
was undergoing a dephasing evolution under the action of the diagonal part.

###  4.2 Wegner and Toda flows

As already advanced, there are several choices of the generator $\eta$ for
diagonalizing a given $N\times N$ matrix $H(0)$. As a possible form, consider

$\eta_{nm}(l)=H_{nm}(l){\rm sgn}\,(m-n),$ (28)

for $m\neq n$ and $\eta_{nn}=0$. Equation (12) then reduces to

$\frac{dH_{nm}(l)}{dl}=\sum_{k}H_{nk}(l)H_{km}(l)\left[{\rm sgn}\,(k-n)-{\rm
sgn}\,(m-k)\right].$ (29)

Further, assume that the matrix $H(l)$ takes a symmetric tridiagonal form.
Then, the equations for diagonal and off-diagonal components are written
respectively as

$\displaystyle\frac{dH_{nn}(l)}{dl}=2(H_{n,n+1}^{2}(l)-H_{n-1,n}^{2}(l))\qquad(n=1,2,\dots,N),$
(30)
$\displaystyle\frac{dH_{n,n+1}(l)}{dl}=H_{n,n+1}(l)(H_{n+1,n+1}(l)-H_{nn}(l))\qquad(n=1,2,\dots,N-1),$
(31)

with $H_{01}=H_{N,N+1}=0$. This set of equations takes a closed form and is
known as the Toda equations in classical nonlinear integrable systems [79, 80,
81]. We thus refer to the Hamiltonian flow generated by (28) as the Toda flow.

We note that Eq. (23) is not satisfied in the present choice of $\eta$.
However, it is still guaranteed that the matrix is diagonalized at large $l$
due to the relation

$\frac{d}{dl}\sum_{n=1}^{k}H_{nn}(l)=2H_{k,k+1}^{2}(l)\geq 0,$ (32)

where $k=1,2,\dots,N$ [82, 83]. The relation for $k=1$ denotes that
$H_{11}(l)$ is a non-decreasing function. Since $\Tr H^{2}(l)$ is independent
of $l$, each component of $H(l)$ is not divergent, if each component of the
original matrix $H_{0}$ takes a finite value. We conclude that
$\lim_{l\to\infty}H_{11}(l)$ converges to a finite value and
$\lim_{l\to\infty}H_{12}(l)=0$. Then, we examine the relation for $k=2$ to
conclude that $\lim_{l\to\infty}H_{22}(l)$ takes a finite value and
$\lim_{l\to\infty}H_{23}(l)=0$. We can repeat the same consideration for the
other values of $k$ to conclude that $H(l)$ is diagonalized at $l\to\infty$
keeping the eigenvalues of the matrix unchanged.

A possible realization of the tridiagonal matrix is the one-dimensional XY
model with isotropic interaction [84],

$H(l)=\frac{1}{2}\sum_{n=1}^{N-1}v_{n}(l)\left(X_{n}X_{n+1}+Y_{n}Y_{n+1}\right)+\frac{1}{2}\sum_{n=1}^{N}h_{n}(l)Z_{n}.$
(33)

In the $z$-basis, the second term represents the diagonal part and the first
term represents the off-diagonal part. The corresponding generator of the time
evolution is

$\eta(l)=\frac{i}{2}\sum_{n=1}^{N-1}v_{n}(l)\left(X_{n}Y_{n+1}-Y_{n}X_{n+1}\right),$
(34)

and the set of coupling functions
$\\{v_{1}(l),v_{2}(l),\dots,v_{N}(l),h_{1}(l),h_{2}(l),\dots,h_{N-1}(l)\\}$
satisfies the Toda equations

$\displaystyle\frac{dh_{n}(l)}{dl}=2(v_{n}^{2}(l)-v_{n-1}^{2}(l)),$ (35)
$\displaystyle\frac{dv_{n}(l)}{dl}=v_{n}(l)(h_{n+1}(l)-h_{n}(l)).$ (36)

This Hamiltonian commutes with the total magnetization $M=\sum_{n=1}^{N}Z_{n}$
and the matrix form of the single flip sector with $M=\pm(N-2)$ takes a
tridiagonal form.

Figure 3:  OQSLs for the Wegner and Toda flows. The upper panels represent
$\theta(l)=\arccos(\Tr(H(0)H(l))/\norm{H(0)}^{2})$ (bold lines) and its bound
$\int_{0}^{l}ds\,||[\eta(s),H(s)]||/||H||$ (thin lines) for Wegner and Toda
flows with $N=3$ (left panel) and $N=10$ (right). We set the initial matrix as
a symmetric tridiagonal form and each component is taken from a uniform random
number between $-1$ and $1$. In the lower panels, we plot the sum of off-
diagonal components $\sum_{m\neq n}H_{mn}^{2}(l)$.

We plot examples of the Wegner and Toda flows in Fig. 3. We take a traceless
symmetric tridiagonal matrix as an initial given Hamiltonian $H(0)$ in which
each nonzero component is taken from a uniform random number between $-1$ and
$1$. The numerical results in Fig. 3 implies that the Toda flow gives a tight
bound for a small $l$ and becomes worse for a large $l$ due to the
nonmonotonic decay of the off-diagonal components.

In Fig. 4, we show how the result is dependent on the dimension of the matrix
$N$. For the Wegner flow, when $N$ is not considerably large, the angle
$\theta(l)$ between $H(0)$ and $H(l)$ grows faster by the $l$-evolution as $N$
becomes large. In the Wegner flow, even though we start the time evolution
from a tridiagonal form, the matrix breaks the band structure during the flow,
which makes $\theta$ a large value. It does not necessarily give the property
that the overlap $(H(0)|H(l))$ decays rapidly as a function of $N$, as we can
see in some many-body systems exhibiting the orthogonality catastrophe. On the
other hand, the Toda flow does not show any growing behavior as a function of
$N$. This is due to the property that the tridiagonal form is kept throughout
the time evolution. As for $l_{\textsc{qsl}}$, a saturating behavior is seen
for the Wegner flow and is not seen for the Toda flow.

Figure 4:  Plot of the $\theta(l)=\arccos(\Tr(H(0)H(l))/\norm{H(0)}^{2})$ (top
panels) and $l_{\textsc{qsl}}$ (bottom panels) for several values of $N$. We
set the initial matrix as a symmetric tridiagonal form and each component is
taken from a uniform random number between $-1$ and $1$. The left panels
represent the Wegner flow, while the right panels correspond to the Toda flow.

The independence of the Toda flow on the matrix dimension implies that we can
find a tight OQSL for specific initial Hamiltonians. As we have discussed in
section 3.1, saturation is possible when the flowing operator has support in
only two of the eigenspaces of the generator. In what follows, we will
consider the condition that the eigenvectors of the generator $\eta(l)$ are
$l$-independent.

Hereafter, we write $H_{nn}(l)=h_{n}(l)$ and $H_{n,n+1}(l)=v_{n}(l)$. The
eigenvalue equation $\eta(l)|\varphi\rangle=i\lambda(l)|\varphi\rangle$, with
a real eigenvalue $\lambda(l)$ and the corresponding eigenvector
$|\varphi\rangle$ is written as

$\left(\begin{array}[]{ccccc}\varphi_{2}&0&&&\\\
-\varphi_{1}&\varphi_{3}&0&&\\\ 0&-\varphi_{2}&\varphi_{4}&&\\\
&&\ddots&\ddots&\\\
&&0&-\varphi_{N-2}&\varphi_{N}\end{array}\right)\left(\begin{array}[]{c}v_{1}(l)\\\
v_{2}(l)\\\ v_{3}(l)\\\ \vdots\\\
v_{N-1}(l)\end{array}\right)=i\lambda(l)\left(\begin{array}[]{c}\varphi_{1}\\\
\varphi_{2}\\\ \varphi_{3}\\\ \vdots\\\ \varphi_{N-1}\end{array}\right),$ (37)

and $-v_{N-1}(l)\varphi_{N-1}=i\lambda(l)\varphi_{N}$, where
$(\varphi_{1},\varphi_{2},\dots,\varphi_{N})$ denotes the $l$-independent
eigenvector $|\varphi\rangle$. When the diagonal components of the matrix on
the left-hand side are nonzero, the matrix is invertible and $v_{n}(l)$ for
any index $n$ is proportional to the same $l$-dependent function $\lambda(l)$.
The possibility that some of the components of $|\varphi\rangle$ are
identically zero is excluded since that condition only results in
$|\varphi\rangle=0$.

As an exceptional case, we can find the eigenvector with $\lambda(l)=0$ when
$N$ is odd. In that case, the eigenvector is written as

$|\varphi\rangle\propto\left(1,0,\frac{v_{1}(l)}{v_{2}(l)},0,\frac{v_{3}(l)v_{1}(l)}{v_{4}(l)v_{2}(l)},0,\dots,0,\frac{v_{N-2}(l)v_{N-4}(l)\cdots
v_{1}(l)}{v_{N-1}(l)v_{N-3}(l)\cdots v_{2}(l)}\right)^{\rm T}.$ (38)

The $l$-independence of $|\varphi\rangle$ gives the conditions
$v_{2k}(l)\propto v_{2k-1}(l)$ with $k=1,2,\cdots,(N-1)/2$.

We note that the eigenvector with $\lambda(l)=0$ is unique if it exists.
Therefore, when we impose the condition that two of the eigenvectors of
$\eta(l)$ are $l$-independent, the dependence of $\eta(l)$ on $l$ is described
by a single function $f(l)$. Each of the nonzero components is written as

$v_{n}(l)=f(l)v_{n}.$ (39)

We insert the condition (39) into Eqs. (30) and (31) to find

$\displaystyle h^{\prime}_{n}(l)=2f^{2}(l)(v_{n}^{2}-v_{n-1}^{2}),$ (40)
$\displaystyle f^{\prime}(l)=f(l)(h_{n+1}(l)-h_{n}(l)),$ (41)

where the prime symbol denotes the derivative with respect to $l$. The second
equation (41) shows that $h_{n}(l)$ is a linear function in $n$. Since the
constant shift $h_{n}(l)\to h_{n}(l)+h_{0}$ does not change Eqs. (30) and
(31), we set $\sum_{n=1}^{N}h_{n}(l)=0$ and obtain

$h_{n}(l)=\frac{f^{\prime}(l)}{f(l)}\left(n-\frac{N+1}{2}\right).$ (42)

We use this form for the first equation (40). Then, $v_{n}^{2}(l)$ is a
quadratic function of $n$ and $f(l)$ obeys the differential equation

$\left(\frac{f^{\prime}(l)}{f(l)}\right)^{\prime}=-2d_{1}f^{2}(l),$ (43)

where $d_{1}$ represents a constant. The corresponding form of $v_{n}$ is

$v_{n}^{2}=\frac{1}{2}d_{1}n(N-n)+d_{0},$ (44)

where $d_{0}$ represents a constant.

To determine $d_{0}$, we look at the following condition, which follows from
the conservation of the norm

$\sum_{n=1}^{N}h_{n}^{2}(l)+2\sum_{n=1}^{N-1}v_{n}^{2}(l)={\rm const}.$ (45)

Without losing the generality, we can put the form $f(l)=\cos\theta(l)$. Using
Eqs. (43) and (45), we find as the possible solution

$\displaystyle d_{0}=0,$ (46)
$\displaystyle\left(\frac{\theta^{\prime}(l)}{\cos\theta(l)}\right)^{2}=2d_{1}.$
(47)

The $l$ dependence of each component is specified as
$h_{n}(l)=h_{n}\sin\theta(l)$ and $v_{n}(l)=v_{n}\cos\theta(l)$. By
representing $d_{1}$ with respect to $h_{1}$, we finally obtain

$\displaystyle
h_{n}(l)=-\frac{2h_{1}}{N-1}\left(n-\frac{N+1}{2}\right)\sin\theta(l),$ (48)
$\displaystyle
v^{2}_{n}(l)=\frac{n(N-n)}{(N-1)^{2}}h_{1}^{2}\cos^{2}\theta(l),$ (49)

and

$\frac{\theta^{\prime}(l)}{\cos\theta(l)}=\frac{2h_{1}}{N-1}.$ (50)

The differential equation for $\theta(l)$ is easily solved as

$\sin\theta(l)=\frac{\sinh\left(\frac{4h_{1}}{N-1}l\right)+\sin\theta(0)\cosh\left(\frac{4h_{1}}{N-1}l\right)}{\cosh\left(\frac{4h_{1}}{N-1}l\right)+\sin\theta(0)\sinh\left(\frac{4h_{1}}{N-1}l\right)}.$
(51)

In Fig. 5, we plot $\\{h_{n}\\}_{n=1,2,\cdots,N}$,
$\\{v_{n}\\}_{n=1,2,\cdots,N-1}$, and $\theta(l)$ for $N=20$.

Figure 5:  Left: The parameters of the initial matrix $H$,
$\\{h_{n}\\}_{n=1,2,\cdots,N}$ and $\\{v_{n}\\}_{n=1,2,\cdots,N-1}$, resulting
in the tight bound in the Toda flow with $N=20$. Right:
$\theta(l)=\arccos(\Tr(H(0)H(l))/\norm{H(0)}^{2})$. We set
$l_{0}=(N-1)/4h_{1}$ and $\sin\theta(0)=0$.

All components of the matrix $H(l)$ are parameterized by a single
$l$-dependent function $\theta(l)$, which implies that the time evolution can
be denoted by a motion along an arc in the Bloch space. In fact, we find that
$\theta(l)$ denotes the operator angle and the dynamics gives the tight bound:

${\rm
arccos}\,\left|\frac{\Tr(H(l)H(0))}{\Tr(H(0)^{2})}\right|=\int_{0}^{l}dt\,\sqrt{\frac{\Tr([\eta(t),H(t)]^{2})}{\Tr(H(0)^{2})}}=\theta(l)-\theta(0).$
(52)

##  5 Operator Growth and Krylov Complexity

In the previous section, we have considered the flow of an observable, the
Hamiltonian, with respect to a parameter different than time. Here we
illustrate another application of the OQSL, pointing out that operator flows
need not necessarily concern an observable. In particular, given a Liouvillian
operator $\mathds{L}=[H,\cdot]$, we show that the geometrical OQSL (7) can
also be applied to the unitary flow of a superoperator, generated under the
action of $\mathds{S}=[\mathds{L},\cdot]$, which can be accordingly viewed as
a “super Liouvillian”. This kind of flow arises naturally in characterizing
the complexity of a given quantum evolution. Specifically, in the context of
operator growth, the notion of Krylov complexity [54, 58, 59, 60, 61, 62] has
recently gained attention as a measure of operator complexity for the
Heisenberg evolution of an observable under the action of a time-independent
Hamiltonian. The evolution of simple, local observables into increasingly
complex and nonlocal ones can be described as the operator spreading in the
so-called Krylov space. As mentioned in section 3.2, the latter provides the
minimal subspace in which the Heisenberg dynamics unfolds and is uniquely
determined by the Hamiltonian of the system and the initial operator $O_{0}$.
Krylov complexity can then be understood as the mean position of the evolving
operator $O_{t}$ in the so-called Krylov basis. It can be expressed as an
expectation value $(O_{t}|\mathds{K}O_{t})$ of a corresponding (super)operator
$\mathds{K}$, known as the complexity operator. At this point, one can again
change representation and let the superoperators, such as $\mathds{K}$, evolve
while keeping the observable $|O)$ fixed. We shall call this representation
the super-Heisenberg picture from the evident analogy with the standard
Heisenberg representation of the quantum evolution in the Hilbert space. In
this picture, the complexity operator evolves accordingly to the equation

$\dot{\mathds{K}}=i[\mathds{L},\mathds{K}]$ (53)

and the corresponding unitary flow is constrained by the speed limit (7), upon
identifying $A$ and $H$ with $\mathds{K}$ and $\mathds{L}$, respectively. We
note that our result (7) holds for finite dimensions and that the dimension of
the Krylov space is always finite whenever the Hilbert space that the
observables are defined over is finite [61].

###  5.1 Quantum dynamics in Krylov space

Let us start by briefly recalling how the Krylov space and the corresponding
notion of complexity are constructed. For a more detailed discussion, we refer
to [54, 58, 59, 60, 61, 62]. The evolution in the Heisenberg picture of an
operator $O_{t}=e^{iHt}O_{0}e^{-iHt}$ can be formally written in terms of the
nested commutators with the Hamiltonian $H$, that is, the powers of the
Liouvillian $\mathds{L}=[H,\cdot]$, as
$O_{t}=\sum_{n=0}^{\infty}\frac{(it)^{n}}{n!}\mathds{L}^{n}O$. The space
explored during this evolution is given by the span of the infinite set
$\\{\mathds{L}^{n}O\\}_{n=0}^{\infty}$ and is precisely the Krylov space. From
this infinite set, one can extract an orthonormal, finite basis
$\\{O_{n}\\}_{n=0}^{D-1}$ by applying the so-called Lanczos algorithm. The
first element of the basis coincides with the initial operator $O_{0}$, which
we will assume to be normalized to one. Then, at each iterative step, the next
orthogonal vector is constructed as
$|A_{n+1})=\mathds{L}|O_{n})-b_{n}|O_{n-1})$, where $b_{n}=\norm{A_{n}}$ is
the $n$-th Lanczos coefficient, and the corresponding element of the Krylov
basis $|O_{n+1})$ is obtained upon normalization as $O_{n}=A_{n}/b_{n}$.
Throughout this section, we will use the Hilbert-Schmidt inner product
$(A|B)=\Tr A^{\dagger}B$ between operators. By making use of the Krylov space,
the unitary evolution of the operator $O_{t}$ is effectively mapped to a
hopping problem on the one-dimensional, semi-infinite chain represented by the
Krylov basis $\\{O_{n}\\}_{n=0}^{D-1}$, where the Lanczos coefficients $b_{n}$
play the role of hopping parameters and the Liouvillian, which takes the
tridiagonal form

$\mathds{L}=\sum_{n=0}^{D-1}b_{n+1}|O_{n+1})(O_{n}|+b_{n}|O_{n-1})(O_{n}|,$
(54)

with $|O_{-1})=|O_{D})=0$, acts as an Hamiltonian for the so-called operator
wavefunction $|O_{t})$. The Krylov complexity operator $\mathds{K}$ is then
defined as the position operator

$\mathds{K}=\sum_{n=0}^{D-1}n|O_{n})(O_{n}|,$ (55)

on this lattice. The most studied object in this context is the expectation
value of the above (super)-operator with respect to $O_{t}$ and is simply
known as the Krylov complexity: $K=(O_{t}|\mathds{K}O_{t})$. Its rate of
growth is constrained by the speed limit

$|\partial_{t}K(t)|\leq 2b_{1}\Delta\mathds{K},$ (56)

introduced in [62] and known as the dispersion bound, given that
$(\Delta\mathds{K})^{2}=(O_{t}|\mathds{K}^{2}O_{t})-(O_{t}|\mathds{K}O_{t})^{2}$
is the variance of $\mathds{K}$ with respect to $O_{t}$. The dispersion bound
is saturated at any time if and only if the structure of Krylov space features
the so-called complexity algebra (57), which we shall introduce below.

It was pointed out in [59] that the Liouvillian in Krylov space, given by Eq.
(54), can be written as the sum $\mathds{L}=\mathds{L}_{+}+\mathds{L}_{-}$ of
raising and lowering operators that act on the Krylov basis as
$\mathds{L}_{+}|O_{n})=b_{n+1}|O_{n+1})$ and
$\mathds{L}_{-}|O_{n})=b_{n}|O_{n-1})$, respectively. It appears then natural
to introduce a super-operator $\mathds{B}=\mathds{L}_{+}-\mathds{L}_{-}$,
conjugated to the Liouvillian, and consider their commutator
$\tilde{\mathds{K}}=[\mathds{L},\mathds{B}]$ [59]. The dispersion bound (56)
is identically saturated if and only if these three operators close an
algebra, which can only take the form [62]

$[\mathds{L},\mathds{B}]=\tilde{\mathds{K}},\quad[\tilde{\mathds{K}},\mathds{L}]=\alpha\mathds{B},\quad[\tilde{\mathds{K}},\mathds{B}]=\alpha\mathds{L}$
(57)

and implies also that $\tilde{\mathds{K}}=\alpha\mathds{K}+\gamma\mathds{1}$,
where $\alpha,\gamma\in\mathbb{R}$ [59]. The Krylov complexity growth rate is
then maximal [62]. It can be shown that $\gamma$ is always a positive number
and $\alpha$ is a real number satisfying the condition
$\alpha=-\frac{2\gamma}{D-1}$ for finite Krylov dimension $D$ and $\alpha\geq
0$ if $D=\infty$ [62]. Moreover, the algebraic closure (57), or equivalently
the saturation of the bound (56), implies that the Lanczos coefficients evolve
according to [59, 62]

$b_{n}=\sqrt{\frac{1}{4}\alpha n(n-1)+\frac{1}{2}\gamma n}.$ (58)

For $\alpha>0$, this dependence captures the asymptotic linear growth
$b_{n}=\sqrt{\alpha}n$ conjectured by Parker et al. to hold in generic non-
integrable systems, leading to the maximal growth of Krylov complexity [54]. A
paradigmatic example of this class of systems is the celebrated Sachdev-Ye-
Kitaev (SYK) model [85].

###  5.2 Saturation of the OQSL by the complexity algebras

In the “super” Heisenberg picture, the observable operator is kept fixed while
the complexity operator $\mathds{K}$ evolves unitarily according to

$\mathds{K}_{t}=e^{-i\mathds{L}t}\mathds{K}_{0}e^{i\mathds{L}t}=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}\mathds{S}^{n}(\mathds{K}_{0})\,t^{n},$
(59)

where $\mathds{S}=[\mathds{L},\cdot]$ will be referred to as super
Liouvillian. In the case of closed complexity algebras, thanks to the
commutation relations (57), all the powers $\mathds{S}^{n}(\mathds{K}_{0})$
reduce to terms proportional to either $\mathds{K}_{0}$ or $\mathds{B}$. In
particular, if $\alpha\neq 0$ one can show that [62]

$\displaystyle\mathds{S}^{2n}(\mathds{K}_{0})$ $\displaystyle=$
$\displaystyle(-1)^{n}\alpha^{n-1}(\alpha\mathds{K}_{0}+\gamma\mathds{1}),$
(60) $\displaystyle\mathds{S}^{2n+1}(\mathds{K}_{0})$ $\displaystyle=$
$\displaystyle(-1)^{n+1}\alpha^{n}\mathds{B},$ (61)

where the first equation clearly holds only for $n>0$, being
$\mathds{S}^{0}(\mathds{K}_{0})=\mathds{K}_{0}$. Conversely, when $\alpha=0$
only $\mathds{S}(\mathds{K}_{0})=-\mathds{B}$ and
$\mathds{S}^{2}(\mathds{K}_{0})=-\gamma\mathds{1}$ survive, being
$\mathds{S}^{n}(\mathds{K}_{0})=0$ for $n>2$. Thus,

$\mathds{K}_{t}=\left\\{\begin{array}[]{ll}(1-\alpha)\mathds{K}_{0}-\gamma\mathds{1}+\cosh(\sqrt{\alpha}t)\left(\mathds{K}_{0}+\frac{\gamma}{\alpha}\mathds{1}\right)+i\sinh(\sqrt{\alpha}t)\frac{1}{\alpha}\mathds{B}&\quad\alpha\neq
0,\\\
\mathds{K}_{0}+i\mathds{B}t+\frac{\gamma}{2}t^{2}\mathds{1}&\quad\alpha=0.\end{array}\right.$
(62)

Equation (59), together with Eqs. (60)-(61), implies that whenever the
dispersion bound (56) is saturated, the full time-evolution of the Krylov
complexity operator $\mathds{K}_{t}$ must be contained in a $3$-dimensional
space, spanned by the identity $\mathds{1}$, the initial complexity
$\mathds{K}_{0}$ and $\mathds{B}=[\mathds{K}_{0},\mathds{L}]$:

$\mathds{K}_{t}\in\text{Span}\\{\mathds{1},\mathds{K}_{0},\mathds{B}\\}.$ (63)

This space defines the “super” Krylov space of Krylov complexity itself and
turns out to be dramatically simplified by the assumption of closed complexity
algebra. From the discussion in section 3.2, we can conclude that
$\mathds{K}_{t}$ having a $3$-dimensional Krylov space is a sufficient
condition for it to saturate the refined OQSL (10). We recall that here
$A_{t}$ and $\mathds{L}$ are replaced by $\mathds{K}_{t}$ and $\mathds{S}$,
respectively. Then Eq. (63) implies that $\mathds{K_{t}}$ has support in only
three eigenspaces of $\mathds{S}$, one of them corresponding to the eigenvalue
$0$: $\mathds{K}_{t}=\mathds{P}_{0}+\mathds{P}_{\omega}+\mathds{P}_{-\omega}$.
Therefore, we conclude that the super operator $\mathds{K}_{t}-\mathds{P}_{0}$
saturates the OQSL (10); that is, it evolves along a geodesic trajectory. In
other words, the maximal growth rate of Krylov complexity, leading to the
saturation of the dispersion bound (56), is equivalent to the geodesic
evolution of the Krylov complexity operator, provided that we subtract its
stationary component $\mathds{P}_{0}$. Indeed, we remark that $\mathds{P}_{0}$
is necessarily different from zero since $\Tr\mathds{K}\neq 0$, and must be
removed to obtain a tight OQSL. From the explicit computation performed below,
we shall conclude that the identity is indeed the only stationary component of
the Krylov complexity, i.e.,
$\mathds{P}_{0}=\Tr\mathds{K}\frac{\mathds{1}}{\|\mathds{1}\|^{2}}$. By doing
so, we shall also explicitly assess the improvement achieved by replacing the
OQSL (7) with its refined counterpart (10).

###  5.3 Computation of the OQSL for the complexity algebras

The geometrical OQSL (7) for the Krylov complexity reads as

$t\geq\|\mathds{K}_{0}\|\frac{\arccos\big(\frac{(\mathds{K}_{0}|\mathds{K}_{t})}{\|\mathds{K}_{0}\|^{2}}\big{missing})}{\|[\mathds{L},\mathds{K}_{0}]\|},$
(64)

where, given a time-independent Liouvillian $\mathds{L}$, the velocity of the
flow $\|[\mathds{L},\mathds{K}_{0}]\|$ is constant. The initial complexity
$\mathds{K}_{0}$ coincides with the usual Krylov complexity operator (55) in
the standard Heisenberg picture. In order to assess its deviation from
saturation, we explicitly evaluate the OQSL (64) in the case of the $SU(2)$
complexity algebra, that is, when the complexity growth saturates the
dispersion bound (56) in finite dimension [62]. Let us define
$\mathcal{V}_{\mathds{K}}\equiv\|\mathds{K}_{0}\|^{-1}\|[\mathds{L},\mathds{K}_{0}]\|$
as the (normalized) velocity of the complexity flow. By explicit computation
in the Krylov basis, one can verify that $\mathcal{V}_{\mathds{K}}$, being
$[\mathds{L},\mathds{K}_{0}]=-\mathds{B}$ by construction, always reduces to

$\mathcal{V}^{2}_{\mathds{K}}=\frac{\|\mathds{B}\|^{2}}{\|\mathds{K}_{0}\|^{2}}=\frac{2}{\|\mathds{K}_{0}\|^{2}}\sum_{n=1}^{D-1}b_{n}^{2},$
(65)

where the norm of the complexity is fixed by the Krylov dimension as

$\|\mathds{K}_{0}\|^{2}=\sum_{n=0}^{D-1}n^{2}=\frac{D(D-1)(2D-1)}{6}.$ (66)

The velocity (65) of the complexity flow is maximized whenever the Lanczos
coefficients growth is maximal, i.e., linear in $n$, which is the case for
maximally chaotic systems according to the universal growth hypothesis [54].
Indeed, by focusing on the initial scrambling period and neglecting the role
of the following plateau and descent in the $b_{n}$’s [61], we conclude that a
sub-polynomial behavior $b_{n}\propto n^{\delta}$ with $0<\delta<1$ always
leads to a smaller velocity $\mathcal{V}_{\mathds{K}}$ than the linear growth
$b_{n}\propto n$. We stress that this observation also holds for infinite-
dimensional Krylov spaces. As shown below, the velocity (65) remains finite in
this limit, and once we fix the proportionality constant, the velocity is
maximized by the linear growth of the $b_{n}$’s. In other words, according to
the universal growth hypothesis [54], complexity flows at the highest speed in
maximally chaotic systems.

Let us now focus on the instances of Krylov dynamics that saturate another
notion of the speed limit for operator growth, namely the above-mentioned
dispersion bound (56). As reviewed above, in such cases, the dynamics of
Krylov complexity is determined by an underlying $3$-dimensional algebra [62]
and the Lanczos coefficients obey Eq. (58). As a result, the velocity (65) of
the complexity flow can be expressed as

$\mathcal{V}^{2}_{\mathds{K}}=\frac{\alpha(D-2)+3\gamma}{2D-1}.$ (67)

Before restricting the analysis to the finite-dimensional case $\alpha<0$,
where the geometrical QSL (7) can be applied, let us stress that this notion
of velocity is well defined also in the limit $D\to\infty$, where $\alpha\geq
0$. In particular, if $\alpha>0$, when the Krylov complexity diverges
exponentially with time as $K(t)\sim e^{\sqrt{\alpha}t}$ [59, 62], we obtain
that $\mathcal{V}_{\mathds{K}}\to\sqrt{\alpha/2}$: the speed of the complexity
operator flow is proportional to the characteristic time scale of the
exponential divergence of the Krylov complexity. Instead, if $\alpha=0$, which
leads to a quadratic divergence of $K$ [59, 62], the above-defined speed of
the flow vanishes. This singular behavior can be understood as the QSL (64),
derived under the assumption of finite dimension $D$, may not have a well-
defined counterpart for $D\to\infty$. In particular, as we shall see below,
the numerator in Eq. (64) also vanishes for $\alpha=0$, resulting in an
indeterminate form $0/0$. Finally, for the Krylov dynamics to saturate the
dispersion bound (56) over a finite-dimensional space, the underlying
complexity algebra must be that of $\textit{SU}(2)$, corresponding to the case
$\alpha<0$ [62]. In such case, given the condition $b_{D}=0$, the parameters
$\alpha$ and $\gamma$ of Eq. (58) are subject to the further constraint
$2\gamma=|\alpha|(D-1)$ [62], which also ensures the expression (67) to be
positive. Indeed, the velocity of the complexity flow generated by the
$\textit{SU}(2)$ algebra reduces to
$\mathcal{V}^{2}_{\mathds{K}}=|\alpha|(D+1)/[2(2D-1)]$. For this class of
models, we compute the QSL (64) exactly, thus establishing a direct comparison
between the geometrical OQSL introduced in the present work and the dispersion
bound that constraints the growth of Krylov complexity [62].

In addition to the velocity of the flow, the other quantity that characterizes
the speed limit is the notion of distance spanned during the evolution, which
appears at the numerator of Eqs. (7) and (64) and is given in terms of the
autocorrelation function, i.e., the operator overlap. In the case of closed
complexity algebras, the autocorrelation function of the complexity

$(\mathds{K}_{0}|\mathds{K}_{t})=\sum_{n=0}^{\infty}\frac{(-i)^{n}}{n!}(\mathds{K}_{0}|\mathds{S}^{n}(\mathds{K}_{0}))\,t^{n}$
(68)

can be explicitly evaluated by making use of Eqs. (60)-(61). We note that,
since $(\mathds{K}_{0}|\mathds{B})=0$, only the even powers of the super
Liouvillian $\mathds{S}$ contribute to the sum in Eq. (68). Moreover, the only
finite-dimensional case where we can straightforwardly apply the OQSL (64) is
that of the $\textit{SU}(2)$ complexity algebra, i.e., when $\alpha<0$. For
this class of models, by substituting Eqs. (60)-(61) into the expression (68)
and recognizing the Taylor expansion of the cosine, we obtain

$(\mathds{K}_{0}|\mathds{K}_{t})=(\|\mathds{K}\|^{2}+\frac{\gamma}{\alpha}\Tr\mathds{K})\cos{\sqrt{|\alpha|}t}-\frac{\gamma}{\alpha}\Tr\mathds{K},$
(69)

where $\Tr\mathds{K}=D(D-1)/2$ and $\|\mathds{K}\|^{2}$ is given by Eq. (66).
Therefore, in the case of the $SU(2)$ complexity algebra, the autocorrelation
function $(\mathds{K}_{0}|\mathds{K}_{t})$ oscillates at the frequency
$\sqrt{|\alpha|}$, which, we note, is half the frequency of oscillation of the
Krylov complexity $K$ itself [62]. By substituting Eq. (69) into Eq. (64) and
using that $2\gamma=|\alpha|(D-1)$, we rewrite the OQSL as

$t\geq\frac{1}{\mathcal{V}_{\mathds{K}}}\arccos[\Bigg{(}1-\frac{3(D-1)}{2(2D-1)}\Bigg{)}\cos{\sqrt{|\alpha|}t}+\frac{3(D-1)}{2(2D-1)}\Bigg{]},$
(70)

where $\mathcal{V}^{2}_{\mathds{K}}=|\alpha|(D+1)/[2(2D-1)]$. As argued above,
we expect this bound not to be tight due to the presence of a stationary
component in the complexity flow given by the non-vanishing of its trace. We
illustrate the deviation from the geodesic trajectory

$(\mathds{K}_{0}|\mathds{K}_{t})|_{\text{geodesic}}=\|\mathds{K}\|^{2}\cos{\mathcal{V}_{\mathds{K}}t}$
(71)

and the divergence of the two sides of Eq. (70) in Fig. 6. In what follows, we
shall explicitly remove the stationary component of the complexity flow, thus
proving the saturation of the refined OQSL (10) and showing its equivalence
with the dispersion bound (56).

Figure 6: We illustrate the deviation from the saturation of the OQSL (64) for
the $SU(2)$ complexity algebra, with Krylov dimension $D=1000$ and
$\alpha=-1$. a We compare the evolution (69) of the complexity autocorrelation
function $(\mathds{K}_{0}|\mathds{K}_{t})$ (red curve) with the geodesic
trajectory $(\mathds{K}_{0}|\mathds{K}_{t})|_{\text{geodesic}}$ (71) (blue
dashed curve). b We show the left and right-hand side of the inequality (70)
(blue dashed and red curves, respectively). The OQSL is tight only near $t=0$,
and the deviation increases with time.

Finally, although the framework of the geometrical QSL requires a finite
dimension, it is instructive to consider the behavior of the quantities
involved in Eq. (64) as $D\to\infty$. In particular, we have already stressed
above that the velocity $\mathcal{V}_{\mathds{K}}$ (65) of the complexity flow
remains finite, is non-zero for $\alpha>0$ and vanishes for $\alpha=0$.
Moreover, from Eq. (68), with analogous steps as for the $\textit{SU}(2)$, we
obtain that for $\alpha=0$ the autocorrelation function behaves as
$(\mathds{K}_{0}|\mathds{K}_{t})=\|\mathds{K}\|^{2}+\frac{\gamma}{2}t^{2}$. At
any finite time, this implies the vanishing of the numerator of the QSL (64),
since the argument of the arccosine approaches $1$ as $D\to\infty$. The
evaluation of the limit of the full expression yields as a result that
$\tau_{QSL}\to 0$ as $D\to\infty$ in the case of the HW algebra, i.e., for
$\alpha=0$. Conversely, the notion of distance employed in our QSL (7) is not
well defined in the case $\alpha>0$, since the normalized autocorrelation
function diverges exponentially with time,
$(\mathds{K}_{0}|\mathds{K}_{t})/\|\mathds{K}\|^{2}\sim\exp{\sqrt{\alpha}t}$.

####  5.3.1 Saturation of the refined OQSL

The fact that the Krylov complexity operator $\mathds{K}_{t}$ cannot saturate
the OQSL (64) is already evident from the observation that
$\Tr\mathds{K}_{t}\neq 0$, as this condition results in a non-zero stationary
component $\mathds{P}_{0}$. In particular, the trace accounts for the
stationary component along the identity. Removing the trace is sufficient to
achieve saturation only if there are no other stationary components of
$\mathds{K}_{t}$, meaning that the zero eigenvalue of the super Liouvillian
$\mathds{S}$ has no degeneracy, such that the corresponding eigenspace is
spanned by the identity. We explicitly show that this is indeed the case for
closed complexity algebras. In this sense, the dispersion bound (56) and the
OQSL (10) provide a unique constraint on the operator growth in Krylov space
and the saturation of the former automatically implies the saturation of the
latter.

Let us, therefore, consider the operator $\overline{\mathds{K}}_{t}$ obtained
by subtracting from the Krylov complexity its component along the identity:

$\overline{\mathds{K}}_{t}=\mathds{K}_{t}-(\mathds{K}_{t}|\mathds{1})\frac{\mathds{1}}{\|\mathds{1}\|^{2}},$
(72)

where $(\mathds{K}_{t}|\mathds{1})=\Tr\mathds{K}$ and $\|\mathds{1}\|^{2}=D$.
The OQSL (7) for $\overline{\mathds{K}}_{t}$ reads as

$t\geq\frac{\arccos\Big(\frac{(\overline{\mathds{K}}_{0}|\overline{\mathds{K}}_{t})}{\|\overline{\mathds{K}}\|^{2}}\Big{missing})}{\mathcal{V}_{\overline{\mathds{K}}}},$
(73)

where
$\mathcal{V}^{2}_{\overline{\mathds{K}}}=\|\mathds{B}\|^{2}/\|\overline{\mathds{K}}\|^{2}$.
Now, by using Eq. (66) and that $\Tr\mathds{K}=D(D-1)/2$, we obtain

$\|\overline{\mathds{K}}\|^{2}=\|\mathds{K}\|^{2}-\frac{(\Tr\mathds{K})^{2}}{D}=\frac{D(D^{2}-1)}{12}.$
(74)

Moreover, in the case of $SU(2)$ complexity algebra, from Eq. (58) with
$\alpha<0$ and $2\gamma=|\alpha|(D-1)$ we find

$\|\mathds{B}\|^{2}=2\sum_{n=1}^{D-1}b_{n}^{2}=|\alpha|\frac{D(D^{2}-1)}{12}.$
(75)

By taking the ratio of the expressions above, we conclude that the velocity of
the $\overline{\mathds{K}}_{t}$ flow for the $SU(2)$ complexity algebra is
$\mathcal{V}_{\overline{\mathds{K}}}=\sqrt{|\alpha|}$. On the other hand, the
autocorrelation function
$(\overline{\mathds{K}}_{0}|\overline{\mathds{K}}_{t})$ can be computed
analogously to the one of $\mathds{K}_{t}$ in Eqs. (68)-(69), with the only
difference that now $\Tr\overline{\mathds{K}}=0$. We thus obtain

$(\overline{\mathds{K}}_{0}|\overline{\mathds{K}}_{t})=\|\overline{\mathds{K}}\|^{2}\cos\sqrt{|\alpha|}t=\|\overline{\mathds{K}}\|^{2}\cos\mathcal{V}_{\overline{\mathds{K}}}t,$
(76)

which implies that the inequality in the OQSL (73) reduces t an identity at
any time. In conclusion, the saturation of the dispersion bound (56) in finite
dimension implies that the Krylov complexity $\mathds{K}_{t}$ also saturates
the refined OQSL (10) with $\mathds{P}_{0}=\mathds{1}\Tr\mathds{K}/D$. We
illustrate this saturation in Fig. (7). From the comparison between Figs. 6
and (7) we can assess the efficiency of the refined OQSL (10) in yielding a
tight evolution by removing the components that do not contribute dynamically
to the flow.

Figure 7: Illustration of the saturation of the OQSL (73) for the $SU(2)$
complexity algebra, with Krylov dimension $D=1000$ and $\alpha=-1$. a The
complexity autocorrelation function
$(\overline{\mathds{K}}_{0}|\overline{\mathds{K}}_{t})$ (76) (red curve) is
shown to match the geodesic trajectory (blue dashed curve). b The left-hand
and right-hand sides of the inequality (73) (blue dashed and red curves,
respectively). The OQSL reduces to an identity at any time.

##  6 Conclusions, Discussion and Outlook

Conventional QSLs bound the minimum time for the completion of a process by
quantifying the distance traveled by the system along the evolution in state
space. OQSLs generalize the scope of conventional QSL to account for processes
described in terms of operator flows, i.e., the change of an operator
resulting from a conjugation by a one-parameter unitary [56]. In this work, we
have introduced a geometric OQSL that holds for arbitrary unitaries, i.e.,
whether the generator of flow is parameter dependent or not. In addition, we
have shown the bound to be tight and identified the required conditions for
its saturation. This has led us to introduce a refined OQSL upon identifying
the subspace in which the dynamics unfolds.

The usefulness of these OQSLs has been illustrated in the context of a
continuous renormalization group formulated as a Wegner Hamiltonian flow for
block diagonalization. In this context, we have shown that Wegner’s choice of
the flow generator leads to a monotonic decay of the off-diagonal elements of
the flowing Hamiltonian towards the target block-diagonal one. However, such a
choice does not saturate the OQSL. By contrast, an alternative choice of the
generator associated with the Toda flow can lead to the saturation of the OQSL
for a specific family of initial Hamiltonians. Beyond the case of Wegner
Hamiltonian flows, we expect our results to apply to other schemes for
Hamiltonian diagonalization, such as those relying on the Schrieffer-Wolff
transformation [86].

We have further discussed the implication of our results in the context of
operator growth in Krylov space. In this representation, the time evolution of
an operator is analogous to the spreading of a particle in the Krylov lattice,
where the mean position is a proxy for operator complexity. The conditions for
maximal operator growth are then associated with the saturation of the
dispersion bound [62], which occurs when the Lanczos coefficients exhibit a
specific dependence on the lattice site index. Here, we have introduced a
“super-Heisenberg” representation of the Krylov complexity operator generated
by a super Liouvillian. Making use of the OQSL in such representation, we have
shown that the saturation of the dispersion bound implies the saturation of
the OQSL for the Krylov complexity operator. The application of OQSL to other
complexity measures, such as the family of q-complexities including out-of-
time-order correlators [54], offers an interesting prospect.

Beyond these examples, we expect OQSLs to find manifold applications in the
characterization of nonequilibrium phenomena, such as the crossing of a
quantum phase transition, the equilibration and thermalization of isolated
many-body systems, quantum thermodynamic processes, quantum control, and
quantum annealing. In addition, OQSL may be used in the study of integrable
systems, using the zero-curvature representation [87], Lax pairs [88], and
Hamiltonian deformations [89, 90, 91]. The generalization of our results to
dissipative quantum systems would be highly desirable, given its prospective
applications, e.g., to the description of open quantum dynamics in
Heisenberg’s representation and the quest for fundamental limits to nonunitary
operator growth [92, 93].

##  7 Acknowledgements

It is a pleasure to acknowledge discussions with Léonce Dupays, Íñigo L.
Egusquiza and Federico Roccati. KT acknowledges support by JSPS KAKENHI grant
No. JP20K03781 and No. JP20H01827.

## Appendices

###  A Proving bijection between positive semi-definite inner-products and
positive semi-definite operators

######  Lemma 1.

Let $\mathcal{P}\in\textrm{End}(\mathcal{B})$ be a positive semi-definite
superoperator. The binary operation
$\langle\cdot,\mathcal{P}\cdot\rangle_{\textsc{h}}:\mathcal{B}\times\mathcal{B}\rightarrow\mathds{R}$
is a positive semi-definite inner product on $\mathcal{B}$.

###### Proof.

We need to show that the map
$\langle\cdot,\mathcal{P}\cdot\rangle_{\textsc{h}}$ satisfies linearity,
Hermitian symmetry, and positive semi-definiteness.

Linearity

Consider any triplet of operators $A$, $B$ and $C$ and a complex number
$\lambda$. We then have

$\displaystyle\langle A,\mathcal{P}\lambda B\rangle_{\textsc{h}}=\langle
A,\lambda\mathcal{P}B\rangle_{\textsc{h}}=\lambda\langle
A,\mathcal{P}B\rangle_{\textsc{h}}$ (77) $\displaystyle\langle
A,\mathcal{P}(B+C)\rangle_{\textsc{h}}=\langle
A,\mathcal{P}B+\mathcal{P}C\rangle_{\textsc{h}}=\langle
A,\mathcal{P}B\rangle_{\textsc{h}}+\langle
A,\mathcal{P}C\rangle_{\textsc{h}},$ (78)

where we have used the linearity of the Hilbert-Schmidt inner product and the
superoperator.

Hermitian symmetry

For any pair of operators $A$ and $B$ we have

$\langle
A,\mathcal{P}B\rangle_{\textsc{h}}=\langle\mathcal{P}B,A\rangle_{\textsc{h}}^{*}=\langle
B,\mathcal{P}^{\dagger}(A)\rangle_{\textsc{h}}^{*}=\langle
B,\mathcal{P}A\rangle_{\textsc{h}}^{*},$ (79)

where we have used the Hermitian symmetry property of the Hilbert-Schmidt
inner product and the fact that a positive semi-definite superoperator is
self-adjoint.

Positive semi-definiteness

From the definition of positive semi-definiteness of a superoperator it
follows directly that

$\langle A,\mathcal{P}A\rangle_{\textsc{h}}\geq 0,$ (80)

for any operator $A$. ∎

######  Proposition 2.

Given the Hilbert space
$(\mathcal{B},\langle\cdot,\cdot\rangle_{\textsc{h}})$, the map
$\mathcal{P}\mapsto\langle\cdot,\mathcal{P}\cdot\rangle_{\textsc{h}}$ is a
bijection between the set of positive semi-definite operators on
$(\mathcal{B},\langle\cdot,\cdot\rangle_{\textsc{h}})$ and the set of positive
semi-definite inner products on $\mathcal{B}$.

###### Proof.

We can prove that the map is a bijection if we can prove that it is injective
and surjective.

Injectivity

Suppose that
$\langle\cdot,\mathcal{P}\cdot\rangle_{\textsc{h}}=\langle\cdot,\eta(\cdot)\rangle_{\textsc{h}}$
for some pair of superoperators $\mathcal{P}$ and $\eta$. It follows that

$\displaystyle\begin{split}\langle\cdot,\mathcal{P}\cdot\rangle_{\textsc{h}}=\langle\cdot,\eta(\cdot)\rangle_{\textsc{h}}&\iff\langle
A,\mathcal{P}B\rangle_{\textsc{h}}=\langle
A,\eta(B)\rangle_{\textsc{h}}\quad\forall A,B\in\mathcal{B}\\\
&\iff\mathcal{P}=\eta.\end{split}$ (81)

Surjectivity

Given any positive semi-definite inner product $(\cdot|\cdot)$, we want to
construct a positive semi-definite superoperator $\mathcal{P}$ such that
$(\cdot|\cdot)=\langle\cdot,\mathcal{P}\cdot\rangle_{\textsc{h}}$. Let
$M_{1},M_{2}\dots M_{n^{2}}$ be an operator basis in $\mathcal{B}$ and $a_{k}$
and $b_{k}$ be the corresponding components of $A$ and $B$. Define $\langle
M_{i},\mathcal{P}M_{j}\rangle_{\textsc{h}}=(M_{i}|M_{j})$. This superoperator
is positive semi-definite since

$\displaystyle\begin{split}\langle
A,\mathcal{P}B\rangle_{\textrm{H}}&=\sum_{i=1}^{n^{2}}\sum_{j=1}^{n^{2}}a^{*}_{i}b_{j}\langle
M_{i},\mathcal{P}M_{j}\rangle_{\textrm{H}}=\sum_{i=1}^{n^{2}}\sum_{j=1}^{n^{2}}a^{*}_{i}b_{j}(M_{i}|M_{j})\\\
&=(A|B)\quad\implies\quad\langle A,\mathcal{P}A\rangle_{\textrm{H}}=(A|A)\geq
0,\end{split}$ (82)

from which it is also clear that $\mathcal{P}$ maps to $(\cdot|\cdot)$. ∎

###  B The kernel of seminorms

######  Proposition 3.

Let $\norm{\cdot}$ be the seminorm induced by the positive semi-definite inner
product $\langle\cdot,\mathcal{P}\cdot\rangle_{\textsc{h}}$. It is then the
case that $\norm{A}=0\iff\mathcal{P}A=0$.

###### Proof.

Let $\\{M_{k}\\}_{k=0}^{n^{2}}$ be an orthonormal eigenbasis of $\mathcal{P}$
such that $\lambda_{k}$ is the eigenvalue corresponding to the eigenvector
$M_{k}$. Let $a_{k}$ be the components of an operator $A$. We then have

$\displaystyle\begin{split}\langle
A,\mathcal{P}A\rangle_{\textrm{H}}&=\sum_{i=1}^{n^{2}}\sum_{j=1}^{n^{2}}a^{*}_{i}a_{j}\langle
M_{i},\mathcal{P}M_{j}\rangle_{\textrm{H}}=\sum_{i=1}^{n^{2}}\sum_{j=1}^{n^{2}}a^{*}_{i}a_{j}\lambda_{j}\delta_{ij}\\\
&=\sum_{k=1}^{n^{2}}\absolutevalue{a_{k}}^{2}\lambda_{k}.\end{split}$ (83)

Since $\lambda_{k}\geq 0$, this sum can only be zero if $a_{k}=0$ for
$\lambda_{k}>0$, in other words, $A$ lies in the kernel of $\mathcal{P}$. ∎

###  C Proof of equation 4

###### Proof.

Define $\tilde{A}=A-\hat{A}$. We can choose an orthogonal basis
$M_{1},M_{2},\dots M_{d}$, $N_{1},N_{2},\dots N_{n-d}$, such that the
operators $M_{k}$ and $N_{k}$ span $\textrm{im}(\mathcal{P})$ and
$\textrm{ker}(\mathcal{P})$ respectively and $d$ is the dimension of
$\textrm{im}(\mathcal{P})$. Let $a_{k}$ be the components of $A$ with respect
to this basis. We have that

$\displaystyle\begin{split}\mathcal{P}\tilde{A}=\mathcal{P}(A-\hat{A})&=\mathcal{P}A-\mathcal{P}\hat{A}\\\
&=\sum_{k=1}^{d}a_{k}\mathcal{P}M_{k}+\sum_{k=1}^{n-d}a_{k}\mathcal{P}N_{k}-\sum_{k=1}^{d}a_{k}\mathcal{P}M_{k}=0.\end{split}$
(84)

Using this together with linearity of $(\cdot|\cdot)$, we get

$(A|B)=(\hat{A}|\hat{B})+(\hat{A}|\tilde{B})+(\tilde{A}|\hat{B})+(\tilde{A}|\tilde{B})=(\hat{A}|\hat{B})=\innerproduct*{\hat{A}}{\hat{B}},$
(85)

where the step to the second equality follows from proposition 3 and the last
step follows from the definition of $\langle\cdot,\cdot\rangle$. ∎

###  D Smallest subspace containing the dynamics

Let $V$ be a real or complex finite dimensional vector space and let $L$ be a
linear endomorphism on $V$. Assuming that $L$ is diagonalizable, we can write
$L=\sum_{i=1}^{d}l_{i}P_{i}$, where $l_{i}$ are the $d$ distinct eigenvalues
of $L$ and $P_{i}$ are the projections onto the corresponding eigenspaces
satisfying $\sum_{i=1}^{d}P_{i}=I$ and $P_{i}P_{j}=\delta_{ij}P_{i}$. Here $I$
is the identity map on $V$. It follows from the definition of the exponential
function that $e^{Lt}=\sum_{i=1}^{d}e^{l_{i}t}P_{i}$. Consider now any initial
vector $v$ in $V$ evolving according to $v(t)=e^{Lt}v$. Let us define the
subspace $W=\textrm{span}\\{v_{i}\\}_{i\in I}$ where $i\in I\iff
v_{i}:=P_{i}v\neq 0$. We then have that $v(t)=\sum_{i\in I}e^{l_{i}t}v_{i}$
and we see that the evolution is entirely contained in $W$. Given a proper
time interval $T\subset\mathbb{R}$, we now ask whether $W$ is the smallest
subspace for which {$v(t)$ : $t\in T$} is contained in.999A proper interval is
an interval in $\mathbb{R}$ excluding the empty set and singletons. The answer
is affirmative. To show this, first, note that the functions $e^{l_{i}t}$ with
domain $T$ are linearly independent given that all eigenvalues $l_{i}$ are
distinct. This implies that

$\sum_{i\in I}c_{i}e^{l_{i}t}=0\textrm{ }\forall t\in T\iff c_{i}=0\textrm{
}\forall i\in I,$ (86)

where $c_{i}\in\mathbb{C}$. We will use proof by contradiction to show that
$W$ must be the smallest subspace. Assume that there exists a subspace
$\tilde{W}$ containing the evolution with a dimension strictly smaller than
$W$. We must then have the evolution contained in the subspace given by the
intersection $F=W\cap\tilde{W}$. By our assumption, $F$ must then have a
dimension strictly smaller than $W$. This implies that there exists a non-zero
linear functional $w$ with domain $W$ for which $F\subset\textrm{ker}(w)$. We
can expand this functional in the basis $\\{f_{i}\\}_{i\in I}$ defined by
$f_{i}(v_{j})=\delta_{ij}$ so that $w=\sum_{i\in I}w_{i}f_{i}$, where
$w_{i}=w(v_{i})$. We get that $w\big{(}v(t)\big{)}=0\textrm{ }\forall t\in
T\iff\sum_{i\in I}w_{i}e^{l_{i}t}=0\textrm{ }\forall t\in T$. This last
expression together with (86) implies that $w_{i}=0$ $\forall i\in I\iff w=0$.
Thus, we have reached a contradiction; hence, $W$ is the smallest subspace
containing {$v(t)$ : $t\in T$}.

The proof can be carried out analogously for the case when $L$ is time
dependent but commutes, i.e., $[L(t_{1}),L(t_{2})]=0$ $\forall t_{1},t_{2}\in
T$.

###  E Optimal refinement

Consider the decomposition $\hat{A}=S+V_{t}$ discussed in section 2.3 and
assume that the subspace $\ker(\mathds{L})\cap\mathcal{H}_{\mathcal{P}}$ does
not change over time. Suppose $V_{t}$ has a non-zero projection $S^{\prime}$
onto $\ker(\mathds{L})\cap\mathcal{H}_{\mathcal{P}}$ then $S^{\prime}$ is
guaranteed to remain unchanged since we have by the assumption that
$\ker(\mathds{L})\cap\mathcal{H}_{\mathcal{P}}$ is time-independent. Let
$V_{t}^{\prime}$ be the orthogonal complement so that
$V=S^{\prime}+V_{t}^{\prime}$. It follows that

$\langle
V,V_{t}\rangle=(V|V_{t})=(S^{\prime}+V^{\prime}|S^{\prime}+V^{\prime}_{t})=(V^{\prime}|V^{\prime}_{t})+\norm{S^{\prime}}^{2}.$
(87)

This implies that

$\real(V^{\prime}|V^{\prime}_{t})=\real(V|V_{t})-\norm{S^{\prime}}^{2}=\real
C(t)-\norm{S}^{2}-\norm{S^{\prime}}^{2}=\real C(t)-\norm{S+S^{\prime}}^{2},$
(88)

where the last equality follows from the assumption $\real\langle
S,V_{t}\rangle=0$. We can thus improve the speed limit further whenever
$S^{\prime}\neq 0$ since we would then have that
$\norm{S+S^{\prime}}>\norm{S}+\norm{S^{\prime}}>\norm{S}$. The operator
$P_{0}=S+S^{\prime}$ is precisely the orthogonal projection of $\hat{A}_{t}$
onto $\ker(\mathds{L})\cap\mathcal{H}_{\mathcal{P}}$.

###  F Proving orthogonality from the preservation of norm

We here want to show that the operators $P_{0}$, $X_{\omega}$ and $Y_{\omega}$
in (11) are orthogonal and that $X_{\omega}$ and $Y_{\omega}$ have the same
norm. Using (11), we get that

$\displaystyle\norm{A_{t}}^{2}$
$\displaystyle=\norm{P_{0}}^{2}+\norm{P_{\omega}}^{2}+\norm{P_{-\omega}}^{2}+2\cos{\theta(t)}\real\langle
P_{0},X_{\omega}\rangle+2\cos(2\theta(t))\real\langle
P_{\omega},P_{-\omega}\rangle$ (89)
$\displaystyle=\norm{P_{0}}^{2}+\norm{P_{\omega}}^{2}+\norm{P_{-\omega}}^{2}+2\sin{\theta(t)}\real\langle
P_{0},Y_{\omega}\rangle+2\cos(2\theta(t))\real\langle
P_{\omega},P_{-\omega}\rangle.$ (90)

At time $t=0$, we have that $\theta(0)=0$ and we get from expression (89) and
(90) that $\real\langle P_{0},X_{\omega}\rangle=0$. Assuming that $\theta(t)$
is not zero over the whole interval $[0,\tau]$, we can conclude from
$\real\langle P_{0},X_{\omega}\rangle=0$, (89) and (90) that $\real\langle
P_{0},Y_{\omega}\rangle=0$ must also hold. This in turn implies that
$\real\langle P_{\omega},P_{-\omega}\rangle=0$ since $\norm{A_{t}}$ must be
constant. This last equality guarantees that the norm of $X_{\omega}$ and
$Y_{\omega}$ are equal. Note that in the case when $\theta(t)=0$ over the
whole interval, we have that the dynamics is stationary.

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