# Geometry of Complexity in Conformal Field Theory

Mario Flory<EMAIL_ADDRESS>Institute of Physics, Jagiellonian
University, 30-348 Kraków, Poland  Michal P. Heller
<EMAIL_ADDRESS>On leave of absence from: _National Centre for
Nuclear Research, 02-093 Warsaw, Poland_ Max Planck Institute for
Gravitational Physics (Albert Einstein Institute), 14476 Potsdam-Golm, Germany

###### Abstract

We initiate quantitative studies of complexity in (1+1)-dimensional conformal
field theories with a view that they provide the simplest setting to find a
gravity dual to complexity. Our work pursues a geometric understanding of
complexity of conformal transformations and embeds Fubini-Study state
complexity and direct counting of stress tensor insertion in the relevant
circuits in a unified mathematical language. In the former case, we
iteratively solve the emerging integro-differential equation for sample
optimal circuits and discuss the sectional curvature of the underlying
geometry. In the latter case, we recognize that optimal circuits are governed
by Euler-Arnold type equations and discuss relevant results for three well-
known equations of this type in the context of complexity.

##  I Introduction

One of the most perplexing recent results in quantum gravity are holographic
complexity proposals. They are convincing albeit speculative relations between
volumes [1, 2, 3] or actions [4, 5] in anti-de Sitter spacetimes [6, 7, 8] and
hardness of preparing states in dual quantum field theories (QFTs).

Holographic complexity proposals not only call for a better understanding and
explicit derivation, but also motivate searching for other manifestations of
complexity in gravity. However, advancing the broadly defined field of
holographic complexity requires expanding our knowledge of complexity in QFT.
The aim of the present work is to initiate _quantitative_ studies of physical
notions of complexity in what arguably is the most universal setting of all –
conformal field theories in 1+1 spacetime dimensions (CFTs1+1) and local
conformal transformations generated by the energy-momentum tensor.

Complexity in its native quantum computing setting concerns hardness of
approximating a given unitary using circuits composed from discrete gates
acting only on a limited number of qubits (circuit complexity) or
approximating a desired quantum state using such circuits acting on a simple
state (state complexity).

Motivated by holographic complexity proposals, [9, 10] initiated studies of
complexity in QFTs. These works view the preparation of a unitary operator $U$
or, upon acting on a reference state $|\mathcal{R}\rangle$, also (target)
state preparation $|\mathcal{T}\rangle=U|\mathcal{R}\rangle$, in a continuous
way as a path-ordered exponential

$U(\tau)=\overleftarrow{{\cal P}}e^{-i\int_{0}^{\tau}Q(\gamma)d\gamma}$ (1)

with $U(\tau=1)$ being equal to some desired unitary $U$. In this equation,
the Hermitian operator $Q(\tau)d\tau$ is a single layer of the circuit
parametrized by the parameter $\tau$ that constructs $U$. The idea used in [9,
10] to _define_ complexity in a QFT appeared earlier in [11, 12, 13] as a way
of bounding complexity of discrete circuits acting on qubits. The relevant
definition assigns a cost to $Q(\tau)$ which reflects the decomposition of
$Q(\tau)$ into more elementary building blocks (_gates_), each with a
specified cost, and minimizes the sum of the contributions from all layers of
the circuit subject to appropriate boundary conditions. Most of the studies to
date were concerned with free QFTs and claimed optimality of circuits with
$Q(\tau)$ being at most quadratic in underlying bosonic or fermionic
operators. Such studies, despite their simplicity, could be fine-tuned to
reproduce predictions of holographic complexity proposals [9, 10, 14, 15, 16,
17, 18].

In our eyes, studies of complexity in QFT are similar to the development of
entanglement entropy in the same context. The latter also arose in connection
with black hole physics [19, 20, 21] and later became an independent subject
with a strong quantum gravity component. One of the most fruitful seeds of
progress in this discipline originated from the studies of entanglement
entropy in the setting of CFTs1+1. This includes in particular the universal
result for single interval entanglement entropy in the vacuum [21, 22].
Drawing a parallel from the field of entanglement entropy, which includes also
the matching of the results of [21, 22] by the holographic entanglement
entropy [23], CFTs1+1 should indeed be viewed as an ideal setting for
accelerating our understanding of complexity in QFTs and holography.

This vision is shared by [24, 25, 26, 27, 28, 29], and the challenges
encountered in these works directly motivate our approach. These include badly
posed variational problems, assigning cost predominantly to trivial $U(1)$
factors in (1), or finding optimal circuits only to leading order for
infinitesimal transformations. In the present work, we overcome these
challenges and present a unified, geometric and _quantitative_ view on
complexity in CFTs1+1.

The universality of CFTs1+1 stems from their stress-energy tensor generating
the Virasoro algebra. Adopting the setting of [25, 29], we will be concerned
with unitary circuits obtained from the exponentiation of the stress-energy
tensor but employ a different way of assigning a cost to the involved
operations. We focus on a CFT1+1 defined on a Lorentzian cylinder, whose
circle has a unit radius and is parametrized by the coordinate $\sigma$. We
will also restrict ourselves to one copy of the Virasoro algebra. This means
we will assign complexities to unitary circuits of the form (1) on a
representation of the group of diffeomorphisms on the circle, where an
operator $U(\tau)$ corresponds to a group element $f(\tau,\sigma)$ that maps
the circle to itself. The function $f(\tau,\sigma)$ hence represents a
sequence of diffeomorphisms of $\sigma$ interpolating between the identity
$f(0,\sigma)=\sigma$ and a desired one $f(1,\sigma)\equiv f(\sigma)$. Note
that we will ignore terms stemming from the central extension of the group, as
these would lead to additional complex phase-factors in (1), which are
generally considered to be irrelevant for physical notions of complexity (a
problem faced in [25, 29]). An infinitesimal layer of the circuit is generated
by

$Q(\tau)=\int_{0}^{2\pi}\frac{d\sigma}{2\pi}\,T(\sigma)\epsilon(\tau,\sigma)\,,$
(2)

where $T(\sigma)$ is the right-moving component of the stress-energy tensor
operator and $\epsilon(\tau,\sigma)$ is an element of the Lie-algebra defined
via

$\displaystyle\epsilon(\tau,f(\tau,\sigma))=\dot{f}(\tau,\sigma).$ (3)

In this letter we discuss two viable instances of cost functions. The first
emerges by taking an energy eigenstate $|h\rangle$ and evaluating the Hilbert
space distance traversed by the circuit defined by (1) and (2). This is the
Fubini-Study complexity defined in [9]. The second instance realizes the
approach of [10] and arises from treating $T(\sigma)$ as a one-parameter set
of elementary contributions to each circuit layer and minimizing the
$L_{2}$-norms of $\epsilon(\tau,\sigma)$, $\epsilon^{\prime}(\tau,\sigma)$, or
a combination of both, averaged over all circuit layers.

Our approach is the first study of complexity in a generic (including large
central charge $c$) CFT1+1 that 1) does not assign cost to trivial factors
(see [29]), 2) has a well-posed variational problem for determining a
transformation between two arbitrary unitaries generated by the insertions of
$T(\sigma)$, 3) sheds light on the underlying geometry of circuits by probing
its sectional curvatures and 4) selects the physically desirable notions of
complexity based on the properties of optimal circuits. While we intend to
present our main results in a concise and self-contained manner, some further
discussions can be found in our companion work [30].

##  II Cost functions and complexity

The Fubini-Study metric a.k.a. fidelity susceptibility arises from considering
an overlap between two nearby states in the Hilbert space [31]. It is
attractive from the point of view of holography and the largely open problem
of physical interpretation of holographic complexity proposals, since it is
known how the overlap between at least certain states in holographic QFTs
manifest itself on the gravity side [26]. For a family of states
$|\psi(\tau)\rangle$ parametrized by $\tau$, we can define

$\displaystyle|\langle\psi(\tau)|\psi(\tau+d\tau)\rangle|\approx
1-G_{\tau\tau}(\tau)d\tau^{2}+\mathcal{O}(d\tau^{3})$ (4)

where $G_{\tau\tau}\geq 0$ is the Fubini-Study-metric. Assume
$|\psi(\tau)\rangle$ is a path on the space of states parametrized by unitary
operators acting on an initial state $\left|h\right\rangle$,

$\displaystyle|\psi(\tau)\rangle\equiv U(\tau)\left|h\right\rangle.$ (5)

The Fubini-Study metric $G_{\tau\tau}$ becomes then the variance of $Q(\tau)$
evaluated in the state $|\psi(\tau)\rangle$ [25] or, introducing

$\displaystyle\tilde{Q}(\tau)=U(\tau)^{\dagger}Q(\tau)U(\tau),$ (6)

equivalently the variance of $\tilde{Q}(\tau)$ evaluated in the state
$\left|h\right\rangle$: $\left\langle
h\right|\tilde{Q}^{2}(\tau)\left|h\right\rangle=\left\langle\psi(\tau)|Q^{2}(\tau)|\psi(\tau)\right\rangle$
by definition, and similarly for the one-point function. Note that the
applications of the operators $U(\tau)$ in (6), using (2), essentially causes
a conformal transformation of the stress-energy tensor. Using the well-known
transformation law and ignoring the Schwarzian term leading to an irrelevant
for us phase factor, we can write [25]

$\tilde{Q}(\tau)=\int_{0}^{2\pi}\frac{d\sigma}{2\pi}\,T(\sigma)\frac{\dot{f}(\tau,\sigma)}{f^{\prime}(\tau,\sigma)}.$
(7)

Each trajectory through state space $|\psi(\tau)\rangle$ parametrized by
$\tau\in[0,1]$ can now be assigned the total cost $\mathrm{L}_{\mathrm{FS}}$

$\displaystyle\mathrm{L}_{\mathrm{FS}}=\int_{0}^{1}d\tau\sqrt{G_{\tau\tau}(\tau)}$
(8)

and complexity arises as its minimum subject to the appropriate boundary
conditions [9]. We should note here that the present discussion is completely
general and concerns complexity of state
$|\mathcal{T}\rangle=U\left|h\right\rangle$ given a reference state
$|\mathcal{R}\rangle=\left|h\right\rangle$. Alternatively, one can view it as
a definition of circuit complexity associated with a circuit representation of
a unitary $U$ in which one decomposes $\tilde{Q}(\tau)$ into elementary
transformations. This is similar to the notion of circuit complexity explored
in [28].

One can alternatively define an a priori inequivalent notion of complexity
based on a variance of $Q(\tau)$ in the state $\left|h\right\rangle$ (instead
of $\left|\psi(\tau)\right\rangle$ as so far), which would be a more faithful
realization of the approach [10]. What we mean by that is that the cost of one
layer in the Fubini-Study metric depends not only on $\epsilon(\tau,\sigma)$
from a given layer, but also on what all previous layers do through the two-
point function of $T$ in the _evolved_ state $|\psi(\tau)\rangle$. On the
other hand, in the approach of [10] the cost of each layer depends only on
$\epsilon(\tau,\sigma)$ as the insertions of $T$ would be assigned the same
weight at each layer.

The discussion so far was completely general, but now we specialize to
circuits defined by (7). Our choice for $\left|h\right\rangle$, as in [25,
29], is that of an energy eigenstate in the CFT1+1 corresponding, via the
operator-state correspondence, to a (quasi-)primary of the chiral dimension
$h$. To evaluate (8) explicitly, we follow [25, 29] and write the variance of
$\tilde{Q}$ as a bi-local integral

$\displaystyle\mathrm{L}=\int_{0}^{1}\frac{d\tau}{2\pi}\sqrt{\iint_{0}^{2\pi}d\sigma
d\kappa\frac{\dot{f}(\tau,\sigma)}{f^{\prime}(\tau,\sigma)}\frac{\dot{f}(\tau,\kappa)}{f^{\prime}(\tau,\kappa)}\Pi(\sigma-\kappa)},$
(9)

where $\Pi$ corresponds to a connected correlator of the stress-energy tensor
in the state $\left|h\right\rangle$ [32]

$\displaystyle\Pi(\sigma-\kappa)=\left\langle
h\middle|T(\sigma)T(\kappa)\middle|h\right\rangle-\left\langle
h\middle|T(\sigma)\middle|h\right\rangle\left\langle
h\middle|T(\kappa)\middle|h\right\rangle$
$\displaystyle=\frac{c}{32\,\sin^{4}{[(\sigma-\kappa)/2]}}-\frac{h}{2\,\sin^{2}{[(\sigma-\kappa)/2]}}.$
(10)

As usual when studying geodesic motion, by requiring affine parametriztion we
can move from a length-functional (8) to an energy-functional where,
essentially, the square-root in (8) and (9) is removed. To do so, we note that
(9) clearly corresponds to a geodesic problem in infinite dimensions, where
summation over indices has been replaced by integration over variables
$\sigma,\kappa$, $f(\tau,\sigma)$ has taken over the role of the coordinate
$X^{\sigma}(\tau)$, and the expression
$\frac{\Pi(\sigma-\kappa)}{f^{\prime}(\tau,\sigma)f^{\prime}(\tau,\kappa)}$
plays the role of the metric $g_{\sigma\kappa}(X(\tau))$. The partial integro-
differential equation (IDE) of motion for the circuit $f(\tau,\sigma)$
extremizing (9) then reads

$\displaystyle\int_{0}^{2\pi}d\sigma\Bigg{(}-\Pi(\sigma-\kappa)\frac{d}{d\tau}\left(\frac{\dot{f}(\tau,\sigma)}{f^{\prime}(\tau,\sigma)f^{\prime}(\tau,\kappa)}\right)+\frac{\dot{f}(\tau,\sigma)}{f^{\prime}(\tau,\sigma)}\partial_{\kappa}\left(\Pi(\sigma-\kappa)\frac{\dot{f}(\tau,\kappa)}{f^{\prime}(\tau,\kappa)^{2}}\right)\Bigg{)}\equiv
0.$ (11)

As expected, this equation is of second order in derivatives with respect to
$\tau$. This is adequate for a boundary value problem in which we envision
being given an initial and final condition, $f(0,\sigma)$ and $f(1,\sigma)$,
and finding the shortest circuit $f(\tau,\sigma)$ connecting these two maps.
This is a notable contrast to the geometric actions studied in [25, 29], which
lead to equations first order in $\tau$, in which generally only one initial
value $f(0,\sigma)$ needs to be provided to fix a solution.

It is the most natural for us to equate the integration kernel $\Pi$ with the
connected stress-energy two-point function in the state $\left|h\right\rangle$
(10), however we have kept our discussion more generic for a reason. Broadly
speaking, our goal in this paper is to define geodesic motion on the Virasoro-
group, and this has of course already been accomplished in the framework of
Euler-Arnold-type partial differential equations (PDEs) such as the Korteweg-
de Vries (KdV), Camassa-Holm (CH), or Hunter-Saxton (HS) equations [33]. They
were already discussed in the context of QFT complexity in [25, 29], see also
[34]. In order to show how our functional (9) and equation (11) fit into this
framework, we note that according to (9), the distance between the identity
map $f=\sigma$ and a map $f=f_{1}(\sigma)$ is identical to the distance
between $f=\sigma$ and the inverse map $F_{1}(\sigma)=f_{1}^{-1}(\sigma)$.
This is easy to show by using invariance under a change of affine parameter
$\tau\rightarrow s=1-\tau$ and invariance under conformal transformations
$f(\tau,\sigma)\rightarrow G(f(\tau,\sigma))$. Hence, replacing the circuit
$f(\tau,\sigma)$ by the inverse circuit $F(\tau,\sigma)$ in (9) and using the
identity [25]

$\displaystyle\epsilon(\tau,\sigma)=-\frac{\dot{F}(\tau,\sigma)}{F^{\prime}(\tau,\sigma)},$
(12)

we can write the inner product in (9) entirely in terms of the kernel $\Pi$
and the Lie-algebra-element $\epsilon$. Note that if $A=B\cdot C$, then
$A^{-1}=C^{-1}\cdot B^{-1}$, hence replacing $f(\tau,\sigma)$ by
$F(\tau,\sigma)$ in (9) corresponds to switching from a left- to a right-
invariant metric on the Lie-group. This would yield the alternative complexity
definition based on the variance of $Q$ (not $\tilde{Q}$) in the state
$\left|h\right\rangle$ discussed below (8).

Now, while our derivation above would suggest $\Pi$ to be the stress-energy
two-point function in the state $\left|h\right\rangle$ [32], another choice of

$\displaystyle\Pi(\sigma-\kappa)=a\,\delta(\sigma-\kappa)+b\,\delta^{\prime\prime}(\sigma-\kappa),$
(13)

with Dirac’s delta function $\delta(\sigma-\kappa)$, would allow us to obtain
the CH equation ($a=b=1$), the HS equation ($a=0,b=1$), and the KdV equation
($a=1,b=0$), ignoring as before the term coming from the central extension.
Through the lens of [10], (13) can be seen as assigning directly a spatially
uniform cost to individual insertions of $T(\sigma)$ via $Q(\tau)$ defined in
(2). In a sense, our complexity functional (9) corresponds to a generalization
of the inner products that led to these well studied PDEs. Likewise, while
these PDEs provide valid choices for a definition of complexity for the
Virasoro group, they may be also regarded as simpler models for the physics
encoded in the optimization problem behind the Fubini-Study complexity defined
by (11) with (10).

In the Fubini-Study case, in order to assign a well-defined finite value to
(9) despite the pole of $\Pi$, we can make use of the method of differential
regularisation [35, 36]. This means we will (implicitly) phrase the divergent
terms of the two-point-function in (9) as derivatives of more mildly-divergent
terms and then carry the derivatives over onto the test-function
$\frac{\dot{f}(\tau,\sigma)}{f^{\prime}(\tau,\sigma)}\frac{\dot{f}(\tau,\kappa)}{f^{\prime}(\tau,\kappa)}$
via integration by parts. The immediate physical consequence of this is that
the metric will be degenerate. For example, if
$\frac{\dot{f}(\tau,\sigma)}{f^{\prime}(\tau,\sigma)}=\mathrm{const}$, the
integrals in (9) will vanish when applying derivatives to this term due to
differential regularisation. Although the cause of some technical problems,
this degeneracy of the metric is a desirable feature as it makes sure that
transformations which only change the reference state by a complex phase will
be assigned zero cost in terms of complexity. For this reason, we believe that
the HS equation, $a=0$ in (13), will be a more realistic model of CFT-
complexity than the KdV equation studied in [25, 29]. We discuss these issues
in more detail in [30].

##  III Optimal circuits for Fubini-Study

The exact solutions and properties of the KdV, CH and HS equations are well
studied [33], and hence we will now focus again on the concrete equations of
motion (11) with $\Pi$ given by (10). For this, apart from trivial circuits
such as $f(\tau,\sigma)=f(\tau+\sigma)$, we do not know exact solutions.
However, for boundary conditions of the form $f(0,\sigma)=\sigma$,
$f(1,\sigma)=\sigma+\frac{\varepsilon}{m}\sin(m\sigma)$ with $m\in\mathbb{N}$,
$\varepsilon\ll 1$, we iteratively construct a circuit $f(\tau,\sigma)$
satisfying (11) order by order in $\varepsilon$. For example, for $m=1$ this
yields

$\displaystyle f(\tau,\sigma)=$
$\displaystyle\sigma+\varepsilon\tau\sin(\sigma)$ (14)
$\displaystyle+\varepsilon^{2}\frac{c\tau^{2}-c\tau+20h\tau^{2}-20h\tau}{4(c+8h)}\sin(2\sigma)+...$

and the square of the on-shell Lagrangian in (9) is

$\displaystyle\iint d\sigma d\kappa\ \Pi(\sigma-\kappa)\
\frac{\dot{f}(\tau,\sigma)}{f^{\prime}(\tau,\sigma)}\frac{\dot{f}(\tau,\kappa)}{f^{\prime}(\tau,\kappa)}$
(15)
$\displaystyle=2\pi^{2}\,\mathit{h}\,\varepsilon^{2}+\frac{\pi^{2}\,\left(3\,\mathit{c}^{2}+56\,\mathit{c}\,\mathit{h}+112\,\mathit{h}^{2}\right)}{96\,(\mathit{c}+8\,\mathit{h})}\varepsilon^{4}+...$

We comment more on this result and how it was obtained in the appendix. Figure
1 shows a graphical representation of the circuit (14).

Figure 1: A graphical representation of the circuit (14), where for simplicity
we have set $c=0,h=1$ and $\epsilon=1$ in order to generate a result that is
visible to the eye. For each fixed value of $\tau$, $f(\tau,\sigma)$ is a map
of the circle to itself, depicted by the colored points unevenly spread out
along the circles according to the function $f(\tau,\sigma)$ with evenly
spaced input values for $\sigma$. The radial ordering (also emphasized by
colour) corresponds to the progression of the parameter $\tau$ from $-1$ to
$1$ in steps of $\Delta\tau=1/4$. The solid circles mark $\tau=-1,0,1$,
respectively. At $\tau=0$, $f(\tau,\sigma)$ is the identity map by
construction. ”Wave breaking” happens when infinitesimally close points are
mapped to the same location because $f^{\prime}(\tau,\sigma)=0$.

##  IV Geometry of complexity

In order to gain a qualitative understanding of the geometry of the Fubini-
Study complexity and compare with other notions, we will instead proceed to
calculate the sectional curvatures $K(u,v)$ at the identity map
$f(\sigma)=\sigma$ for tangent-vectors of the form $u=sin(m\sigma)$,
$v=\sin(n\sigma)$, $m,n\in\mathbb{N}$ and $m\neq n$. Following the analogy
with finite dimensional geodesic motion, first and second derivatives of the
metric can then be defined as functional derivatives, e.g. $\frac{\partial
g_{\sigma\kappa}}{\partial x^{\eta}}\rightarrow\frac{\delta}{\delta
f(\eta)}\frac{\Pi(\sigma-\kappa)}{f^{\prime}(\sigma)f^{\prime}(\kappa)}$
$=-\Pi(\sigma-\kappa)\left(\frac{\delta^{\prime}(\sigma-\eta)}{f^{\prime}(\sigma)^{2}f^{\prime}(\kappa)}+\frac{\delta^{\prime}(\kappa-\eta)}{f^{\prime}(\sigma)f^{\prime}(\kappa)^{2}}\right)$.
In order to calculate the sectional curvatures,

$\displaystyle
K(u,v)=\frac{R_{\sigma\kappa\eta\omega}u^{\sigma}u^{\eta}v^{\kappa}v^{\omega}}{(u\cdot
u)(v\cdot v)-(u\cdot v)^{2}},$ (16)

we would still need an analogue of the inverse metric which is needed in the
definition of the Riemann-tensor $R_{\sigma\kappa\eta\omega}$. Note that in
(16) the scalar product is taken using the metric $g_{\sigma\kappa}$ and we
assume Einstein’s summation formula involving integration. The inverse metric
in question does not exist because our metric is degenerate. However, for
circuits of the type (14) which satisfy the condition $\int
d\sigma\frac{\dot{f}(\tau,\sigma)}{f^{\prime}(\tau,\sigma)}=0$ for all $\tau$,
it is possible to show that the equation of motion (11) is left invariant when
adding a non-zero constant to the two-point function $\Pi$. This creates a
metric which is invertible and yet has identical (i.e. independent of the
added constant) sectional curvatures to the original metric in the tangent-
planes spanned by $u=sin(m\sigma),v=\sin(n\sigma)$. We find

$\displaystyle
K(u,v)=\frac{3}{\pi^{2}(m+n)}\left(\frac{(2m+n)(m+2n)}{24h+c(m+n-1)(m+n+1)}-\frac{(2m-n)(m+n)^{2}}{m\left(24h+c\left(m^{2}-1\right)\right)}\right)\text{
for $m>n$.}$ (17)

For $0\leq h/c<\frac{4}{13}$, all $K(u,v)$ in (17) are negative, for
$h/c\rightarrow+\infty$, all $K(u,v)$ are positive, and for generic $h>0,c>0$,
only a finite number of $K(u,v)$ for small $m,n$ will be positive, while the
infinitely many remaining ones will be negative. Due to a theorem from [37],
we know that there should always be some tangent-planes in which $K>0$, but
the result above suggests that the generic curvature felt by the geodesic
curves we are investigating will be negative unless $h/c\rightarrow\infty$.
This is important, as in the study of Euler-Arnold-type geodesic equations
negative sectional curvatures are related to a strong dependence of geodesics
on initial conditions [38]. For models of complexity, the necessity of
negative sectional curvatures has been discussed in [39, 40]. Curiously, the
metrics underlying the KdV and CH equations lead to sectional curvatures of
non-definite sign [41, 42], while the geometry underlying the HS equation is
positively curved [43]. Furthermore, it was shown in [43] that the geometry
underlying the HS equation maps the group-manifold to an open subset of an
$L^{2}$-sphere, implying geodesic incompleteness: Geodesics can leave the
space of invertible maps on the circle in finite affine parameter $\tau$. From
the point of view of the HS equation as a wave equation, this is related to
the phenomenon of wave breaking, but from a complexity perspective this
phenomenon is harder to interpret. It would be fascinating to develop a better
understanding of the conditions on a generic $\Pi$ under which equations of
the form (11) allow for such wave-breaking in finite time $\tau$. We
conjecture that this should not be the case for physically sensible choices of
complexity.

##  V Summary and outlook

We addressed the problem of complexity of unitary operators resulting from
exponentiation of the right- (or left-) moving component of the stress-energy
tensor in CFT1+1, see (1) and (2). There are three important features of the
complexity notions we consider: Firstly, they lead to equations of motion
second order in derivatives of $\tau$, which allows to search for optimal
circuits connecting two elements of the Virasoro group. Secondly, they are
based on only counting nontrivial (neglecting phase factors) elementary
operations with non-negative weights. It would be interesting to see if there
are circumstances under which the neglected phase factor can be interpreted as
a geometric phase [44], see e.g. [45]. Thirdly, the whole construction uses
explicitly spatially local gates generated by the stress-energy operator and
counts either linear combinations of it via the Fubini-Study approaches or
directly the usage of local operators itself.

We started with the definition of complexity based on the Fubini-Study
distance (4), as it seems to be easier to embed in holography and has other
attractive properties. This naturally led us to the Euler-Arnold type
nonlinear equation (11). Despite the intricate form of this equation of motion
in an infinitely-dimensional space, we were able to find approximate solutions
interpolating between the identity and its perturbation containing a single
Fourier mode, see (14). The leading term there reproduces the result of [24,
26] for complexity change under infinitesimal conformal transformations, which
is a double integral of two test functions (related to $\dot{f}$) integrated
against the stress-energy two-point function as integration kernel. Higher
order terms in (14) and (15) are new predictions originating from the
nonlinear nature of the equation of motion. We also evaluated the sectional
curvatures at the identity and found that for the Fubini-Study metric in
physically relevant cases it is negative in most directions, see (17).

Subsequently, we looked at another possibility of defining circuit complexity
that is based on explicit counting of appearances of the stress-energy tensor.
This approach can be thought of as originating from a state in which the
correlation function of the stress-energy tensor is ultra-local (13), and it
would be exciting to pursue this analogy further, perhaps along the lines of
[46]. Optimal circuits in this case would be exactly described by the KdV, CH
or HS equations, which were suggested as models for complexity in [25, 29].
However, the geometries underlying these equations may have undesirable
properties from a complexity perspective, such as positive sectional
curvatures or geodesic incompleteness [43].

One intriguing open problem is realizing circuits given by (1) and (2) in
holography, and their relation to holographic complexity proposals [1, 2, 3,
4, 5]. A strong hint in this direction is the agreement noted in [26] for
infinitesimal conformal transformations using the results from [24].

Another way forward is to understand the circuit (1) with $Q(\tau)$ given by
(2) as being realized by a CFT1+1 in a curved geometry with $\tau$ as the
physical time. This brings a parallel with the path-integral optimization
program [47, 48, 28], which was, however, predominantly presented in the
context of non-unitary circuits originating from the Euclidean time evolution.
Another interesting issue is the question of permissible cost functions being
covariant functionals of the underlying metric [28]. Finally, it is clearly
important to see if inclusion of primary operators and their descendants in
circuits containing the stress-energy tensor can lead to short-cuts, see [34].

###### Acknowledgements.

We would like to thank Volker Schomerus for being involved in the initial part
of this collaboration. In the context of this work and its companion paper
[30] we particularly benefited from correspondence with Diptarka Das and
discussions with Paweł Caputa on [25]. Furthermore, we gratefully acknowledge
Martin Ammon, Johanna Erdmenger, Marius Gerbershagen, Romuald Janik, and Anna-
Lena Weigel for conversations and correspondence, as well as Alexandre Belin,
Minyong Guo, Ro Jefferson, Javier Magán, Ali Naseh, Onkar Parrikar, Blagoje
Oblak, Gábor Sárosi and Claire Zukowski for very useful comments on both
papers. The Gravity, Quantum Fields and Information (GFQI) group at AEI is
supported by the Alexander von Humboldt Foundation and the Federal Ministry
for Education and Research through the Sofja Kovalevskaja Award. MF was
supported by the Polish National Science Centre (NCN) grant
2017/24/C/ST2/00469. MF is also grateful to the organizers of the GQFI
Workshop IV, where this work was presented for the first time, and the GFQI
group at AEI for its hospitality.

## Appendix

In this extra section, we will further elaborate on the iterative procedure
which led us to the approximate solution (14), see also [30] for further
details.

Suppose we want to calculate the geodesic circuit from $f_{0}(\sigma)=\sigma$
at $\tau=0$ to some target map of our choice, written as

$f_{1}(\sigma)=\sigma+\varepsilon\,g(\sigma)$ (18)

at $\tau=1$ with $\varepsilon\ll 1$. To lowest order in $\varepsilon$, the
geodesic circuit connecting these two maps will just be the linear
interpolation between them, and for higher order corrections in $\varepsilon$
we write

$\displaystyle
f(\tau,\sigma)=\sigma+\tau\,\varepsilon\,g(\sigma)+\varepsilon^{2}\,f^{(2)}(\tau,\sigma)+...\,.$
(19)

This ansatz alone stills lead to integro-differential equations for $f^{(n)}$
at increasing orders in $\varepsilon$. In addition, we represent
$f^{(n)}(\tau,\sigma)$ as a Fourier series

$\displaystyle f^{(n)}(\tau,\sigma)=b_{n,0}(\tau)$
$\displaystyle+\sum_{j\in\mathbb{N}}b_{n,j}(\tau)\cos(j\,\sigma)$ (20)
$\displaystyle+\sum_{j\in\mathbb{N}}a_{n,j}(\tau)\sin(j\,\sigma).$

Instead of one integro-differential equation of motion for
$f^{(n)}(\tau,\sigma)$, we obtain an infinite number of coupled ordinary
differential equations (ODEs) for the modes $b_{n,0}(\tau)$, $b_{n,j}(\tau)$
and $a_{n,j}(\tau)$. The term $b_{n,0}(\tau)$ can be set equal to zero at
every order without loss of generality for reasons explained in [30] . The
other modes should be solved for subject to the conditions

$a_{n,j}(0)=a_{n,j}(1)=b_{n,j}(0)=b_{n,j}(1)=0.$ (21)

For the moment, we make the additional assumption that $g(\sigma)$ in (18) is
proportional to one single Fourier-mode: $g(\sigma)=\sin(m\sigma)$ with some
integer $m$. The benefit of this assumption is that the on-shell (squared)
Lagrangian under the ansatz (19) can be calculated explicitly order by order
in $\varepsilon$, and equations of motion can then be calculated for the modes
$a_{n,j}(\tau),b_{n,j}(\tau)$ by varying this expression for the Lagrangian.
For the form of $g(\sigma)$ which we have chosen, and in fact for any
$g(\sigma)$ which has a Fourier series that exactly terminates after a finite
mode-number $m$, it can be shown that there will be an integer number $M(m,n)$
such that the modes $a_{n,j}(\tau),b_{n,j}(\tau)$ decouple from all other
modes for $j\geq M(m,n)$, obtaining trivial equations
$\ddot{a}_{n,j}=0=\ddot{b}_{n,j}$, which, subject to (21), simply imply
$a_{n,j}=0=b_{n,j}$.

Hence, at any finite order $n$ of $\varepsilon$, we only have to solve a
finite system of coupled ordinary differential equations for the
$a_{n,j}(\tau),b_{n,j}(\tau)$ with $j<M(m,n)$, which is tedious but
straightforwardly doable. Higher and higher orders in $\varepsilon^{n}$ can
then be added iteratively. For the specific choice of $g(\sigma)=\sin(\sigma)$
($m=1$), we obtained the solution (14) in this way.

What is now the distance between $f(0,\sigma)=\sigma$ and
$f(1,\sigma)=\sigma+\varepsilon\sin(\sigma)$? As the Lagrangian is conserved
by affine parametrisation and the curve is parametrised such that it reaches
its destination at $\tau=1$, the square of the Fubini-Study distance (8) is
just (up to a factor $4\pi^{2}$) equal to the value $\mathrm{L}_{sq}$ of the
square of the Lagrangian from (9), i.e. (15). As expected, for small
$\varepsilon$ the distance $\mathrm{L}_{FS}\propto\sqrt{\mathrm{L}_{sq}}$ will
be linear in $\varepsilon$.

We will now list a few important observations about this result and the
iterative method in general:

• While we assume $\varepsilon\ll 1$, in principle we can go to arbitrarily
high orders in $\varepsilon^{n}$. This means that even when the target map
(18) is not infinitesimally close to the identity, we can calculate the
extremal circuit to arbitrary precision, modulo possible issues arising in the
presence of conjugate points, which we leave for future study. Furthermore, as
the extremal circuit is a minimizer of length, even finite accuracy
approximations could be useful in providing upper bounds on the distance which
become increasingly tight as more and more orders of $\varepsilon$ are added.

• Of course, if only an approximation to the circuit with finite precision is
needed, the equations of motion might be solved numerically instead. Besides
some technical problems pointed out in [30], this would require a shooting or
relaxation method while our iterative method is by design well adapted to the
situation where initial and final conditions are given for the circuit
(instead of initial position and velocity), which is the most physical setup
for a problem of this kind. Also, the iterative method produces analytical
results (as series-expansions in $\varepsilon$), which may be inherently an
advantage in some circumstances.

• The first point above concerns the issue of what the convergence radius is
for the series-expansion in (15). For $\varepsilon=1$,
$f^{\prime}(1,\sigma)=1+\cos(\sigma)$ can be zero for some
$\sigma=\sigma_{0}$, and such maps with $f^{\prime}(\sigma_{0})=0$ are not
proper group elements as they would not be invertible one-to-one maps of the
circle to itself. A very important question is whether this boundary of the
space of allowed maps can be reached in finite distance, which is indeed
possible in the case of the HS equation as shown in [43]. Based on our
physical intuition for complexity, we would like to conjecture that this is
not the case. This would require that the series in (15) has a finite
convergence radius $\varepsilon<1$ and describes an analytic function with a
pole at $\varepsilon=1$. We hope to study this issue in more detail in the
future.

• In the above we have set $g(\sigma)=\sin(m\sigma)$ as a simple example and
proof of concept. However, our iterative procedure is much more versatile.
Firstly, we could straightforwardly use any function $g(\sigma)$ for which the
Fourier series terminates after a finite mode number (even though in practice
it adds a lot of workload).

But even more generally, consider a target map with non-terminating Fourier
series

$\displaystyle g(\sigma)=\sum_{m\in\mathbb{Z}}c_{m}e^{im\sigma}$ (22)

For this to converge to a smooth function $g\in C^{\infty}$, we need
$\sum_{m}|c_{m}||m|^{N}<\infty$ for any integer $N$. For concreteness, we can
consider the case $c_{m}=\hat{c}_{m}q^{m}$ with some $0<q<1$ and
$\hat{c}_{m}=\mathcal{O}(1)$ for any $m$. In this case, we can simply replace
$q$ with $\varepsilon$, and do the iterative procedure as before, imposing
adequate boundary conditions on all modes at any given order. Essentially, as
$\lim_{m\rightarrow\infty}c_{m}=0$, the Fourier series (22) can be interpreted
as always effectively terminating for any given finite level of accuracy
(determined by the magnitude of $\varepsilon^{n}$).

• Of course, in our eyes there is no ”best” or ”right” choice of $g(\sigma)$.
The most important possible achievement would be a comparison with bulk
results, for example along the lines of the comparison between the leading
order results of [24] and [26], but for higher orders in $\varepsilon$. For
this task a simple function like $g(\sigma)=\sin(m\sigma)$ would be sufficient
for obtaining non-trivial results, and we see no reason why a more complicated
function $g(\sigma)$ would give any additional benefits beyond that.

• In (15) we worked with the square of the original Lagrangian, which is also
more convenient for deriving the equations of motion. However, we have so far
left ambiguous whether the actual value of the complexity should be calculated
from this squared Lagrangian as in (15), or the original Lagrangian including
the square root (9). The reason for this is that assuming affine
parametrization, the equations of motion following from both kinds of
Lagrangian are equivalent, and so are hence their solutions and the
sensitivity of the geodesic problem to initial conditions (related to the
sectional curvatures (17)), which was a focus of our paper. It is our
philosophy that the study of such geometric properties of a given complexity
proposal is as important as the motivation of the employed cost function
itself, in contrast to some parts of the holographic literature that seem to
end their investigations at the point where they have motivated a choice of
cost function, without analysing the equations of motion, their generic
solutions and underlying geometry in detail.

See [49] for a discussion of the issue of distance functionals (i.e.
Lagrangians including a square-root) versus energy functionals (i.e. squared
Lagrangians). In our case, if we have in mind a comparison with bulk results
for the complexity=volume proposal along the lines of [24, 26], the squared
Lagrangian seems more relevant as then complexity scales linearly with the
central charge instead of its square-root. In fact, our leading order result
in $\varepsilon$ then trivially reproduces the results for complexity change
of the ground state ($h=0$) calculated in [24, 26] for the complexity=volume
proposal and conformal transformation in only one copy of the Virasoro group
(which we have exclusively considered in this paper). Extending our
methodology to two copies of the Virasoro group in the most trivial way
$\mathcal{C}_{total}\equiv\mathcal{C}_{Vir_{1}}+\mathcal{C}_{Vir_{2}}$ would
unfortunately fail to reproduce the general results of [24, 26] already at
leading order. However, we leave it to future studiy whether for
transformations in only one copy of the Virasoro group the volume change
matches with our results beyond leading order.

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