aainstitutetext: Institute for Theoretical Physics, University of Amsterdam,
PO Box 94485, 1090 GL Amsterdam, The Netherlandsbbinstitutetext: Institute of
Physics, Jagiellonian University, 30-348 Kraków, Polandccinstitutetext:
Instituto de Física Téorica IFT-UAM/CSIC, Universidad Autonoma de Madrid,
28049, Madrid, Spainddinstitutetext: Max Planck Institute for Gravitational
Physics (Albert Einstein Institute), 14476 Potsdam-Golm, Germany

# Spacetime as a quantum circuit

A. Ramesh Chandra a Jan de Boer b,c Mario Flory d,1 Michal P. Heller,111On
leave of absence from: National Centre for Nuclear Research, 02-093 Warsaw,
Poland a Sergio Hörtner a and Andrew Rolph<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We propose that finite cutoff regions of holographic spacetimes represent
quantum circuits that map between boundary states at different times and
Wilsonian cutoffs, and that the complexity of those quantum circuits is given
by the gravitational action. The optimal circuit minimizes the gravitational
action. This is a generalization of both the “complexity equals volume”
conjecture to unoptimized circuits, and path integral optimization to finite
cutoffs. Using tools from holographic $T\bar{T}$, we find that surfaces of
constant scalar curvature play a special role in optimizing quantum circuits.
We also find an interesting connection of our proposal to kinematic space, and
discuss possible circuit representations and gate counting interpretations of
the gravitational action.

##  1 Introduction

Quantum information theoretic concepts such as entanglement entropy have
proven to be of fundamental importance for our understanding of quantum
gravity, most notably in the context of the AdS/CFT correspondence Ryu:2006bv
; Maldacena:1997re . However, it has also been claimed that “entanglement is
not enough” Susskind:2014moa , such as in the inability of holographic
entanglement entropy to probe the late time linear growth of the Einstein-
Rosen bridge in eternal black holes, and that other concepts, in particular
state complexity, are needed for a more complete understanding.

The notion of state complexity in quantum mechanics refers to a setup where
one is given an initial state, a final state, a margin of error and a list of
allowed unitary operations. The smallest number of unitaries needed to obtain
the final state from the initial state up to the margin of error is an
indication of how difficult it is to obtain the final state from the initial
state. If the initial state is a fixed and very simple state, i.e. an
unentangled product state, one can simply refer to the complexity of the final
state as state complexity.

The idea to build interesting quantum states using a limited set of operations
has been successful in condensed matter physics leading to e.g. tensor network
representations of states Orus:2013kga . Moreover, a specific relation between
tensor networks and gravity has been proposed Swingle:2009bg ; Swingle:2012wq
in which one interprets constant time slices of AdS spacetime as a so-called
multiscale entanglement renormalization (MERA) tensor network Vidal:2008zz .
As a first piece of evidence for this relation one notices that the tensor
network in question indeed closely resembles a lattice discretization of an
equal time slice of AdS. If one imagines that this MERA tensor network is the
optimal network to obtain the ground state of a (discretized) CFT, then the
number of tensors needed equals the volume of the equal time slice. This
subsequently led to a much more general “complexity equals volume” proposal
Stanford:2014jda  where one proposes that the complexity of any viable state
in holographic quantum field theories can be obtained from the minimal volume
of a slice of the geometry which is anchored at the relevant fixed boundary
time slice. Other complexity proposals include those where complexity is
computed from the action Brown:2015bva ; Brown:2015lvg  or spacetime volumes
Couch:2016exn  evaluated in bulk Wheeler-deWitt (WdW) patches. All these
holographic complexity proposals share certain qualitative features that any
notion of state complexity should possess, while still lacking a precise
microscopic definition in the dual CFT.

A continuum version of MERA Haegeman:2011uy  was an important factor in the
realization of Jefferson:2017sdb ; Chapman:2017rqy , that a natural way to
count gates and define complexity in QFT is by assigning a metric to a
suitable group underlying the state preparation of interest Nielsen1133 .
While this is arguably the most promising way to define and prove holographic
complexity proposals, or to find other gravitational manifestations of
complexity, in this paper we will not directly attempt to find a precise
microscopic definition of complexity in holographic CFTs. Instead, we will
propose a significant refinement of the relation between geometry and
complexity as follows: we suggest that any spacetime region can be interpreted
as a quantum circuit, with the gravitational action providing a notion of
complexity for this particular quantum circuit.

Figure 1: We consider a subregion $M$ of Euclidean Poincaré AdS3. We introduce
two time-slices $t=t_{i}$ and $t=t_{f}$ corresponding to the field theory
ground states $|0\rangle_{z_{i}}$ and $|0\rangle_{z_{f}}$, which are prepared
for different values of the radial cutoff. The radial boundary is at finite
cutoff, $z=\rho(t)$. Our proposal is that the complexity of the circuit that
maps between these ground states with different finite Wilsonian cutoffs is
given by the gravitational action on $M$.

Let us make our proposal more precise. Take an Euclidean asymptotically AdS
geometry with radial coordinate $z$, the asymptotic boundary being at $z=0$,
and a spacetime region $M$ given by $t_{i}\leq t\leq t_{f}$ and $z\geq\rho(t)$
for some function $\rho(t)$, see figure 1. The AdS geometry describes the time
evolution of a given state, and the region $z\geq\rho(t)$ knows about the
state, but only up to a UV cutoff set by $\rho(t)$. Here, we use the well-
known relation between the radial distance in AdS and the UV cutoff in the
theory Susskind:1998dq  and its recent refinement McGough:2016lol . The latter
development links gravity in AdS spacetimes with a finite radial cutoff to
finite irrelevant deformations of dual CFTs by a $T\bar{T}$ operator, and this
will be a useful way of thinking about the UV cutoff in the remainder of this
paper. The bulk geometry of interest can be thought of as describing a
sequence of states, which are related to each other by both Euclidean time
evolution and a change of UV cutoff. One can ask what the complexity of this
particular process is, i.e. how many operations would be required to recover a
(discretized version of) this sequence of states. For previous work combining
holographic $T\bar{T}$ and complexity, see Jafari:2019qns ; Geng:2019yxo ;
Chen:2019mis  (see also Chakraborty:2020fpt  for a related development). We
propose that the number of operations, the complexity, is given by the
gravitational action itself, evaluated with suitable boundary condition on the
boundary of the spacetime region.

Our proposal, after specializing to two boundary dimensions, and keeping only
the first two orders in a Taylor expansion of the Wilsonian cutoff, coincides
with the path integral optimization proposal Caputa:2017urj ; Caputa:2017yrh ;
Czech:2017ryf ; Takayanagi:2018pml ; Camargo:2019isp , and so for holographic
CFTs can be seen as generalization of that proposal to any dimension and
finite cutoff. In path integral optimization one considers different
preparations of a fixed state using CFT path integrals on background
geometries differing by a Weyl factor. The proposal is to regard the change in
the unnormalized path integral measure – which for AdS3 is given by the
Liouville action Polyakov:1981rd  – as a cost function. However, as recognized
in Camargo:2019isp , minimization of such a cost function is not consistent
with keeping only the terms which remain finite as the UV cutoff is removed.
Our proposal avoids this problem altogether by considering the full
gravitational action, with the Liouville action merely capturing the leading
two terms in the limit where one takes the cutoff to infinity.

Coming back to our tensor network motivation, it is interesting to note that
in the course of the past several years substantial progress has been achieved
in obtaining MERA from a systematic coarse-graining procedure rather than
using it merely as an efficient variational ground state ansatz for critical
spin chains. The key idea is to employ an entanglement-based coarse-graining
of the discretized Euclidean path-integral Evenbly_2015 . This procedure is
closely related to one where one puts the corresponding conformal field
theories, which arise in the continuum limit of the critical spin chains, on
curved geometries Milsted:2018yur ; Milsted:2018san . We find the apparent
connection between these ideas and our setup quite intriguing. In particular,
in the language of Milsted:2018yur  one would be tempted to call part of our
circuit associated with moving in $t$ as composed from “euclideons”, whereas
the motion in the $z$ direction would have to do with “isometries” and
possibly “disentanglers”.

It would be certainly very interesting to make this association more
quantitative, perhaps using the results of Kruthoff:2020hsi , as there are
currently several distinct proposals for associating geometries to MERA. It
has been suggested to connect MERA to a hyperbolic geometry of an equal time
slice of AdS as mentioned above, to a light-cone Milsted:2018san  and to an
auxiliary dS geometry Czech:2015kbp  as in Beny:2011vh . The latter was
motivated by work trying to probe the bulk geometry using non-local CFT
observables such as entanglement entropy of spherical subregions, which gave
rise to the kinematic space program Czech:2015qta ; Czech:2016xec ;
deBoer:2016pqk . This confusing state of affairs was one of our motivations to
try to sharpen the relation between gravity and quantum circuits. It should be
also noted that a more precise relation between gravity and (other than MERA)
tensor networks was proposed recently in Bao:2018pvs ; Bao:2019fpq .

It is important to point out that we are not considering arbitrary circuits:
all circuits are essentially composed of time evolution and changes in the
local cutoff starting with a given initial state. One could certainly imagine
more general circuits, but these would not be captured by a single semi-
classical geometry and would require off-shell gravitational configurations.
The latter are typically exponentially suppressed and we will not consider
them in this paper. One can still try to find the optimal circuit within a
given semi-classical geometry, by varying over $t_{i}$ and over $\rho(t)$. In
particular, taking $\rho(t_{i})=\infty$ corresponds to a CFT state where the
CFT has a momentum cutoff brought down to zero, so this is akin to making the
state at $t=t_{f}$ from “nothing” Hartle:1983ai . We find that optimizing
complexity over this restricted set of circuits gives results quite similar to
other holographic complexity proposals. Perhaps in these holographic
situations, there is nothing to be gained (from a complexity point of view) by
considering circuits that involve different semiclassical geometries.

Finally, let us emphasize that there was earlier work, such as Nozaki:2012zj ;
Miyaji:2015fia ; Takayanagi:2018pml ; Belin:2018bpg ; Belin:2020oib ;
Boruch:2020wax ; Caputa:2020fbc , advocating for the relation between quantum
circuits and holographic geometry. In the present work we are building on
these earlier developments to bring in three important new elements into this
discussion: being explicit about an initial state, realizing the need of
keeping UV cut-off finite and interpreting it in terms of a $T\bar{T}$
deformation and, last but not least, making a connection with the kinematic
space program.

The outline of this paper is as follows. In section 2, we will first describe
the general setup and then give an explicit example in vacuum AdS3, where we
will find that our notion of complexity agrees with the complexity equals
volume proposal and, in the limit $\rho(t)\rightarrow 0$, also with path-
integral optimization. Subsequently, we discuss various finer points
associated with our proposal in sections 2.2 and 2.3. This brings us to
section 3, where we describe the relation between a change in the spacetime
region and $T\bar{T}$-deformations using bulk flow equations. Considerations
in this section will also allow us to argue that complexity is optimized if
the boundary of the spacetime region has constant scalar curvature. Finally,
we will discuss some ideas to more directly connect the gravitational action
to a gate counting procedure in section 4, and end with some conclusions and
suggestions for future work in section 5.

##  2 Vacuum preparation using gravity

The main idea of our construction is as follows: We can produce states in a
CFT using path integrals over Euclidean manifolds with a boundary and operator
insertions. Similarly, path integrals over manifolds with two boundaries can
be interpreted as objects mapping states to states. We would like to think of
these path integrals as describing a circuit which prepares states or maps
states to states, and associate a notion of cost function to them which
measures the effort it takes to perform these CFT operations in a given way.
To define such a cost function it seems inevitable to introduce some sort of
cutoff in the field theory. This cutoff defines a local lattice spacing and
provides the natural length scale at which to define tensors which make up an
approximate tensor network description of the CFT operation. The cutoff could
in principle be space and time dependent.

To determine the complexity, we are going to propose to use the unnormalized
CFT path integral. An important issue is how to incorporate the space and time
dependent cutoff in this computation. In the field theory, one could try to
implement this by including a space and time dependent $T\bar{T}$-deformation
in the CFT which is known to implement a particular type of finite cutoff
Smirnov:2016lqw ; Cavaglia:2016oda . It seems difficult to compute the
required path integral directly in the CFT, but luckily, for CFTs with a
holographic dual we can use the AdS/CFT correspondence to do the computation.
Following McGough:2016lol , the relevant AdS setup is one where we move the
boundary of AdS a finite distance inward, with the time (and possibly space)
dependent radial position corresponding to the cutoff or coefficient of the
$T\bar{T}$ deformation. The partition function of the CFT with cutoff is then
computed, to leading order, by the on-shell value of the gravitational action
with a finite instead of an asymptotic boundary.

There are some aspects of this proposal that require clarification. One is the
choice of boundary for the gravitational path integral away from the surface
where the cutoff CFT lives. For example, if the cutoff CFT lives on a
hemisphere, we need to fill in the boundary of the hemisphere in AdS. There is
in general no canonical slice in the bulk where the state “lives”. In the
example that we consider below, there are always natural time-symmetric
surfaces in the bulk which are the natural surfaces where to bound the bulk
path integral. Another issue with the construction is whether or not to
include the standard counterterms for AdS/CFT for co-dimension one boundaries
when evaluating the bulk action. Due to the existence of a finite cutoff,
there is no strict need to do so, and not including them appears to be the
most natural thing to do as we discuss below. A closely related issue comes
from the fact that the full bulk region has corners, and one may need to
include corner terms when evaluating the bulk action. We will also address
this issue below.

###  2.1 Action calculation

For simplicity and concreteness, we are going to consider the preparation of
the ground state of a 2d CFT on a line using the Euclidean path integral. To
this end, we take the standard Euclidean AdS solution, with the curvature
scale $l_{AdS}=1$,

$\displaystyle ds^{2}=\frac{dz^{2}+dt^{2}+dx^{2}}{z^{2}},$ (1)

and the partition function of the CFT equals the exponent of minus the on-
shell bulk action

$\displaystyle
I=\frac{1}{\kappa}\int_{M}d^{3}x\,\sqrt{G}\left(R+2\right)+\frac{2}{\kappa}\int_{\partial
M}d^{2}x\,\sqrt{g}K+I_{c}.$ (2)

$M$ is the bulk region bounded by $\rho(t)\leq z\leq\infty$ and $t_{i}\leq
t\leq t_{f}$, as shown in figure 1. The a priori finite function $\rho(z)$
interpolates between the values $z=z_{i}$ at $t=t_{i}$ and $z=z_{f}$ at
$t=t_{f}$, with $t_{i}\leq t_{f}$ and $z_{f}<z_{i}$. For simplicity we also
take the setup to be independent of the transverse direction $x$. Furthermore,
we write $\kappa=16\pi G_{N}$, $G$ for the 3d metric on $M$, $g$ for the
induced 2d metric on $\partial M$, and $K$ is the trace of the extrinsic
curvature. $\partial M$ is only piecewise smooth and has a kink or joint at
$t=t_{f}$ and $t=t_{i}$ as shown in figure 1. Each joint contributes a term

$\displaystyle I_{c}=\frac{2}{\kappa}\int dx\sqrt{j}\ \alpha$ (3)

to the gravitational action. Herein, $\sqrt{j}$ is the length element along
the joint and $\alpha$ is simply the angle between the two normal vectors of
the two surfaces coming together at the joint (which may have either sign).
Joint-terms of this type were studied by Hayward in Hayward:1993my ;
Brill:1994mb , but in the Euclidean setting, which is of interest here, this
was already done earlier in Hartle:1981cf , see also the discussion in
Lehner:2016vdi .

As discussed above, we are going to interpret the on-shell value of the bulk
effective action of the region $M$ as the complexity of the circuit defined by
the surface $z=\rho(t)$ which maps the vacuum state $|0\rangle_{z_{i}}$ with
cutoff $z_{i}$ to the vacuum state $|0\rangle_{z_{f}}$ with cutoff $z_{f}$. If
we use the relation between a finite radial cutoff and the coefficient $\mu$
of the $T\bar{T}$ deformation via McGough:2016lol ,

$\mu(t)=\kappa\,\rho(t)^{2},$ (4)

we can reinterpret the states $|0\rangle_{\rho(t)}$ as ground states of the
$T\bar{T}$ deformed CFT with a time-dependent coefficient $\mu(t)$.

Concretely, the induced line element on the boundary surface is

$\displaystyle ds^{2}=\frac{(1+\dot{\rho}^{2})dt^{2}+dx^{2}}{\rho^{2}},$ (5)

its Ricci scalar reads

$\displaystyle
R^{(d-1)}=\frac{2(\rho\ddot{\rho}-\dot{\rho}^{2}(1+\dot{\rho}^{2}))}{(1+\dot{\rho}^{2})^{2}},$
(6)

the trace of the extrinsic curvature reads

$\displaystyle
K=\frac{\rho\ddot{\rho}+2(1+\dot{\rho}^{2})}{(1+\dot{\rho}^{2})^{3/2}},$ (7)

and from (2) we obtain

$\displaystyle I$
$\displaystyle=\frac{-4}{\kappa}\int_{M}d^{2}x\int_{z=\rho}^{\infty}\frac{dz}{z^{3}}+\frac{2}{\kappa}\int_{\partial
M}d^{2}x\,\frac{\rho\ddot{\rho}+2(1+\dot{\rho}^{2})}{\rho^{2}(1+\dot{\rho}^{2})}+I_{c}$
$\displaystyle=\frac{2V_{x}}{\kappa}\int_{t_{i}}^{t_{f}}dt\,\frac{\rho\ddot{\rho}+(1+\dot{\rho}^{2})}{\rho^{2}(1+\dot{\rho}^{2})}+I_{c}$
(8)

for the on-shell bulk action, where we have introduced $V_{x}=\int dx$. For
the corner term, we also find

$\displaystyle
I_{c}=\frac{2V_{x}}{\kappa}\left(\frac{\pi/2-\arctan{\dot{\rho}(t_{f})}}{z_{f}}+\frac{\pi/2+\arctan{\dot{\rho}(t_{i})}}{z_{i}}\right).$
(9)

Integrating by parts, this action can be written only using first derivatives
of $\rho$, yielding

$\displaystyle I=$
$\displaystyle\frac{2V_{x}}{\kappa}\int_{t_{i}}^{t_{f}}dt\,\left(\frac{1}{\rho^{2}}+\frac{\dot{\rho}\arctan{\dot{\rho}}}{\rho^{2}}\right)+\frac{\pi
V_{x}}{\kappa}\left(\frac{1}{z_{f}}+\frac{1}{z_{i}}\right).$ (10)

The terms which are independent of $\rho$ do not affect the equations of
motion, and can always be removed by a suitable counter term, which we will
assume to be done from now on. We believe this is justified, as it is known
Brill:1994mb ; Lehner:2016vdi  that the joint term can spoil the additivity of
the action under combining bulk regions, which besides the formulation of a
well defined variational principle is usually the second main reason for
adding boundary terms to the action (2).222 Note that in our Euclidean
setting, where spacelike surfaces have spacelike normal vectors, the joints
under consideration are more similar to the timelike joints discussed in
Brill:1994mb ; Lehner:2016vdi  than spacelike ones in a Lorentzian setting.

The equations of motion obtained by extremizing (10) read

$\displaystyle\frac{\rho\ddot{\rho}+(1+\dot{\rho}^{2})}{\rho^{3}(1+\dot{\rho}^{2})^{2}}=0.$
(11)

The most immediately visible solution to this equation is the one where we
formally take the limit $\dot{\rho}\rightarrow\infty$. This corresponds to the
boundary surface turning into an equal-time slice, which is in fact where,
based on the intuition surrounding holographic complexity and tensor networks,
we expect the most optimised circuit preparing the state $|0\rangle_{z_{f}}$
to live, see e.g. Boruch:2020wax . The generic solution to (11) reads

$\displaystyle\rho(t)=\sqrt{\mathcal{R}^{2}-(t-t_{0})^{2}}$ (12)

and describes circular arcs of radius $\mathcal{R}$ centered on the boundary
point at $t=t_{0}$. The formal solution $\dot{\rho}\rightarrow\infty$
corresponds to the limit of infinite radius.

Our proposal is that the Euclidean action (10) (excluding the
$\rho$-independent remnants of the joint terms) is a measure of the complexity
of preparing the state $|0\rangle_{z_{f}}$ from the state $|0\rangle_{z_{i}}$
using the circuit described by $\rho(t)$. The optimal circuit, with fixed
Euclidean time distance $\Delta t=|t_{f}-t_{i}|$, is then of the form (12),
and the complexity of this circuit is given by evaluating the Euclidean action
on this solution. With the explicit boundary conditions being
$\rho(t_{f})=z_{f}$ and $\rho(t_{i})=z_{i}$, the value of the Euclidean action
in the first term of (10) is

$\displaystyle
I=\frac{2V_{x}}{\kappa}\left(\frac{1}{z_{f}}\arctan{\frac{z_{i}^{2}-z_{f}^{2}+\Delta
t^{2}}{2z_{f}\Delta
t}}-\frac{1}{z_{i}}\arctan{\frac{z_{i}^{2}-z_{f}^{2}-\Delta
t^{2}}{2z_{i}\Delta t}}\right).$ (13)

Note that this result comes entirely from the corner terms, as the first term
in (8) exactly vanishes on-shell. Interpreting it as a function of the
variable $t_{i}\leq t_{f}$ while keeping $z_{i}\neq z_{f}$ fixed, we can
verify that the above expression is minimized by $t_{i}=t_{f}$. This
corresponds to the limit $\mathcal{R}\rightarrow\infty$ or
$\dot{\rho}\rightarrow\infty$ and hence the equal time slice that is
intuitively expected to play a special role in describing the complexity of
the state $|0\rangle_{z_{f}}$. Using $1/\kappa=c/24$ Brown:1986nw , the
minimum value is given by

$\displaystyle
I_{min}=\frac{c{\pi}V_{x}}{24}\left(\frac{1}{z_{f}}-\frac{1}{z_{i}}\right),$
(14)

which is proportional to the spatial volume of the strip $z_{f}\leq z\leq
z_{i}$ on the equal time slice at $t=t_{f}$. Of course, if we send
$z_{f}\rightarrow\epsilon\ll 1$ and $z_{i}\rightarrow\infty$, this reproduces
the standard result of the volume proposal for the complexity of the CFT
ground state. Clearly, this result also vanishes if $z_{i}=z_{f}$, which we
take as a non-trivial consistency check and further justification for
excluding the remnants of the joint terms in (10). 333As an illustrative
example, imagine a Euclidean axisymmetric spacetime, with a spacetime region
in the shape of a regular prism that breaks rotational symmetry around the
axis to a discrete subgroup. In the limit where the radius of the prism goes
to zero, the action on that region may not go to zero, as while bulk and
surface terms vanish in this limit due to the vanishing of bulk volume and
surface area, the joint terms will lead to a contribution proportional to an
integral along the axis of symmetry. This remnant term is the analogue of the
last bracket in (10).

To close this section, let us compare our results to the ones that can be
obtained from the Liouville action. For $\dot{\rho}\ll 1$, equation (10) can
be approximated as

$I=\frac{2V_{x}}{\kappa}\int
dt\left(\frac{1}{\rho^{2}}+\frac{\dot{\rho}^{2}}{\rho^{2}}\right),$ (15)

which, assuming no $x$-dependence, is equivalent to the Liouville Lagrangian

$S_{L}=\frac{c}{24\pi}\int dt\int
dx\left(\eta\,e^{2\omega}+\left(\partial_{t}\omega\right)^{2}+\left(\partial_{x}\omega\right)^{2}\right).$
(16)

after a change of variables $\rho(t)\to(1/\sqrt{\eta})\,e^{-\omega(t)}$. Note
that the physically interesting solution $\dot{\rho}\rightarrow\infty$ falls
outside of the range of applicability of the approximation necessary to obtain
the Liouville action from (10). The equations of motion derived from (15) take
the form

$\displaystyle\frac{\rho\ddot{\rho}+(1-\dot{\rho}^{2})}{\rho^{3}}=0.$ (17)

As we will see below, these field equations also arise if we introduce a new
time coordinate in order to bring the induced metric on the boundary into
conformal gauge.

###  2.2 Conformal time and extremizing the action

There is a subtle but crucial difference between our setup discussed in the
previous subsection and the calculations of Boruch:2020wax , which we will
discuss in this subsection in order to avoid confusion.

In order to do so, we note that Boruch:2020wax  investigates a setup similar
to the one depicted in figure 1, and up to notation (8) also appears in the
appendix of that paper. Following Boruch:2020wax , we can now introduce a
conformal time $u$, with

$\displaystyle du=\sqrt{1+\dot{\rho}(t)^{2}}dt,$ (18)

such that the line element (5) is transformed into the conformal gauge form

$ds^{2}=\frac{du^{2}+dx^{2}}{\varrho(u)^{2}}.$ (19)

Here, we have introduced a new variable such that $\varrho(u(t))=\rho(t)$.
Under (18), the action (10) changes to Boruch:2020wax

$I=\frac{2V_{x}}{\kappa}\int_{u_{i}[\varrho]}^{u_{f}[\varrho]}du\left(\frac{\sqrt{1-\varrho^{\prime
2}}+\varrho^{\prime}\arcsin{\varrho^{\prime}}}{\varrho^{2}}\right).$ (20)

If we were to just identify the integrand in (20) as a Lagrangian and compute
naively the Euler equations, we arrive at

$\displaystyle\frac{\varrho\varrho^{\prime\prime}+2(1-\varrho^{\prime}{}^{2})}{\varrho^{3}(1-\varrho^{\prime}{}^{2})^{2}}=0,$
(21)

which, up to notation and the addition of a nonzero tension term, are the
equations which where studied in Boruch:2020wax .

The subtlety announced at the beginning of the subsection is that (18) is a
reparametrization of time which is dependent on the variable with respect to
which we want to vary the action, hence formally in going from (10) to (20)
the integration bounds $u_{i}$ and $u_{f}$ become themselves functionals of
$\varrho$, and will lead to a nontrivial contribution according to Leibniz’s
rule when varying the action. In fact it can be checked that introducing (18)
and $\varrho(u)$ in the equation of motion (11) gives a result

$\displaystyle\frac{\varrho\varrho^{\prime\prime}+(1-\varrho^{\prime}{}^{2})}{\varrho^{3}}=0$
(22)

that is inequivalent to (21). Interestingly, (22) has the form of the
Liouville equation (17), just for $\varrho(u)$ instead of $\rho(t)$.

The most commonly known example where a field-dependent reparametrization can
be useful is the Lagrangian for geodesic motion, which becomes a constant when
introducing affine parametrisation. Of course, this does not mean that the
equations of motion degenerate, as the full information about the value of the
action – i.e. the length of the curve – is now entirely encoded in the
integration domain. Unfortunately, the expression (20) rather inelegantly
falls into a middle ground between the two possible extremes, as both the
integrand and the integration bounds are functionals of the variable
$\varrho$, and for this reason we found it intractable to work with.

This does not mean that either our work or Boruch:2020wax  are wrong, just
that we are studying a different variational problem. We work with the action
(10) where explicitly we assume Dirichlet boundary conditions for $\rho(t)$ at
the fixed values $t=t_{f}$ and $t=t_{i}$, while Boruch:2020wax  works with the
action (20) with the implicit assumption of Dirichlet boundary conditions for
the field $\varrho$ at fixed values of $u_{i},u_{f}$, which is an inequivalent
mathematical exercise.

###  2.3 Comparison to AdS/BCFT models

We can investigate this issue a bit further. So far, we have essentially
considered what amounts to minisuperspace models, by plugging in an ansatz
into the action and deriving equations of motion for the function
parametrizing that ansatz, instead of first deriving general equations of
motion and then simplifying them with a given ansatz. How can we write our
equations of motion in a form that is more suggestive for their general
meaning and potential origin? We will do this in the next section, but as an
aside, we will now demonstrate that the semicircle solutions that we found can
also be obtained if we interpret the boundary of the bulk domain as an “end of
the world brane” with an energy-momentum tensor describing matter with a very
specific equation of state. The covariant equations of motion of this end of
the world brane will imply the general equation that we will derive in the
next section. The derivation in the next section does not rely on an end of
the world brane interpretation, and it remains to be seen whether this
agreement is more than a technical coincidence.

We should also point out that the work of Boruch:2020wax  was strongly
influenced by the type of AdS/boundary CFT (BCFT) models introduced in
Takayanagi:2011zk ; Fujita:2011fp . In such models the boundary of the space
on which the BCFT lives is also extended into the bulk spacetime in the form
of an end of the world brane, on which Neumann boundary conditions are
imposed. Besides the bulk Einstein equations, this leads to an equation of
motion of the form

$\displaystyle K_{\mu\nu}-Kg_{\mu\nu}=\frac{\kappa}{2}T_{\mu\nu}$ (23)

which determines the embedding of the end of the world brane into the ambient
space. These models allow for considerable bottom-up toy-model building
freedom, and $T_{\mu\nu}$ is the energy-momentum tensor of any matter that
lives in the brane worldvolume. In practice, it is often set to be a constant
tension term

$\displaystyle T_{\mu\nu}=\lambda\,g_{\mu\nu}$ (24)

with tension $\lambda$. As reported in Boruch:2020wax , their equation of
motion is consistent with (23). As we ignore tension terms, we would set the
right hand side of (23) to zero, and apart from the equal time slice obtained
by $\dot{\rho}\rightarrow\infty$, our semicircular embeddings do not satisfy
this equation.

Interestingly, in a Lorentzian AdS/BCFT context, semicircular embeddings into
Poincaré AdS were derived in Erdmenger:2014xya  for a simple model of
$T_{\mu\nu}$ given by a perfect fluid with equation of state $p=a\sigma$
($p=$pressure, $\sigma$=energy density) in the limit $a\rightarrow\infty$. So
we see that semicircular embeddings into a Poincaré AdS do satisfy an equation
of the form (23), just with a specific non-trivial right hand side. Due to the
peculiar limit in the parameter $a$, $T_{\mu\nu}$ satisfies the condition

$\displaystyle\det[T_{\mu\nu}]=0$ (25)

or equivalently

$\displaystyle T_{\mu\nu}T^{\mu\nu}-T^{2}=0,$ (26)

and hence

$\displaystyle\det\left[K_{\mu\nu}-Kg_{\mu\nu}\right]=0,$ (27)

respectively

$\displaystyle(K_{\mu\nu}-Kg_{\mu\nu})(K^{\mu\nu}-Kg^{\mu\nu})-\operatorname{Tr}[K_{\mu\nu}-Kg_{\mu\nu}]^{2}=K_{\mu\nu}K^{\mu\nu}-K^{2}=0$
(28)

for our semicircular embeddings (12), even though they were not derived from
an AdS/BCFT ansatz in this paper. We will give a direct derivation of equation
(28) as a flow equation for our complexity proposal in the following section.

##  3 Bulk action and $T\bar{T}$

We have considered the on-shell action of a cutout region of Poincaré AdS3,
and interpreted it as a complexity functional of states in $T\bar{T}$-deformed
holographic CFTs. The relation (4) between the coefficient of the $T\bar{T}$
deformation and the radial location has been derived for constant radial
cutoff McGough:2016lol ; Taylor:2018xcy ; Hartman:2018tkw , but not for time-
dependent $\rho(t)$. In this section we consider the flow equations which
describe movement of the cutoff surface in a fixed background. By integrating
these flow equations we should be able to derive a more precise relation
between the coefficient of the $T\bar{T}$ deformation and the location of the
bulk surface. In addition, these flow equations will tell us how complexity
changes as we change the surface locations, and for which surfaces complexity
is optimized while keeping the initial and final state fixed.

###  3.1 Excluding counterterms

The relevant flow equation can most easily be derived using the ADM formalism
ADM:1962 . We will keep the number of spacetime dimensions free in what
follows, and write the metric as

$ds^{2}=N^{2}dr^{2}+g_{\mu\nu}(x,r)(dx^{\mu}+N^{\mu}dr)(dx^{\nu}+N^{\nu}dr).$
(29)

This contains the usual lapse and shift functions, for which one can locally
choose a convenient gauge $N=1$ and $N^{\mu}=0$. Following ADM and choosing
units so that $\kappa=1$, we now write the Lagrangian in terms of canonical
variables

${\cal
L}=\sqrt{g}\left(\pi^{\mu\nu}\partial_{r}{g}_{\mu\nu}-NH-N^{\mu}H_{\mu}\right),$
(30)

where the lapse and shift functions appear as Lagrange multipliers enforcing
the Hamiltonian and momentum constraints

$H=H^{\mu}=0.$ (31)

The canonical momenta are given by Brown.York:1993

$\begin{split}\pi_{\mu\nu}&=\frac{1}{\sqrt{g}}\frac{\partial S}{\partial
g^{\mu\nu}}\\\ &=-(K_{\mu\nu}-Kg_{\mu\nu})\\\
&=-\frac{1}{2}\left(\partial_{r}g_{\mu\nu}-g_{\mu\nu}g^{\rho\sigma}\partial_{r}g_{\rho\sigma}\right),\end{split}$
(32)

where in the second step we used the fact that metric variations are given by
the Brown-York tensor, and in the last step we used the explicit form of the
extrinsic curvature for the metric (29) in the gauge $N=1$, $N^{\mu}=0$. Of
course, the same result can also be obtained by explicitly rewriting the
action as in (30). Using (32), we find for the radial derivative

$\partial_{r}g_{\mu\nu}=-2\pi_{\mu\nu}+\frac{2}{d-2}g_{\mu\nu}\pi^{\rho}_{\rho}$
(33)

where $d$ is the total number of bulk spacetime dimensions (in this paper we
are predominantly interested in $d=3$). The Hamiltonian constraint can be
computed from (30) and, for unit AdS radius, one finds

$\begin{split}H&=R^{(d-1)}-2\Lambda-(K^{2}-K^{\mu\nu}K_{\mu\nu})\\\
&=R^{(d-1)}+(d-1)(d-2)+\pi^{\mu\nu}\pi_{\mu\nu}-\frac{1}{d-2}(\pi^{\rho}_{\rho})^{2}.\end{split}$
(34)

It is fairly straightforward to include matter fields in the discussion; the
Hamiltonian constraint will then also contain the Hamiltonian of the matter
sector, but we will for simplicity restrict to the purely gravitational case.
To describe the flow we imagine starting with a surface at constant $r$ and
moving the cutoff slightly so that $r\rightarrow r+\epsilon(x)$. For any
surface, we can always locally find coordinates such that the surface is
located at fixed value of $r$ and the metric is in the ADM gauge, so there is
no loss of generality in this assumption. Then

$\begin{split}\delta_{\epsilon}S&=\int\epsilon(x)\partial_{r}g^{\mu\nu}\frac{\partial{S}}{\partial
g^{\mu\nu}}\\\ &=\int\sqrt{g}\epsilon(x)\partial_{r}g^{\mu\nu}\pi_{\mu\nu}\\\
&=2\int\sqrt{g}\epsilon(x)\left({\pi}^{\mu\nu}{\pi}_{\mu\nu}-\frac{1}{d-2}({\pi}^{\rho}_{\rho})^{2}\right),\end{split}$
(35)

where we used equation (33) for the radial dependence of the metric in terms
of momenta. Interestingly, this is precisely of $T\bar{T}$ form, but with $T$
and $\bar{T}$ defined with respect to the metric variations of the finite
surface, not the boundary at infinity. A more coordinate independent way of
stating the result is that as we move a surface in a given AdS background, we
turn on a local $T\bar{T}$-deformation with a coefficient given by the
orthogonal distance between the original and deformed surface. If we could
relate the local $T$ and $\bar{T}$ on a given surface to the $T$ and $\bar{T}$
as defined at infinity, we could integrate these flow equations and write the
final result in terms of a finite $T\bar{T}$ deformation of the theory at
infinity. We leave a further exploration of this interesting question to
future work but thinking of finite $T\bar{T}$ deformations in terms of a
change in the boundary conditions for the metric we expect it to involve the
linearized Einstein equations around the background Guica:2019nzm .

Clearly, using (32) for $d=3$ the variation of the action vanishes if equation
(28) is satisfied. As is clear from the Hamiltonian constraint, this condition
can also be phrased as $R^{(d-1)}+(d-1)(d-2)=0$, i.e. the boundary surface has
constant scalar curvature. Therefore, to optimize the complexity of the
process we should use constant scalar curvature surfaces; the metric on a
Euclidean AdSd-1 manifold precisely has the required scalar curvature. This is
consistent with the observation that complexity is minimized if we take
$t_{i}=t_{f}$ and consider a purely radial surface at the $t=t_{f}$ constant
timeslice in section 2.

###  3.2 Including counterterms

So far the discussion has used the standard bulk AdS action without the
inclusion of additional counterterms, which would render the on-shell value of
the action finite as one takes the surface to the asymptotic boundary. As
alluded to in the beginning, in the original appearance of Liouville theory as
defining path integral complexity, the absence of the volume counterterm was
important. Here we briefly discuss what happens if we add a volume term for
the boundary surface with an arbitrary coefficient. In our discussion of the
on-shell value of the action, it would add an extra term

$S_{c.t}=-2\lambda\int d^{2}x\sqrt{g}=-2\lambda\int
dtdx\frac{\sqrt{1+\dot{\rho}^{2}}}{\rho^{2}}.$ (36)

Adding the counterterm modifies the field equations to

$\displaystyle\frac{1}{\rho^{3}(1+\dot{\rho}^{2})}\left((\rho\ddot{\rho}+1+\rho^{2})-\lambda\sqrt{1+\dot{\rho}^{2}}\left(\frac{1}{2}\rho\ddot{\rho}+1+\dot{\rho}^{2}\right)\right)=0$
(37)

We can also reconsider the flow equations in the presence of the volume
counterterm. Denoting the volume counterterm as

$S_{\rm vol}=-2\lambda(d-2)\int_{\partial M}\sqrt{g}$ (38)

so that $\lambda=1$ is precisely the counterterm which would cancel the volume
divergence near the AdS boundary, we now introduce
$\tilde{\pi}_{\mu\nu}=\pi_{\mu\nu}-\lambda(d-2)g_{\mu\nu}$ so that these are
precisely the canonical momenta in the presence of the extra boundary volume
term. The Hamiltonian constraint can be rewritten as

$\begin{split}H&=R^{(d-1)}+(1-\lambda^{2})(d-1)(d-2)+\tilde{\pi}^{\mu\nu}\tilde{\pi}_{\mu\nu}-\frac{1}{d-2}(\tilde{\pi}^{\rho}_{\rho})^{2}-2\lambda\tilde{\pi}^{\rho}_{\rho}=0\end{split}$
(39)

We can now consider two types of flows. We can consider the variation of the
action as we change the radial surface in a given background, but we can also
consider the variation of the action as we perform a conformal rescaling of
the metric on the radial surface. In $d=3$, the latter does not require an
adjustment of the bulk geometry, but in higher dimensions this is no longer
true. It is therefore not clear whether conformal rescalings of the induced
metric on the boundary surface are in general compatible with keeping the
initial and final states fixed in $d>3$. Regardless, the change of the action
under the first type of flow now reads

$\begin{split}\delta_{\epsilon}S&=\int\epsilon(x)\partial_{r}g^{\mu\nu}\frac{\partial{\tilde{S}}}{\partial
g^{\mu\nu}}\\\
&=\int\sqrt{g}\epsilon(x)\partial_{r}g^{\mu\nu}\tilde{\pi}_{\mu\nu}\\\
&=2\int\sqrt{g}\epsilon(x)\left({\tilde{\pi}}^{\mu\nu}{\tilde{\pi}}_{\mu\nu}-\frac{1}{d-2}({\tilde{\pi}}^{\rho}_{\rho})^{2}-\lambda{\tilde{\pi}}^{\rho}_{\rho}\right)\end{split}$
(40)

and for the second type of flow with $\delta g^{\mu\nu}=\epsilon(x)g^{\mu\nu}$

$\begin{split}\delta_{\epsilon}\tilde{S}&=\int\epsilon(x)g^{\mu\nu}\frac{\partial\tilde{S}}{\partial
g^{\mu\nu}}\\\ &=\int\sqrt{g}\epsilon(x)\tilde{\pi}^{\rho}_{\rho}\\\
&=\frac{1}{2\lambda}\int\sqrt{g}\epsilon(x)\left(R^{(d-1)}+(1-\lambda^{2})(d-1)(d-2)+\tilde{\pi}^{\mu\nu}\tilde{\pi}_{\mu\nu}-\frac{1}{d-2}(\tilde{\pi}^{\rho}_{\rho})^{2}\right).\end{split}$
(41)

We see that both flows take the form of $T\bar{T}$ deformations, with various
extra terms such as the scalar curvature and the trace of the stress tensor.
Just as in the case without counterterm ($\lambda=0$) it would be interesting
to integrate these flows to finite flows starting at the AdS boundary.

The first flow is extremized when the surface obeys

$R^{(d-1)}+(d-1)(d-2)-\lambda(d-2)K=0$ (42)

which still holds for an AdSd-1 equal time slice in AdSd. As expected, for our
setup (42) is equivalent to (37). The second flow, on the other hand, is
extremized when $K=\lambda(d-1)$. This does not have an extremum for an AdSd-1
equal time slice in AdSd unless $\lambda=0$. Moreover, as we indicated above,
it is not clear whether the initial state and final state are kept fixed along
the flow, and therefore the precise interpretation of this flow is somewhat
unclear. In any case, it would be interesting to explore whether surfaces
obeying (42) or $K=\lambda(d-1)$ have the potential to define a new notion of
complexity.

Finally, we notice that it is also possible to add higher order counterterms,
but for those the connection to $T\bar{T}$ deformations becomes more
complicated.

##  4 Towards counting elementary operations

###  4.1 Gravitational action from counting stress tensor insertions

The bulk computation from section 2 and illustrated in figure 1 can be viewed
in the light of the results from the previous section as the following non-
unitary circuit acting on the initial state

$|0\rangle_{z_{f}}=P\exp\left[-\int_{t_{i}}^{t_{f}}dt\,(H_{\rho(t)}+\dot{\rho}\,[T\bar{T}]_{\rho(t)})\right]|0\rangle_{z_{i}}.$
(43)

In the above expression, $H_{\rho(t)}$ represents Euclidean time evolution in
a CFT with cutoff specified by $\rho(t)$, which in the bulk would correspond
to moving in the $t$-direction while keeping $\rho(t)$ fixed. The other term
represents the operator that implements a change in the scale of the theory,
which we have schematically denoted by $[T\bar{T}]_{\rho(t)}$. In the bulk
this would correspond to changing $\rho$ while keeping $t$ fixed. What is
important is that each layer of (43) in general uses operators from a
different theory. We assume that $H_{\rho(t)}$ and $[T\bar{T}]_{\rho(t)}$ can,
at least in principle, be written and understood as operators in the
undeformed field theory, say by explicitly solving the Lagrangian $T\bar{T}$
flow equation, and that the action of those operators on states in the
undeformed theory is well-defined.

Following gate counting ideas of Nielsen1133 ; Jefferson:2017sdb ;
Chapman:2017rqy ; Camargo:2019isp , due to spatial homogeneity of our setup
one might be tempted to regard $H_{\rho(t)}$ and $[T\bar{T}]_{\rho(t)}$ as two
classes of elementary operations with $\rho(t)$ playing two independent roles.
The first role played by $\rho(t)$ lies in labelling the elementary operations
we are using (as already mentioned, both $H_{\rho(t)}$ and
$[T\bar{T}]_{\rho(t)}$ are different operators for each value of $\rho(t)$).
The second role stems from $dt\,\dot{\rho}$ being related to the number of
times the operator $[T\bar{T}]_{\rho(t)}$ is applied in a given layer of the
circuit. Correspondingly, $H_{\rho(t)}$ is applied simply $dt\,1$ number of
times.

As a result, a naïve way of counting insertions of $H(\rho_{t})$ would be
$\int_{t_{i}}^{t_{f}}dt\,1$ and $\int_{t_{i}}^{t_{f}}dt\,|\dot{\rho}|$ when it
comes to $[T\bar{T}]_{\rho(t)}$ with the total number of insertions being the
sum of the two contributions. Note that since $\rho(t)$ is just a label, in
principle the contribution from every layer can be weighted by some non-
negative function of $\rho(t)$ – a penalty factor that weights hardness of
applications of particular transformations. The above logic was based on an
$L_{1}$ norm of the vector $\\{1,\dot{\rho}\\}$, but in principle any norm
would do.

However, our proposal is to view the action (8) or (10) as a cost function for
the circuit (43). It seems quite straightforward to associate a suitable
weight to $H_{\rho(t)}$. If we imagine a CFT with a fixed cutoff or lattice
spacing $\sim\rho$, and we count the number of lattice points in a given
Euclidean volume (which we interpret as suitable tensor operations), then we
immediately obtain an answer proportional to $\int dtdx\rho^{-2}=\int
dtV_{x}\rho^{-2}$, which is indeed proportional to the potential term in the
action (10). Within the logic outlined in the previous paragraph, this
corresponds to using $L_{1}$ norm with the penalty factor equal to
$V_{x}\rho^{-2}$.

The second term in the action (10) is tricky to interpret within the framework
of Nielsen1133 ; Jefferson:2017sdb ; Chapman:2017rqy . Following the above
logic, one would be naturally inclined to associate this term with the
presence of $[T\bar{T}]_{\rho(t)}$ insertions in the circuit (43), however,
this is difficult. Writing the relevant contribution as
$\frac{V_{x}\,|\arctan{\dot{\rho}}|}{\rho^{2}}\,|\dot{\rho}|$ one does not
recognize a standard penalty factor in front of $|\dot{\rho}|$ within an
$L_{1}$ norm. To this end, the penalty factor is not supposed to know about
what circuits do at other layers, and its dependence on $\dot{\rho}$ via
$\arctan{\dot{\rho}}$ induces such a dependence. This is very much reminiscent
of the discussion in Camargo:2019isp  about viewing the Liouville action as a
bona fide cost function.

Following this thread, our interpretation of the “potential” and “kinetic”
terms in the gravity action (10) is quite similar to an earlier discussion
about the aforementioned qualitative interpretation of the Liouville action as
a complexity of a tensor network Czech:2017ryf ; Caputa:2017yrh . In our
study, the “potential” term studies euclideons, whereas the “kinetic” term is
associated with changes in cut-off and might be related to isometries or full
layers of MERA.

Note also that the circuit (43) is very similar to the expression usually
written down for cMERA Haegeman:2011uy , which takes the form of a path-
ordered exponent of infinitesimal unitaries and dilatation operators. An
immediate issue with this expression is the precise meaning of the operator
$[T\bar{T}]_{\rho(t)}$ in the CFT, as we have seen that a careful construction
of this operator requires one to integrate the flow equations from the AdS
boundary to the bulk surface. If we knew the precise definition of this
operator in the CFT, we could try to assign a number to this path-ordered
exponent, for example by computing the length of the trajectory in a suitable
space of operators. It is not inconceivable that such a computation is
possible, as we know the commutation relations between the Hamiltonian and the
$T\bar{T}$ operator in the CFT, and it would be interesting to explore this
further.

Furthermore, it is intriguing to note that $\arctan\dot{\rho}$ is the angle
that the surface makes in the $z,t$-plane, so it looks like this term is
measuring the amount of effort it takes to rotate the surface in the
$z,t$-plane. It would be very interesting to understand this observation
better.

Since our interpretation of the gravity action in terms of gate counting is
more on the qualitative side, in the following we want to propose another way
of arriving at (8).

###  4.2 Relation to kinematic space

In the above, we have often tacitly assumed that the information about the
bulk surface $z=\rho(t)$ is encoded locally in the boundary theory. However,
as our discussion of flows shows, it is highly questionable whether this is a
reasonable assumption. A better way to encode the information of the surface
$z=\rho(t)$ in the boundary theory is through pairs of points
$(t_{1}(t),t_{2}(t))$ (with $x$=0) on the boundary, such that the geodesic
that starts at $t_{1}(t)$ and ends at $t_{2}(t)$ is tangent to the bulk
surface at the point $(z=\rho(t),t,0)$, see figure 2.

Figure 2: We can parametrize a generic bulk curve $\rho(t)$ by the pairs of
boundary points $(t_{1}(t),t_{2}(t))$, such that a bulk geodesic connecting
these two points is tangent to the bulk curve at $z=\rho(t)$. This way, the
profile $\rho(t)$ is encoded as a path in kinematic space, the space of bulk
geodesics.

This construction has the benefit of being covariant, and viewing Euclidean
time as another spatial coordinate, these geodesics encode precisely the
entanglement wedges which touch the surface but do not cross it. In other
words, they precisely encode the information about those regions of spacetime
we try to omit in our bulk path integral construction. One can ask whether
there is a natural geometry associated to the pairs of points of this type,
and the answer is yes. Conformal invariance produces a natural metric on the
space of pairs of points, also known as kinematic space Czech:2015qta . For
the case at hand it is given up to an undetermined constant prefactor by the
2d de Sitter metric

$ds_{ks}^{2}=\frac{-dt_{1}\,dt_{2}}{(t_{1}-t_{2})^{2}}.$ (44)

In the spirit of defining complexity by assigning a metric to a group of
transformations Nielsen1133 ; Jefferson:2017sdb ; Chapman:2017rqy , we can now
ask what the length of the path in this geometry associated with $\rho(t)$ is.
To compute it explicitly, we need the explicit form of $t_{1}(t)$ and
$t_{2}(t)$. These are given by

$t_{1,2}(t)=t+\rho\dot{\rho}\pm\rho\sqrt{\dot{\rho}^{2}+1}.$ (45)

Consider now the action

$S_{ks}\sim\int\frac{dx}{\rho}ds_{ks}(t),$ (46)

where we included the coordinate $x$ in units of the cutoff $\rho$, and the
distance $ds$ obtained from (44) upon inserting (45). This results in

$S_{ks}\sim\int
dtdx\left|\frac{\rho\ddot{\rho}+(1+\dot{\rho}^{2})}{\rho^{2}(1+\dot{\rho}^{2})}\right|,$
(47)

which agrees precisely with the bulk action in the form (8) as long as
$\ddot{\rho}\geq-\rho^{-1}(1+\dot{\rho}^{2})$. This is related to the fact
that the kinematic space is a Lorentzian manifold and the condition in
question is the one that one moves there along a timelike path.

This strongly suggests that the relevant circuit geometry for these types of
finite bulk surface computations is a version of kinematic space444Note that
this analysis did not include corner terms contributions to the action.
However, in the case of open bulk curves, in the kinematic space framework
there are additional contributions that we have not included. It would be
certainly interesting to see if they reproduce the corner terms. We would like
to thank Bartek Czech for bringing this up.. Note that on-shell, (47) vanishes
exactly for the semi-circular arcs that solve (11), as they are also geodesics
in AdS-space. In other words, for these solutions the path traversed in
kinematic space shrinks to a point. We will come back to this point in section
5.

Note that alternatively one may use the standard kinematic space prescription
built around entanglement entropy of intervals on constant $t$ time slices.
The metric (44) is the same but now with $t_{1}$ and $t_{2}$ replaced simply
by $x_{1}$ and $x_{2}$ with

$x_{1,2}(t)=\pm\rho(t).$ (48)

Using again (46) gives this time

$S_{ks^{\prime}}\sim\int dtdx\frac{|\dot{\rho}|}{\rho^{2}}.$ (49)

This is clearly a different expression than (47), which however bears a
striking similarity with the gate counting approach of Nielsen1133 ;
Jefferson:2017sdb ; Chapman:2017rqy  when the latter uses a Manhattan norm.

Let us mention that generalizing the kinematic space consideration leading to
(47) to more complicated geometries is not obvious as minimal geodesics do not
necessarily penetrate the whole spacetime. In the case of geodesics computing
the entanglement entropy, these are entanglement shadows
Balasubramanian:2014sra  and they appear, for example, in the case of double-
sided black holes.

Finally, let us mention that the relation between kinematic space and
complexity was explored earlier in two different instances in Abt:2017pmf  and
Chen:2020nlj , however, these proposals are distinct from ours and use a
standard entanglement-based kinematic space.

##  5 Outlook

In this paper we have discussed the idea that finite spacetime regions
correspond to quantum circuits with a complexity given by the on-shell value
of the gravitational action. We found several intriguing results, but much
more work remains to be done to put our results on a firmer footing. Perhaps
the most pressing of these is to find a more precise circuit interpretation
along the lines we discussed in the previous section. Some other obvious
aspects to explore are the impact of counterterms, higher derivative terms and
matter fields on the computations. We list some further open issues and ideas
for future work below.

### Relation with tensor network renormalization

As we described in the introduction, one of the long-standing questions in
quantum information aspects of holography is understanding the relation
between holographic geometries and MERA tensor network. With the advent of
McGough:2016lol , it is natural to expect that the tensor network description
should include $T\bar{T}$ deformations in some way. It would be certainly very
interesting to explore to what extent this is the case in the existing
formulation of MERA and to what extent this calls for an alternative approach,
see also Kruthoff:2020hsi . In particular, the discussion of geometric
interpretation of MERA in Milsted:2018san  interprets latticized CFT on a
hyperbolic disc geometry as intertwined layers of MERA and Euclideons
(Euclidean time evolution). On the contrary, in the present paper, the
hyperbolic disk geometry of the maximum volume bulk time slice arises from the
absence of Euclidean time evolution in the circuit defined by (43).

### Global AdS and trivial initial state

It is straightforward to repeat our computations in global AdS, as opposed to
the Poincaré patch of AdS. There are no major conceptual changes, except that
we can now choose a smooth surface without the need to pick an initial state.
Stated differently, we have chosen a trivial initial state in the CFT with
infinite cutoff, or equivalently, we have a no-boundary type construction of
the state at later times. The optimization proceeds exactly as in Poincaré
coordinates, and complexity is optimized if the spacetime region collapses
onto an equal time disc, with complexity proportional to the volume of the
disc.

### Choice of time slice

In our proposal we chose to bound our bulk spacetime subregion by constant
Poincaré time slices. One could ask whether we could have made a different
choice of boundaries and still have gotten an action that could be reasonably
interepreted as circuit complexity. In the first place, our choice satisfies
the physical requirement that the complexity of the circuits which do nothing
(no evolution in Euclidean time, or change in cutoff) should be zero. This
would not have be the case had we bounded our region by say constant $z$
surfaces rather than constant $t$. We also could have considered a wiggly
spacelike boundary, which along with our timelike boundary would be
dynamically determined by extremisation of the gravitational action. However,
for Poincaré AdS we showed in section 3.1 that constant $t$ slices are just
such an extremum of the action, so one can consider them to have been
dynamically determined from the perspective of this modified proposal. It is
not clear whether this modification would always give sensible results in
general asymptotically AdS spacetimes, and this is an interesting direction to
consider for future work.

For stationary spacetimes, it seems reasonable to take fixed time slices, but
it is not clear what to do for more general spacetimes. States in gravity are
not associated to a unique time slice. In some sense, states are associated to
complete causal diamonds in the Lorentzian signature. There is therefore no
canonical choice of initial and final time slices which bound the spacetime
region. Our proposal is to use slices with vanishing extrinsic curvature $K$,
as these are covariantly defined, and lead to a vanishing contribution of the
Gibbons-Hawking boundary term. This choice will give rise to corner
contributions, but those seem unavoidable for any choice, and as we saw in the
case where the spacetime region collapses to a disc, they are a feature rather
than a bug. One could, alternatively, try to extend the spacetime region
indefinitely into the past or future, and subtract the contributions of these
semi-infinite pieces later, but this procedure has exactly the same ambiguity
in it. It would be interesting to have a better understanding of the various
choices one can make for the future and past boundaries and what the
implications of these choices are. It might for example also be natural to
take time slices with constant scalar curvature as complexity is locally
extremized for that choice of time slice.

### A finite deformation of Liouville

The effective action for a finite bounded region in AdS is of independent
interest, as it computes the partition function for the CFT with a cutoff and
particular curved manifolds. In the limit where the bounded region approaches
the boundary of AdS, we recover the CFT partition function (including
divergent terms), which in 2d is given by the Polyakov action, and in
conformal gauge becomes the Liouville action. It is interesting to see that
(20) is apparently a finitely deformed version of Liouville theory for a
space-independent Liouville field $\varrho(u)=\exp(-\phi(u))$. If we insert
this and take $\phi(u)\rightarrow\infty$, we indeed recover Liouville theory,
see also the discussion in Boruch:2020wax . One might think that (20)
describes a finite $T\bar{T}$ deformation of Liouville theory and it would be
interesting to make that connection precise. A possible route to address this
matter is to cast the on-shell action in a form involving the scalar curvature
of the cutoff surface, which seems feasible in the ADM formalism, compare with
Polyakov’s non-local form of the effective action, and identify the relevant
deformation.

### Other dimensions

In higher dimensions, the computation is more or less the same, and we will
not present the relevant details here. An exception is AdS2, where after a
partial integration the action becomes proportional to $I\sim\int
dt\rho^{-1}$, which suggests that the coarse graining operation has no cost
associated to it. This is perhaps a consequence of the peculiar nature of the
AdS2/CFT1 correspondence, where AdS2 is merely dual to the ground states of
the CFT1 and is of limited relevance. It would be interesting to repeat the
computation for JT gravity Teitelboim:1983ux ; Jackiw:1984je  and to compare
to flows in spaces of Hamiltonians, which are much easier to control than
$T\bar{T}$ deformations in higher dimensions, and might lead to a more precise
gate counting interpretation.

In $T\bar{T}$-deformed quantum mechanics, the Hamiltonian is mapped to a
function of itself Gross:2019ach ; Gross:2019uxi ,

$H\mapsto f(H).$ (50)

Suppose we wish to quantify the complexity of the circuit created by Euclidean
time evolution,

$U(t)=\exp(-H\,t).$ (51)

Given $U(t)$ of the undeformed theory, we in principle know the operator
$U_{f}(t)$ of the deformed theory,

$U_{f}(t)=\exp(-tf(-\partial_{t}\log U(t))),$ (52)

but even given this simple relation, it is not clear how to relate the
complexities of $U(t)$ and $U_{f}(t)$.

One puzzle arises when combining complexity and holographic $T\bar{T}$.
Increasing the $T\bar{T}$ deformation is dual to bringing in the cutoff
surface, which reduces the volume of the maximal boundary anchored volume
slice, and by the CV conjecture would say implies that the complexity of the
state similarly reduces. Assuming the volume of the maximal volume bulk slice
monotonically decreases as the boundary is brought in, then this implies that
the complexity is monotonically decreasing too under the flow. Is there
something special about the holographic $T\bar{T}$ deformation such that the
complexity of geometric states monotonically decreases under its flow, or is
the CV proposal incorrect at finite cutoff?

### Lorentzian geometries

We could repeat our computation in Lorentzian signature, but then several new
features arise. First, there is the qualitative difference of whether or not
$z=\rho(t)$ describes a timelike or spacelike surface. In the timelike case
the region is delimited by the lightfronts $t=\pm z$, and the on-shell action
takes the form

$\displaystyle I=\frac{2}{\kappa}\int_{\partial
M}d^{2}x\,\frac{\rho\ddot{\rho}+(1-\dot{\rho}^{2})}{\rho^{2}(1-\dot{\rho}^{2})}$
(53)

Integration by parts yields

$\displaystyle I=\frac{2}{\kappa}\int_{\partial
M}d^{2}x\left[\frac{1}{\rho^{2}}-\frac{1}{2}\left(\text{log}(1-\dot{\rho})-\text{log}(1+\dot{\rho})\right)\right]$
(54)

It is easy to see that this expression diverges in the limit
$\dot{\rho}\rightarrow\pm 1$, i.e. when the surface is tangent to the
lightfronts $t=\pm z$. It is possible to properly define gravitational actions
in the presence of null boundaries Lehner:2016vdi , and in order for our
proposal to make sense we should modify it so that in the null limit it
approaches the answer of Lehner:2016vdi . With this modification we would then
be in agreement with the complexity equals action proposal.

If we start with a spacelike surface and start optimizing, then there are two
possibilities, we either find a constant scalar curvature surface, or we
encounter the same null boundaries as in the previous timelike case. Which of
the two optimizes the gravitational action depends on whether we choose $+S$
of $-S$ to optimize, and since it is $e^{iS}$ which appears in the path
integral, it is not a priori clear which one of the two we should take in the
absence of a precise gate counting interpretation. One would be inclined
though to pick the sign such that the term proportional to $1/\rho^{2}$ and
independent of $\dot{\rho}$ has a positive sign so that time evolution at
fixed cutoff has positive complexity. Regardless, we seem to universally find
either constant scalar curvature surfaces or null surfaces as extrema of the
extremization problem.

### BTZ black hole

Based on general arguments, there are several key features that measures of
complexity should possess, such as aforementioned asymptotic linear growth in
time in black hole backgrounds and the switchback effect Susskind:2014jwa . As
a first heuristic check, we can investigate constant scalar curvature slices
(with the right value for the scalar curvature) in the BTZ black hole. In
Kruskal coordinates, the BTZ black hole looks like

$\displaystyle
ds^{2}=-\frac{4\,du\,dv}{(1+uv)^{2}}+\frac{(1-uv)^{2}}{(1+uv)^{2}}d\phi^{2}$
(55)

with the asymptotic AdS boundaries located at $uv=-1$. The relevant constant
curvature slices turn out to take the simple form $uv+\lambda u+\mu v-1=0$.
Consider the special case $uv+(u+v)/\sinh\xi-1=0$ which intersects the
boundary at $u=e^{\xi}$, $v=-e^{-\xi}$ and on the other boundary of the
eternal black hole at the point obtained by interchanging $u,v$. Shifting
$\xi$ is therefore like shifting time upwards on both asymptotic boundaries,
and we are interested in the behavior at late $\xi$. The midpoint of the slice
is at $u=v=\tanh\xi/2$, which indeed moves towards the singularity at $uv=1$
as $\xi\rightarrow\infty$. Therefore, constant scalar curvature slices do
correctly probe the growing Einstein-Rosen bridge. The optimal spacetime
region in this case is the region between the maximal volume slice with $K=0$
(which is where we propose to end the spacetime region, as discussed above)
and the constant curvature slice. We have not computed the gravitational
action associated to this region but expect it to reproduce the required late
time growth. As the maximal volume slice is also explicitly computable
Carmi:2017jqz , we leave this interesting exercise to future work.

### Higher curvature corrections

We have proposed that the complexity of the circuit that maps between ground
states in two EFTs with different finite cutoffs is given by the on-shell
gravitational action. Considering the effect of higher curvature corrections
on the gravitational action, and therefore complexity, would be a natural
extension to this proposal. Higher curvature corrections to the holographic
complexity=volume proposal were recently studied in Hernandez:2020nem . The
simplest example to study is Gauss-Bonnet (GB) gravity in AdS5. The GB
correction

$\mathcal{L}^{GB}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$
(56)

is a constant in the vacuum AdS geometry we have studied in this work, so
simply rescales the contribution from the volume of $M$ to the on-shell
action. In general, however, we expect non-trivial contributions from the
correction to the boundary Lagrangian, and from the bulk Lagrangian
$\mathcal{L}^{GB}$ for perturbed geometries. Seeing whether the resulting on-
shell action could be interpreted as the complexity of a circuit in
$T\bar{T}$-deformed CFT4 with $a\neq c$ would be interesting.

To our knowledge the generalization of holographic $T\bar{T}$ to higher
curvature gravity has not been studied. There is general method to determine
the deformation to a holographic CFT needed to put the dual gravity theory at
finite cutoff Taylor:2018xcy ; Hartman:2018tkw . To start we note that in a
theory with only one dimensionful parameter $\mu$ (assuming it to have no
spacetime dependence) such as a $T\bar{T}$ deformed CFT, the effective action
changes under infinitesimal length rescaling as

$\Delta_{\mu}\mu\partial_{\mu}W=\int
d^{d}x\sqrt{\gamma}\langle\operatorname{Tr}T\rangle$ (57)

where $\Delta_{\mu}$ is the scaling dimension of $\mu$. The flow of the
boundary action under changing $\mu$ is thus determined by the trace of the
boundary stress tensor, which is related to the bulk Brown-York stress tensor
through $T_{ij}=r_{c}^{d-2}\tilde{T}_{ij}$. The essence of the method is to
use the Hamiltonian constraint to eliminate extrinsic curvature terms in the
Brown-York stress tensor appearing in (57). Following this procedure for pure
Einstein gravity gives the flow equation

$\frac{\partial W}{\partial\mu}=\int
d^{d}x\sqrt{\gamma}\left(T^{ij}T_{ij}-\frac{1}{d-1}(T^{i}_{i})^{2}\right)$
(58)

This is the field theory deformation flow equation for Einstein gravity at
finite cutoff. When we add higher curvature corrections, the method of
substituting out extrinsic curvature terms in the Brown-York stress tensor for
powers of the stress tensor using the Hamiltonian constraint does not change,
but the Brown-York stress tensor and the radial Hamiltonian constraint do
Davis:2002gn , and the final result will differ from (58).

### Relation to entanglement wedge reconstruction

Finally, it is interesting to ask whether it is a mathematical coincidence or
not that the solutions (12) to (11) are semicircular arcs, just like geodesics
in the Poincaré AdS-background, even though in our setup they describe the
embeddings of co-dimension one surfaces, not entanglement entropy. In section
4.2, we addressed this question from a kinematic space perspective. There, we
showed that a generic bulk profile $\rho(t)$ can as well be described as a
path in kinematic space (neglecting a possible $x$-dependence as throughout
the paper), but equations (11) and (12) enforce this path to shrink down to a
length zero point for fixed boundary conditions $t_{i},t_{f},z_{i},z_{f}$.

Assuming that our results can be generalised to the Lorentzian case, it will
be interesting to investigate whether these semicircular embeddings are a
consequence of entanglement wedge reconstruction or entanglement wedge nesting
Wall:2012uf ; Czech:2012bh ; Akers:2016ugt . For example, one might imagine
that the optimization of the path integral (subject to the boundary conditions
$t_{i},t_{f},z_{i},z_{f}$) drives the bulk surface as deep into the bulk as
possible until it leaves the entanglement wedge and cannot move further. In
the fully Lorentzian case, it would be interesting to determine whether this
only reproduces the extremal surface at the edge of the entanglement wedge, or
also its null boundaries. At least extremal area surfaces are expected to play
a role of special importance in quantum gravity for quite generic reasons
Camps:2019opl . As said in section 2.2, such semicircular boundaries have also
been found in Erdmenger:2014xya  as solutions of an AdS/BCFT toy-model with
non-trivial matter content living in the worldvolume of the end of the world
brane. In that paper, it was shown as a consequence of physical energy
conditions on the worldvolume matter fields that the corresponding branes
generally have to be extremal surface barriers in the sense of
Engelhardt:2013tra , and the branes with semicircular embedding profile where
precisely the ones staying as close to the boundary as possible without
violating this condition.

While all these different observations seem to point to a nontrivial quantum
information theoretic reason for the embeddings (12) being the correct ones,
we leave further investigation of this to future work. Besides extending our
work to the Lorentzian case, investigating similar setups in higher dimensions
or on nontrivial backgrounds such as BTZ may yield further insight.

###### Acknowledgements.

We would like to thank Shira Chapman and Ignacio Reyes for being involved in
the initial part of this collaboration and Bartek Czech for useful
discussions. The Gravity, Quantum Fields and Information (GQFI) group at AEI
is supported by the Alexander von Humboldt Foundation and the Federal Ministry
for Education and Research through the Sofja Kovalevskaja Award. AR is
supported by the Stichting Nederlandse Wetenschappelijk Onderzoek Instituten
(NWO-I). JdB is supported by the European Research Council under the European
Unions Seventh Framework Programme (FP7/2007-2013), ERC Grant agreement ADG
834878\. The work of MF was supported by the Polish National Science Centre
(NCN) grant 2017/24/C/ST2/00469 until November 16th 2020 and through the
grants SEV-2016-0597 and PGC2018-095976-B-C21 from MCIU/AEI/FEDER, UE since
December 1st 2020. RC acknowledges support from the Netherlands Organisation
for Scientific Research (NWO-I). SH holds a fellowship from the Ramon Areces
Foundation (Spain).

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