Operator Complexity for Quantum Scalar Fields and Cosmological Perturbations
S. Shajidul<EMAIL_ADDRESS>Chandan Jana
<EMAIL_ADDRESS>Bret
<EMAIL_ADDRESS>
a High Energy Physics, Cosmology & Astrophysics Theory Group
and
The Laboratory for Quantum Gravity & Strings
Department of Mathematics and Applied Mathematics,
University of Cape Town, South Africa b Mandelstam Institute for Theoretical
Physics, Witwatersrand University, Johannesburg, South Africac Department of
Physics, Pacific Lutheran University, Tacoma, WA 98447

###### Abstract

We calculate the operator complexity for the displacement, squeeze and
rotation operators of a quantum harmonic oscillator. The complexity of the
time-dependent displacement operator is constant, equal to the magnitude of
the coherent state parameter, while the complexity of unitary evolution by a
generic quadratic Hamiltonian is proportional to the amount of squeezing and
is sensitive to the time-dependent phase of the unitary operator. We apply
these results to study the complexity of a free massive scalar field, finding
that the complexity has a period of rapid linear growth followed by a
saturation determined by the UV cutoff and the number of spatial dimensions.
We also study the complexity of the unitary evolution of quantum cosmological
perturbations in de Sitter space, which can be written as time-dependent
squeezing and rotation operators on individual Fourier mode pairs. The
complexity of a single mode pair at late times grows linearly with the number
of e-folds, while the complexity at early times oscillates rapidly due to the
sensitivity of operator complexity to the phase of unitary time evolution.
Integrating over all modes, the total complexity of cosmological perturbations
scales as the square root of the (exponentially) growing volume of de Sitter
space, suggesting that inflation leads to an explosive growth in complexity of
the Universe.

###### Contents

  1. 1 Introduction
  2. 2 Operator Complexity
  3. 3 Displacement Operator Complexity
  4. 4 Squeezing Operator Complexity
    1. 4.1 Free Harmonic Oscillator
    2. 4.2 Inverted Harmonic Oscillator
    3. 4.3 General Squeezing and Rotation
  5. 5 Scalar Field Complexity
    1. 5.1 Free Scalar Field
    2. 5.2 Quantum Scalar Cosmological Perturbations
  6. 6 Discussion
  7. A Alternative Matrix Representations

##  1 Introduction

As a concept, quantum circuit complexity is a measure of the number of simple
quantum operations necessary to build a given unitary target operator – which
transforms some reference state into a target state – from the identity. For
quantum systems in which the set of available operators form a continuum,
quantum circuit complexity (hereafter denoted simply as complexity) can be
interpreted in a geometric way as the length of a minimal geodesic in the
space of operators [1, 2, 3]. Because of their wide-ranging applications, we
are interested in the class of quantum systems built from the quantum harmonic
oscillator. In order to make progress in finding the complexity of a target
operator, it is often necessary to characterize the action of the operator on
a restricted subspace of reference and target states, such as Gaussian
wavefunctions, as in e.g. [4, 5, 6, 7, 8]. Such restrictions, however, make it
difficult to make general statements about target unitaries, and can be
difficult to extend to quantum field theory or interacting systems.

In this paper, we will consider the complexity of the target unitary operator
directly, following the group manifold approach of [9, 10, 11, 12] (see also
[13]) and extending it to the quantum harmonic oscillator. In this
formulation, the target unitary is generated from a set of fundamental
operators, which form a Lie algebra. Upon the choice of a suitable right-
invariant metric, the minimal geodesic on the corresponding group manifold of
the Lie algebra is found as a solution to the Euler-Arnold equation [14, 9,
10]. The advantage of this group-theoretic approach is that the geometry is
determined by the generators of the Lie algebra (up to the choice of the
metric), and is manifestly independent of the reference and target states.
Further, this approach can potentially be extended to the study of complexity
for interacting systems by suitably generalizing the Lie algebra of the
fundamental operators beyond those considered here.

The target unitaries we consider here will be the displacement, squeeze and
rotation operators of a quantum harmonic oscillator. These operators are not
only a starting point towards applying this formalism for complexity to more
complex systems, they are also interesting from the perspective of continuous
variable quantum information [15, 16, 17]. In particular, coherent and
squeezed states of light are ubiquitous ingredients in quantum optics [18],
and may play a supporting role in enhancing quantum computation algorithms
(see e.g. [15, 16, 19]). Squeezed states also show up in interesting
applications in quantum field theories on curved backgrounds (such as [20, 21,
22, 23]), and these states and the corresponding operators that generate them
can lead to new quantum information theoretic perspectives on phenomena such
as cosmological perturbations and Hawking radiation [24, 25, 26, 27, 28].
Since the complexity of these operators has been studied in state-based
approaches before, we are able to compare our state-independent to computing
complexity with those methods.

Complexity is particularly interesting when applied to many-body systems,
where it may serve as a diagnostic for quantum chaos [29, 30, 31, 32, 8, 33,
34, 35] or topological phase transitions [36, 37, 38]. To this end, the
operator complexity of a single quantum harmonic oscillator is readily
extended to a free massive quantum scalar field by writing the field as a sum
over (decoupled) Fourier modes up to a UV cutoff. While still integrable, this
simple system can serve as a testing ground for many-body operator complexity
as well as uncover general features of the UV divergences that appear in
complexity when moving to quantum field theory. Another particularly
interesting model is the unitary evolution of quantum cosmological
perturbations in the presence of a time-dependent background. At leading
order, the Fourier modes of quantum cosmological perturbations behave as
quantum harmonic oscillators with time-dependent frequencies [39]; the
corresponding unitary time evolution operator can be written as a product of
time-dependent squeezing and rotation operators [20, 21, 22, 23, 26]. The
state complexity of cosmological perturbations has been studied before [24,
25, 28], but being state-based, these calculations all require some specific
choices for reference and target states. Operator complexity instead is
independent of the particular reference and target states, resulting in a more
general statement about the complexity of the unitary evolution of the
cosmological background.

The rest of the paper is organized as follows. In Section 2 we review the
state-independent method of operator complexity from [9, 10]. In Section 3 we
apply this technique to the displacement operator, finding that the operator-
space geometry is 3-dimensional hyperbolic space, and the complexity is simply
proportional to the magnitude of the coherent state parameter of the
displacement operator. In Section 4 we consider a generic unitary that is
quadratic in the creation and annihilation operators, which can be decomposed
as a product of the squeeze and rotation operators. After describing the
geometry, we explore the upright and inverted harmonic oscillators as simple
examples before finding the complexity for a generic product of squeeze and
rotation operators. In Section 5 we extend our analysis to quantum field
theories with a UV cutoff. Specifically, we first consider a massive free
scalar field, calculating the resulting complexity of unitary time evolution
and its leading UV divergence. We then analyze the complexity of unitary
evolution of quantum cosmological perturbations in a de Sitter background for
individual Fourier modes as well as the integral over all Fourier modes,
comparing our results to corresponding calculations of state complexity. In
Section 6, we conclude with a discussion of our results and future directions.
Finally, in Appendix A we demonstrate that our results for the complexity for
these groups is independent of the choice of matrix representation.

##  2 Operator Complexity

We start by reviewing the continuum circuit complexity construction we will
use throughout the rest of the paper; see also [9, 10]. Our quantum circuit is
a unitary operator that transforms a given reference state $|\psi\rangle_{\rm
R}$ to a specified target state $|\psi\rangle_{\rm T}$

$\displaystyle|\psi\rangle_{\rm T}=\hat{\mathcal{U}}_{\rm target}\
|\psi\rangle_{\rm R}$ (1)

The target unitary $\hat{\mathcal{U}}_{\rm target}$ consists of a continuum of
operations parameterized by a parameter $`s^{\prime}$ that controls the level
of the circuit

$\displaystyle\hat{\mathcal{U}}_{\rm target}={\mathcal{P}}\,{\rm
exp}\left[-i\int_{0}^{1}V^{I}(s)\hat{\mathcal{O}}_{I}\ ds\right]\,,$ (2)

where the operators $\\{\hat{\mathcal{O}}_{I}\\}$ are some set of fundamental
operators, the $V^{I}(s)$ are vectors that specify the path of the sequence of
operators, and the path-ordering ${\mathcal{P}}$ ensures that the operators
are applied sequentially from $s=0$ to $s=1$. It is convenient to introduce
the $s$-dependent unitary

$\displaystyle\hat{U}(s)={\mathcal{P}}\,{\rm
exp}\left[-i\int_{0}^{s}V^{I}(s^{\prime})\hat{\mathcal{O}}_{I}\
ds^{\prime}\right]\,,$ (3)

which is a solution to the differential equation

$\displaystyle\frac{d\hat{U}(s)}{ds}=-iV^{I}(s)\ \hat{\mathcal{O}}_{I}\
\hat{U}(s)\,,$ (4)

subject to the boundary conditions

$\displaystyle\hat{U}(0)=\mathds{1}\,\hskip 14.45377pt\mbox{and}\hskip
14.45377pt\hat{U}(1)=\hat{\mathcal{U}}_{\rm target}\,.$ (5)

In principle, there are many different paths (“circuits”) $V^{I}(s)$ that can
be used to build the target unitary through (4). In order to identify the
“optimal” path, we will characterize each path realizing the unitary (3) by
its circuit depth444There are other choices for the cost function of the
circuit depth [1, 2, 3, 4]. We will choose the geodesic cost function for its
relative simplicity and geometric interpretation.

$\displaystyle{\mathcal{D}}\left[V^{I}\right]=\int_{0}^{1}\sqrt{G_{IJ}V^{I}V^{J}}\
ds\,,$ (6)

corresponding to the geodesic length in the space of operators. The metric
$G_{IJ}$ identifies the operational “cost” or weight to building the path with
any particular operator $\hat{\mathcal{O}}_{I}$; as we will discuss, while a
natural choice is the Cartan-Killing form of the Lie algebra of the operators
$\\{\hat{\mathcal{O}}_{I}\\}$, $G_{IJ}=K_{IJ}$, leading to a bi-invariant
metric, this choice will not be possible for the non-compact groups we are
interested in here, as we will discuss in the following sections. Instead, for
the examples we consider in this paper we will choose a flat metric on the
operators $G_{IJ}=\delta_{IJ}$, leading to a right-invariant metric, so that
there are no preferred directions in operator space (though we leave $G_{IJ}$
as arbitrary for the remaining parts of this section, for clarity). More
generally, the metric may be constructed either phenomenologically, by
including the difficulty of preparing a particular gate in the lab, or by
including a theoretical bias for “simple” versus “composite” operators. The
choice of metric $G_{IJ}$, together with (4), leads to a notion of distance on
this space [4]

$\displaystyle ds^{2}=G_{IJ}\ dV^{I}dV^{J}$ (7)

where

$\displaystyle dV^{I}={\rm Tr}\left(d\hat{U}\ \hat{U}^{-1}\
\hat{\mathcal{O}}_{I}^{\dagger}\right)\,.$ (8)

The optimal quantum circuit is the one with minimal circuit depth, so that the
complexity of (2) is the minimization of the circuit depth

$\displaystyle{\mathcal{C}}_{\rm
target}=\min_{\\{V^{I}\\}}{\mathcal{D}}\left[V^{I}\right]=\min_{\\{V^{I}\\}}\int_{0}^{1}\sqrt{G_{IJ}V^{I}V^{J}}\
ds\,,$ (9)

over all possible paths $\\{V^{I}(s)\\}$ realizing the target operator (2).
The minimal path $V^{I}(s)$ is therefore a geodesic on the space (7), which
solves the Euler-Arnold equation [9, 10, 14]

$\displaystyle G_{IJ}\frac{dV^{J}}{ds}=f_{IJ}^{P}\ V^{J}G_{PL}V^{L}\,,$ (10)

where the $f_{IJ}^{P}$ are the structure constants of the operators,

$\displaystyle\left[\hat{\mathcal{O}}_{I},\hat{\mathcal{O}}_{J}\right]=if_{IJ}^{P}\
\hat{\mathcal{O}}_{P}\,.$ (11)

In the following sections, we will proceed as follows. First, we identify the
target unitary $\hat{\mathcal{U}}_{\rm target}$ and select a set of basis
operators $\\{\hat{\mathcal{O}}_{I}\\}$, with associated Lie group, that we
use to construct this unitary. Paths that solve (10) define a set of geodesics
$\\{V^{I}(s)\\}$ on this space. We then restrict this set of geodesics to
those that realize the target unitary through (4) and the boundary conditions
(5). Finally, we use the resulting optimal construction of the unitary to
calculate the complexity (9), and determine its dependence on the parameters
of the target unitary.

##  3 Displacement Operator Complexity

As a simple application, we begin by analyzing the circuit complexity
associated with the _displacement operator_

$\displaystyle\hat{\mathcal{U}}_{\rm target}=\hat{\mathcal{D}}(\alpha)={\rm
exp}\left[\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}\right]\,,$ (12)

which generates the coherent state
$|\alpha\rangle=\hat{\mathcal{D}}(\alpha)|0\rangle$ from the vacuum. More
generally, we can consider the _time-dependent displacement operator_

$\displaystyle\hat{\mathcal{U}}_{\rm
target}(t)=\hat{\mathcal{D}}(\alpha,t)=e^{i\hat{H}_{0}t}\hat{\mathcal{D}}(\alpha)e^{-i\hat{H}_{0}t}=\hat{\mathcal{D}}\left(\alpha
e^{-i\omega t}\right)\,,$ (13)

in which the displacement operator (12) is time-evolved by a free Hamiltonian
$\hat{H}_{0}=\omega\hat{a}^{\dagger}\hat{a}$. The displacement operator is a
standard starting point for many approaches to calculating complexity [5, 40,
34, 41], and as such will provide a good place to start in applying the
formalism developed in Section 2 as applied to infinite-dimensional Hilbert
spaces.

A natural set of fundamental operators to use for generating (12) through the
construction (2)

$\displaystyle\hat{\mathcal{D}}(\alpha,t)={\mathcal{P}}\ {\rm
exp}\left[-i\int_{0}^{1}V^{I}(s)\hat{e}_{I}\ ds\right]\,,$ (14)

are the Hermitian operators

$\displaystyle\hat{e}_{1}=\frac{1}{\sqrt{2}}\left(\hat{a}+\hat{a}^{\dagger}\right),\hskip
14.45377pt\hat{e}_{2}=\frac{i}{\sqrt{2}}\left(\hat{a}-\hat{a}^{\dagger}\right),\hskip
14.45377pt\hat{e}_{3}=\hat{\mathds{1}}\,,$ (15)

which obey the standard Heisenberg Lie algebra
$[\hat{e}_{1},\hat{e}_{2}]=-i\hat{e}_{3}$, with all other commutators
vanishing. As a result, the only non-zero structure constant (up to
permutations) is $f_{12}^{3}=-1$, so that the Cartan-Killing form vanishes
$K_{IJ}=f_{IK}^{L}f_{JL}^{K}=0$. In terms of the $\\{\hat{e}_{I}\\}$, we can
write our target operator as

$\displaystyle\hat{\mathcal{U}}_{\rm target}(t)={\rm exp}\left[\sqrt{2}i\ {\rm
Im}[\alpha(t)]\ \hat{e}_{1}+\sqrt{2}i\ {\rm Re}[\alpha(t)]\
\hat{e}_{2}\right]\,.$ (16)

The $s$-dependent vectors $V^{I}(s)$ in (14) parameterize the path through the
space of operators, from the identity (at $s=0$) to the target operator (at
$s=1$). The associated circuit depth (6) measures the geodesic length of any
particular path, given a metric $G_{IJ}$ on the space of operators. Since the
Cartan-Killing form vanishes, a natural choice for the metric is a diagonal
metric $G_{IJ}=\delta_{IJ}$ so that the infinitesimal circuit depth
$G_{IJ}V^{I}V^{J}=\left(V^{1}\right)^{2}+\left(V^{2}\right)^{2}+\left(V^{3}\right)^{2}$
is non-zero.

The path from the identity to our target state with the smallest circuit depth
is thus obtained as a solution to the Euler-Arnold equation (10), which
becomes

$\displaystyle\frac{dV^{1}}{ds}$ $\displaystyle=$
$\displaystyle-V^{2}V^{3}\,;$ $\displaystyle\frac{dV^{2}}{ds}$
$\displaystyle=$ $\displaystyle V^{1}V^{3}\,;$ (17)
$\displaystyle\frac{dV^{3}}{ds}$ $\displaystyle=$ $\displaystyle 0\,.$

Combining the first two equations of (17), it is straightforward to see that
$(V^{1})^{2}+(V^{2})^{2}$ is independent of $s$. Since the third equation of
(17) leads to $V^{3}(s)=c_{3}$, a constant, we have the following general
solution for a depth-minimizing path

$\displaystyle V^{1}(s)$ $\displaystyle=$ $\displaystyle
v_{1}\cos(v_{3}s)+v_{2}\sin(v_{3}s)\,;$ $\displaystyle V^{2}(s)$
$\displaystyle=$ $\displaystyle v_{1}\sin(v_{3}s)-v_{2}\cos(v_{3}s)\,;$ (18)
$\displaystyle V^{3}(s)$ $\displaystyle=$ $\displaystyle v_{3}\,;$

where $v_{1},v_{2},v_{3}$ are constants that will be determined by boundary
conditions. The resulting circuit complexity along this minimal path is then
simply

$\displaystyle{\mathcal{C}}_{\rm
target}=\int_{0}^{1}\sqrt{G_{IJ}V^{I}(s)V^{J}(s)}\
ds=\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}\,.$ (19)

Before imposing the boundary conditions to fix the constants $\\{v_{I}\\}$ in
terms of the target unitary operator, let us make a simplification in our
construction of the unitary operator in terms of the $\hat{e}_{I}$. Since the
operator $\hat{e}_{3}$ is a center that commutes with all other elements, its
contribution to (14) is just an overall time-independent phase when operating
on states

$\displaystyle{\rm exp}\left[-i\int_{0}^{1}V^{I}\hat{e}_{I}\
ds\right]\,|\psi\rangle_{\rm R}$ $\displaystyle=$ $\displaystyle{\rm
exp}\left[-i\int_{0}^{1}\left(V^{1}(s)\hat{e}_{1}+V^{2}(s)\hat{e}_{2}\right)ds\right]e^{-iv_{3}\hat{\mathds{1}}}\,|\psi\rangle_{\rm
R}$ (20) $\displaystyle=$ $\displaystyle{\rm
exp}\left[-i\int_{0}^{1}\left(V^{1}(s)\hat{e}_{1}+V^{2}(s)\hat{e}_{2}\right)ds\right]\,e^{-iv_{3}}\,|\psi\rangle_{\rm
R}\,.$

We will therefore set $v_{3}=0$ to avoid this additional phase ambiguity. The
resulting solutions (18) for the depth-minimizing path simplify so that the
solutions (18) become $V^{1}(s)=v_{1},V^{2}(s)=v_{2},V^{3}(s)=0$ with
corresponding complexity

$\displaystyle{\mathcal{C}}_{\rm target}=\sqrt{v_{1}^{2}+v_{2}^{2}}\,.$ (21)

Our target operator (12) is the $s=1$ boundary condition of the $s$-dependent
unitary operator (3)

$\displaystyle\hat{U}(s)={\mathcal{P}}\,{\rm
exp}\left[-i\int_{0}^{s}V^{I}(s^{\prime})\hat{e}_{I}\ ds^{\prime}\right]\,,$
(22)

which is a solution to the differential equation (4)

$\displaystyle\frac{d\hat{U}(s)}{ds}=-iV^{I}(s)\ \hat{e}_{I}\ \hat{U}(s)\,.$
(23)

In order to find an explicit solution to (23), we will use an explicit 3 x 3
upper-triangular matrix representation555We obtain identical results for other
matrix representations, indicating that the precise choice of matrix
representation is not important. See Appendix A. of the Heisenberg Lie algebra
generators as

$\displaystyle\hat{e}_{1}=\begin{pmatrix}0&1&0\cr 0&0&0\cr
0&0&0\cr\end{pmatrix}\,,\hskip 14.45377pt\hat{e}_{2}=\begin{pmatrix}0&0&0\cr
0&0&1\cr 0&0&0\cr\end{pmatrix}\,,\hskip
14.45377pt\hat{e}_{3}=\begin{pmatrix}0&0&i\cr 0&0&0\cr
0&0&0\cr\end{pmatrix}\,.$ (24)

Within this representation, a general element of the Heisenberg group can be
written in terms of group elements $a,b,c$

$\displaystyle\hat{U}(s)=\begin{pmatrix}1&ia(s)&c(s)\cr 0&1&ib(s)\cr
0&0&1\end{pmatrix}\,.$ (25)

Following (8), we can use (25) and the explicit representation of the
fundamental operators $\\{\hat{e}_{i}\\}$ to construct the operator-space
geometry of the Heisenberg group

$\displaystyle ds^{2}=G_{IJ}\ dV^{I}\
dV^{J}=da^{2}+db^{2}+\left(bda+dc\right)^{2}\,,$ (26)

which has constant negative curvature. The operator-space geometry associated
with the Heisenberg group is thus $3$-dimensional hyperbolic space.

With the explicit representation (25) and the solutions for the $V^{I}(s)$,
the differential equation (23) is solved by the parameterizations

$\displaystyle a(s)$ $\displaystyle=$ $\displaystyle a_{0}-v_{1}s\,;$
$\displaystyle b(s)$ $\displaystyle=$ $\displaystyle b_{0}-v_{2}s\,;$ (27)
$\displaystyle c(s)$ $\displaystyle=$ $\displaystyle
c_{0}+b_{0}v_{1}s-\frac{1}{2}v_{1}v_{2}s^{2}\,,$

where the $a_{0},b_{0},c_{0}$ are constants. Imposing the boundary condition
that the unitary reduce to the identity operator at $s=0$,
$\hat{U}(s=0)=\hat{\mathds{1}}$ leads to the unitary

$\displaystyle\hat{U}(s)=\begin{pmatrix}1&-iv_{1}s&-\frac{1}{2}v_{1}v_{2}s^{2}\cr
0&1&-iv_{2}s\cr 0&0&1\end{pmatrix}\,.$ (28)

The boundary condition at $s=1$, $\hat{U}(s=1)=\hat{\mathcal{U}}_{\rm
target}$, with (16), now leads to a solution for the constants
$v_{1}=-\sqrt{2}\ {\rm Im}[\alpha(t)],v_{2}=-\sqrt{2}\ \rm{Re}[\alpha(t)]$.
The resulting complexity for the time-dependent displacement operator is thus

$\displaystyle{\mathcal{C}}_{\rm Heis}=\sqrt{2}\,|\alpha|\,,$ (29)

and is simply proportional to the time-independent magnitude of the coherent
state parameter. This result for the complexity of the displacement operator
is similar to previous results [5, 40] for the complexity of a corresponding
coherent state, and stands in contrast to other measures of complexity of the
displacement operator [34, 41], in which the complexity is time-dependent.

It is interesting to interpret the result (29) in terms of the average number
density – or equivalently, the average energy – of a vacuum coherent state

$\displaystyle\langle
E\rangle\sim\bar{N}_{\alpha}=\langle\alpha|\hat{a}^{\dagger}\hat{a}|\alpha\rangle=|\alpha|^{2}\,.$
(30)

From this perspective, the complexity of the displacement operator (29) scales
as the square root of average energy (average number of particles)

$\displaystyle{\mathcal{C}}_{\rm displacement}\sim\sqrt{\langle E\rangle}\,.$
(31)

In realizations of quantum information protocols, the energy needed to prepare
a state or set of gates can be an important resource [15, 19]. Because of the
scaling (31), this implies that the energy required to build a coherent state
with some fixed complexity ${\mathcal{C}}_{*}$ grows quadratically with that
complexity $\langle E\rangle\sim{\mathcal{C}}_{*}^{2}$. It would be
interesting to study further whether these scalings have general lessons for
building quantum information protocols with continuous variables in the lab.

##  4 Squeezing Operator Complexity

Let us now turn our attention to a unitary quantum circuit that is constructed
from a generic quadratic combination of creation and annihilation operators

$\displaystyle|\psi\rangle_{\rm T}=\hat{U}_{2}|0\rangle\,,$ (32)

where $\hat{U}_{2}$ is the unitary operator

$\displaystyle\hat{U}_{2}={\rm exp}\left[-i\int\hat{H}_{2}\ dt\right]\,$ (33)

defined in terms of a quadratic Hamiltonian

$\displaystyle\hat{H}_{2}=\Omega\
\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\left(\Delta\,\hat{a}^{2}+\Delta^{*}\
\hat{a}^{\dagger 2}\right)\,.$ (34)

The unitary operator (33) can also be written more generally in the factorized
form [18]

$\displaystyle\hat{U}_{2}=\hat{S}(r,\phi)\hat{R}(\theta)$ (35)

in terms of the squeeze and rotation operators

$\displaystyle\hat{S}(r,\phi)=e^{\frac{r}{2}(e^{-2i\phi}\hat{a}^{2}-e^{2i\phi}\hat{a}^{\dagger
2})}\hskip 14.45377pt\text{and}\hskip
14.45377pt\hat{R}(\theta)=e^{-i\theta\frac{\hat{a}^{\dagger}\hat{a}+\hat{a}\hat{a}^{\dagger}}{2}}\,,$
(36)

where $r$, the squeezing parameter, characterizes the amount of squeezing,
$\phi$ is the squeezing angle, and $\theta$ is the rotation angle. The
formulation (35) in terms of the squeezing and rotation operators can be quite
useful, particularly when the parameters $\Omega,\Delta$ of the target
quadratic Hamiltonian (34) are time-dependent, as often happens for
interesting physical applications. Thus, we can represent our quantum circuit
more generally as the _squeezed state_

$\displaystyle|\psi\rangle_{\rm T}=\hat{\mathcal{U}}_{\rm
target}|\psi\rangle_{\rm R}=\left[\hat{S}(r,\phi)\
\hat{R}(\theta)\right]|\psi\rangle_{\rm R}\,,$ (37)

where $|\psi\rangle_{\rm R}$ is an arbitrary reference state. From another
perspective, a Bogoluibov transformation between two sets
$(\hat{a},\hat{a}^{\dagger})\leftrightarrow(\hat{b},\hat{b}^{\dagger})$ of
creation and annihilation operators can be written as a transformation with
respect to the squeeze and rotation operators

$\displaystyle\hat{a}=\alpha\,\hat{b}+\beta^{*}\,\hat{b}^{\dagger}=\hat{\mathcal{U}}_{\rm
target}^{\dagger}\,\hat{b}\,\hat{\mathcal{U}}_{\rm target}\,.$ (38)

From the form of $\hat{H}_{2}$, the natural set of fundamental operators
$\\{\hat{\mathcal{O}}_{I}\\}$ to use in building the target unitary (2) are
the Hermitian operators

$\displaystyle\hat{e}_{1}=\frac{\hat{a}^{2}+\hat{a}^{\dagger
2}}{4}\,,\qquad\hat{e}_{2}=\frac{i(\hat{a}^{2}-\hat{a}^{\dagger
2})}{4}\,,\qquad\hat{e}_{3}=\frac{\hat{a}\hat{a}^{\dagger}+\hat{a}^{\dagger}\hat{a}}{4}\,.$
(39)

These operators satisfy the $\mathfrak{su}(1,1)$ Lie algebra,

$\displaystyle[\hat{e}_{1},\hat{e}_{2}]=-i\hat{e}_{3},\
[\hat{e}_{3},\hat{e}_{1}]$ $\displaystyle=$ $\displaystyle i\hat{e}_{2},\
[\hat{e}_{2},\hat{e}_{3}]=i\hat{e}_{1}\,,$ (40)

therefore the target operator $\hat{\mathcal{U}}_{\rm target}$ is a generic
element of SU(1,1). The structure constants
$[\hat{e}_{i},\hat{e}_{j}]=if_{ij}^{k}\hat{e}_{k}$

$\displaystyle f^{3}_{12}=-1,\ f^{2}_{31}=1,f^{1}_{23}=1$ (41)

define the $\mathfrak{su}(1,1)$ Cartan-Killing form
$K_{ij}=f_{ik}^{l}f^{k}_{jl}$

$\displaystyle K=\begin{pmatrix}2&\ \ 0&\ 0\\\ 0&\ \ 2&\ 0\\\ 0&\ \
0&-2\end{pmatrix}\,.$ (42)

Following the discussion in Section 2, a natural choice for a metric on the
space of operators of $\mathfrak{su}(1,1)$ would be proportional to the bi-
invariant Cartan-Killing form $G_{ij}=K_{ij}$. However, because of the
negative eigenvalue in (42) such a metric would be Lorentzian, potentially
leading to a zero or imaginary complexity (9) for “lightlike”
($G_{ij}V^{i}V^{j}=0$) or “timelike” ($G_{ij}V^{i}V^{j}<0$) paths,
respectively. Since we would like to interpret the complexity (9) as a
continuum version of the number of gates needed to build the target quantum
circuit, negative or imaginary complexities are undesirable. Instead, here we
will again choose a diagonal Riemannian metric with equal cost factors
$G_{ij}=\delta_{ij}$ so that
$G_{ij}V^{i}V^{j}=\left(V^{1}\right)^{2}+\left(V^{2}\right)^{2}+\left(V^{3}\right)^{2}$
is non-negative. This means that all of our fundamental operators are “easy,”
in contrast to the SU(2) analysis of [9] which chose one of the directions to
be a “hard” direction with an increased cost factor.

Given our characterization of the target circuit in terms of the
$\mathfrak{su}(1,1)$ generators $\\{\hat{e}_{i}\\}$

$\displaystyle\hat{\mathcal{U}}_{\rm
target}=\hat{S}(r,\phi)\hat{R}(\theta)={\mathcal{P}}\,{\rm
exp}\left[-i\int_{0}^{1}V^{i}(s)\ \hat{e}_{i}\ ds\right]\,,$ (43)

we now need to find paths $\\{V^{1}(s),V^{2}(s),V^{3}(s)\\}$ that minimize the
circuit depth (6). The geodesic equation (10) becomes the set of three
equations for the $V^{i}(s)$

$\displaystyle\frac{dV^{1}}{ds}$ $\displaystyle=$
$\displaystyle-2V^{2}V^{3}\,;$ $\displaystyle\frac{dV^{2}}{ds}$
$\displaystyle=$ $\displaystyle 2V^{1}V^{3}\,;$ (44)
$\displaystyle\frac{dV^{3}}{ds}$ $\displaystyle=$ $\displaystyle 0\,.$

Solutions to (44) take the form

$\displaystyle V^{1}(s)$ $\displaystyle=$ $\displaystyle
v_{1}\cos(2v_{3}s)-v_{2}\sin(2v_{3}s)\,;$ $\displaystyle V^{2}(s)$
$\displaystyle=$ $\displaystyle v_{1}\sin(2v_{3}s)+v_{2}\cos(2v_{3}s)\,;$ (45)
$\displaystyle V^{3}(s)$ $\displaystyle=$ $\displaystyle v_{3}\,,$

where $v_{1},v_{2},v_{3}$ are constants that we will determine by matching the
form of the target operator (43). With this solution (45) the magnitude of the
vector $V^{i}$ takes the simple form

$\displaystyle
G_{ij}V^{i}V^{j}=\left(V^{1}\right)^{2}+\left(V^{2}\right)^{2}+\left(V^{3}\right)^{2}=v_{1}^{2}+v_{2}^{2}+v_{3}^{2}=v^{2}+v_{3}^{2}$
(46)

where we defined $v^{2}=v_{1}^{2}+v_{2}^{2}$ for convenience. The complexity
(9), the minimal circuit depth evaluated on the solution to the Euler-Arnold
equation (45), becomes

$\displaystyle{\mathcal{C}}_{\rm target}=\sqrt{v^{2}+v_{3}^{2}}\,.$ (47)

The constants $v_{i}$ are determined by matching to the target operator (3),
which solves (4)

$\displaystyle\frac{d\hat{U}(s)}{ds}=-iV^{I}(s)\ \hat{e}_{i}\ \hat{U}(s)\,,$
(48)

subject to the boundary conditions (5) $\hat{U}(0)=\hat{\mathds{1}}$,
$\hat{U}(1)=\hat{\mathcal{U}}_{\rm target}=\hat{S}(r,\phi)\hat{R}(\theta)$. In
order to solve (48), we work with the following 2 x 2 representation of the
$\mathfrak{su}(1,1)$ generators,

$\displaystyle\hat{e}_{1}=\frac{1}{2}\begin{pmatrix}0&-1\\\ 1&\
0\end{pmatrix},\ \hat{e}_{2}=\frac{1}{2}\begin{pmatrix}0&i\\\ i&\
0\end{pmatrix},\hat{e}_{3}=\frac{1}{2}\begin{pmatrix}1&\ 0\\\
0&-1\end{pmatrix}\,.$ (49)

Within the 2 x 2 representation, a general element of SU(1,1) takes the form

$\displaystyle\hat{U}(s)=\begin{pmatrix}q(s)&p(s)^{\ast}\\\
p(s)&q(s)^{\ast}\end{pmatrix}\qquad\text{such that}\qquad|q|^{2}-|p|^{2}=1\,.$
(50)

We can further write this in terms of some $\alpha,\beta,\gamma$, which
parametrize the group

$\displaystyle\hat{U}(s)=\begin{pmatrix}\cosh(\rho)+i\gamma\frac{\sin(\rho)}{\rho}&(\alpha-i\beta)\frac{\sin(\rho)}{\rho}\\\
(\alpha+i\beta)\frac{\sin(\rho)}{\rho}&\cosh-i\gamma\frac{\sin(\rho)}{\rho}\end{pmatrix}\,,$
(51)

where $\rho^{2}=\alpha^{2}+\beta^{2}-\gamma^{2}$. Following (8), we can use
(51) and the explicit representation of the fundamental operators
$\\{\hat{e}_{i}\\}$ to construct the non-compact operator-space geometry of
the $\mathfrak{su(1,1)}$ generators

$\displaystyle ds^{2}_{\mathfrak{su(1,1)}}=$ $\displaystyle
4\frac{\sinh^{4}\rho}{\rho^{4}}\left[(\beta^{2}+\gamma^{2}+\rho^{2}\coth^{2}\rho)d\alpha^{2}+(\alpha^{2}+\gamma^{2}+\rho^{2}\coth^{2}\rho)d\beta^{2}\right.$
(52)
$\displaystyle\left.+(\alpha^{2}+\beta^{2}+\rho^{2}\coth^{2}\rho)d\gamma^{2}-2\alpha\beta
d\alpha d\beta-2(2\beta\rho\coth\rho+\alpha\gamma)d\alpha d\gamma\right.$
$\displaystyle\left.+2(2\alpha\rho\coth\rho-\beta\gamma)d\beta
d\gamma\right]\,.$

The corresponding curvature

$\displaystyle
R_{\mathfrak{su(1,1)}}=-8(\rho^{2}+2\gamma^{2})\tanh^{2}\rho-\rho^{2}$ (53)

is non-positive.

Returning to the form (50), the differential equation (48) then becomes
equivalent to the differential equations for $q(s)$ and $p(s)$

$\displaystyle\frac{dq}{ds}$ $\displaystyle=$
$\displaystyle\,-\frac{iv_{3}}{2}q+\frac{(v_{2}+iv_{1})}{2}e^{-2iv_{3}s}p\,;$
(54) $\displaystyle\frac{dp}{ds}$ $\displaystyle=$
$\displaystyle\,\frac{iv_{3}}{2}p+\frac{(v_{2}-iv_{1})}{2}e^{2iv_{3}s}q\,,$
(55)

with solutions

$\displaystyle q(s)$ $\displaystyle=$ $\displaystyle
e^{-iv_{3}s}(c_{1}e^{\lambda s/2}+c_{2}e^{-\lambda s/2})\,;$ (56)
$\displaystyle p(s)$ $\displaystyle=$
$\displaystyle\frac{v_{2}-iv_{1}}{v^{2}}e^{iv_{3}s}\left(c_{1}(\lambda-
iv_{3})e^{\lambda s/2}-c_{2}(\lambda+v_{3})e^{-\lambda s/2}\right)\,,$ (57)

where again we used $v_{1}^{2}+v_{2}^{2}=v^{2}$ for notational simplicity, and
$\lambda=\sqrt{v^{2}-v_{3}^{2}}$ can be either real or imaginary, depending on
the relative sizes of $v$ and $v_{3}$. Demanding the boundary condition
$\hat{U}(0)=\hat{\mathds{1}}$ implies $q(0)=1,p(0)=0$, which fixes
$c_{1}=\frac{1}{2\lambda}(\lambda+iv_{3}),c_{2}=\frac{1}{2\lambda}(\lambda-
iv_{3})$, so that $\hat{U}(s)$ becomes

$\displaystyle\hat{U}(s)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}e^{-iv_{3}s}\left(\cosh\left(\frac{\lambda
s}{2}\right)+i\frac{v_{3}}{\lambda}\sinh\left(\frac{\lambda
s}{2}\right)\right)&\,\,e^{-iv_{3}s}\,\frac{(v_{2}+iv_{1})}{\lambda}\sinh\left(\frac{\lambda
s}{2}\right)\\\
e^{iv_{3}s}\,\frac{(v_{2}-iv_{1})}{\lambda}\sinh\left(\frac{\lambda
s}{2}\right)&\,\,e^{iv_{3}s}\left(\cosh\left(\frac{\lambda
s}{2}\right)-i\frac{v_{3}}{\lambda}\sinh\left(\frac{\lambda
s}{2}\right)\right)\end{pmatrix}$ (58)

Imposing the remaining boundary condition at $s=1$,
$\hat{U}(1)={\mathcal{U}}_{\rm target}$ will allow us to determine the
geodesic constants $v_{i}$ in terms of the target operator quantities.
However, before we do this in the general case, let us examine limiting cases
in order to gain an intuition for the results.

###  4.1 Free Harmonic Oscillator

As our first example, let us consider time evolution under the free harmonic
oscillator Hamiltonian as our target operator

$\displaystyle|\psi\rangle_{\rm T}=e^{-i\hat{H}_{0}t}|\psi\rangle_{\rm R}$
(59)

with
$\hat{H}_{0}=\frac{\omega}{2}(\hat{a}^{\dagger}\hat{a}+\hat{a}\hat{a}^{\dagger})=2\omega\hat{e}_{3}$.
In terms of the squeeze and rotation operators (36) this corresponds to
vanishing squeezing $r=0$ and a time-dependent rotation angle
$\theta(t)=\omega t$. The resulting target operator is then

$\displaystyle\hat{\mathcal{U}}_{\rm
target}=e^{-i\hat{H}_{0}t}=\begin{pmatrix}e^{-i\omega t}&0\\\ 0&e^{i\omega
t}\end{pmatrix}\,.$ (60)

Imposing the boundary condition $\hat{U}(1)=\hat{\mathcal{U}}_{\rm target}$ on
the operator (58), we find $v_{1}=v_{2}=0$ so that $\lambda=iv_{3}$, and

$\displaystyle e^{-i\omega t}=e^{-iv_{3}/2}\,.$ (61)

Since the complexity becomes ${\mathcal{C}}_{\rm
free}=\sqrt{v_{1}^{2}+v_{2}^{2}+v_{3}^{2}}=|v_{3}|$, the simple solution
$v_{3}=2\theta(t)=2\omega t$ to (61) results in a complexity that is unbounded
and grows linearly with time ${\mathcal{C}}_{\rm free}=2\omega t$. However,
the complexity should be a _minimization_ over the $v_{i}$, and this is not
the minimal solution to (61). For $\omega t>\pi$, we can solve (61) with the
smaller angle $|v_{3}|=2(2\pi-\omega t)$, up until $\omega t>2\pi$, at which
point $v_{3}=4(\omega t-2\pi)/3$ (and so on for increasing windings). The
resulting complexity as a function of time oscillates between positive and
negative slopes of $\pm 4\omega/3$, as seen in Figure 1 Indeed, since the
operator (60) is identical after the revival time $\hat{\mathcal{U}}_{\rm
target}(t=0)=\hat{\mathcal{U}}_{\rm target}(t=T)$ for $T=2\pi/\omega$, we
expect the complexity to show a similar periodicity.

Figure 1: (Left) When $\omega t>\pi$, the minimal length geodesic for
$|v_{3}|$ to satisfy the boundary condition (61) becomes
$|v_{3}|=2(2\pi-\omega t)$. (Right) Because of this “shortcut”, the complexity
of the time evolution operator $e^{-i\hat{H}_{0}t}$ for the free harmonic
oscillator oscillates between positive and negative slopes $\pm 2\omega$ with
a period given by the quantum revival time $T=2\pi/\omega$.

###  4.2 Inverted Harmonic Oscillator

Next we consider the inverted harmonic oscillator, with Hamiltonian

$\displaystyle\hat{H}_{I}=-\frac{\Omega}{2}\left(\hat{a}^{2}+{\hat{a}}^{\dagger
2}\right)=-2\Omega\,\hat{e}_{1}\,.$ (62)

Taking the time evolution operator as our target operator again, with
representation

$\displaystyle\hat{\mathcal{U}}_{\rm target}=e^{-i\hat{H}_{I}t}=e^{2i\Omega
t\hat{e}_{1}}=\begin{pmatrix}\cosh(\Omega t)&-i\sinh(\Omega t)\\\
i\sinh(\Omega t)&\cosh(\Omega t)\end{pmatrix}\,,$ (63)

we see this as a squeeze operator with linearly increasing squeezing
$r(t)=\Omega t$, constant squeeze angle $\phi=\pi/4$, and vanishing rotation
angle $\theta=0$. Matching $\hat{U}(1)=\hat{\mathcal{U}}_{\rm target}$ leads
to the conditions

$\displaystyle
e^{-iv_{3}}\left(\cosh\left(\frac{\lambda}{2}\right)+i\frac{v_{3}}{\lambda}\sinh\left(\frac{\lambda}{2}\right)\right)\,=\cosh(\Omega
t)\,,\qquad\frac{v_{2}+iv_{1}}{\lambda}\sinh\left(\frac{\lambda}{2}\right)\,e^{-iv_{3}}=-i\sinh(\Omega
t)\,,$ (64)

which are solved by $\lambda=v_{1}=2r(t)=2\Omega t$, $v_{2}=v_{3}=0$. The
resulting complexity for the inverted harmonic oscillator grows linearly with
time

$\displaystyle{\mathcal{C}}_{\rm Invert}=2\Omega t\,.$ (65)

Unlike the free harmonic oscillator of the previous subsection, the complexity
for the inverted harmonic oscillator grows without bound, reflecting the
instability of the inverted oscillator.

###  4.3 General Squeezing and Rotation

Having spent some time on the simplified special cases of the previous two
subsections, we now consider a more general element of SU(1,1). The most
general target operator can be written as a product of squeeze and rotation
operators

$\displaystyle\hat{\mathcal{U}}_{\rm
target}=\hat{S}(r,\phi)\hat{R}(\theta)\,.$ (66)

As discussed above, a generic quadratic time-evolution operator (33) can
always be decomposed into a product of this form, in which the parameters
$r(t),\phi(t),\theta(t)$ all inherit time-dependence through the Heisenberg
equation of motion. Using the forms of the squeezing and rotation operators
(36), and the generators (39) with the representation (49), we write our
target operator as

$\displaystyle\hat{\mathcal{U}}_{\rm target}=\begin{pmatrix}e^{-i\theta}\cosh
r&e^{i(2\phi+\theta)}\sinh r\\\ e^{-i(2\phi+\theta)}\sinh r&e^{i\theta}\cosh
r\end{pmatrix}\,.$ (67)

Matching the operator (58) at the boundary condition
$\hat{U}(1)=\hat{\mathcal{U}}_{\rm target}$ with (67) we obtain the conditions

$\displaystyle e^{-i\theta}\cosh
r=e^{-iv_{3}}\left(\cosh\left(\frac{\lambda}{2}\right)+i\frac{v_{3}}{\lambda}\right)\,,\qquad
e^{-i(2\phi+\theta)}\sinh
r=\frac{e^{iv_{3}}(v_{2}-iv_{1})}{\lambda}\sinh\left(\frac{\lambda}{2}\right)\,.$
(68)

Solutions to (68) for
$v=\sqrt{v_{1}^{2}+v_{2}^{2}}=v(r,\theta),v_{3}=v_{3}(r,\theta)$ as functions
of the squeezing parameter and rotation angle can be found analytically in the
limits of small $r\ll 1$ and large $r\gg 1$ squeezing666The relative sizes of
$v_{1}$ and $v_{2}$ are determined from the squeezing angle $\phi$ from (68),
but will not be important in the rest of our analysis.:

$\displaystyle v$ $\displaystyle\approx$ $\displaystyle 2r\,;$ (69)
$\displaystyle|v_{3}|$ $\displaystyle=$
$\displaystyle\begin{cases}2\theta_{\rm min}&\mbox{ for }r\ll 1\cr\theta_{\rm
min}&\mbox{ for }r\gg 1\end{cases}$ (70)

where

$\displaystyle\theta_{\rm min}=\begin{cases}\theta-2\pi n&\mbox{ for }2\pi
n<\theta<\pi(2n+1)\cr 2\pi n-\theta&\mbox{ for }\pi(2n-1)<\theta<2\pi
n\end{cases}\mbox{ for some }n\,,$ (71)

represents the minimization of the rotation angle discussed in Section 4.1.
More generally, the equations (68) can be solved numerically as seen in Figure
2, illustrating that this analytic solution holds quite generally, with
$v_{3}$ interpolating between $2\theta_{\rm min}$ and $\theta_{\rm min}$ for
intermediate values of the squeezing $r$.

(a)

(b)

Figure 2: Numerical solutions to the boundary condition (68) for the geodesic
parameters (a) $v=\sqrt{v_{1}^{2}+v_{2}^{2}}$, illustrating that $v\approx
2r$, and (b) $v_{3}$, illustrating the behavior (70).

The resulting complexity of a generic squeezed state is thus given by

$\displaystyle{\mathcal{C}}_{\rm Squeeze}\approx\sqrt{4r^{2}+v_{3}^{2}}\,,$
(72)

where $v_{3}$ is given by (70). Since $v_{3}$ is bounded by $2\pi$, for large
squeezing $r\gg 1$ the complexity is linear in the squeezing parameter

$\displaystyle{\mathcal{C}}_{\rm Squeeze}\approx 2r\,.$ (73)

Interestingly, the complexity of the target operator (66) consisting of the
product of the squeeze and rotation operators depends periodically on the
rotation angle $\theta$ through $v_{3}$, but is entirely independent of the
squeezing angle $\phi$.

The complexity of a _Gaussian_ squeezed state, generated by acting the
squeezing operator (66) on a vacuum state, has been considered before using
different techniques. For example, in [24] the complexity of a single-mode
squeezed state was calculated by parameterizing the effect of a squeezing
operator on Gaussian wavefunctions, using the approach of [4, 7]. In contrast
to our results above (73) and (73), the complexity of a single-mode squeezed
state in [24] depends sensitively on the squeezing angle. Further, the
Gaussian wavefunction approach of [4, 7] ignores normalization factors and
phases of the Gaussian wavefunction, and cannot be sensitive to the rotation
angle $\theta$ in the way found in (72), even though such states are
technically distinct. The differences between the two approaches is heightened
by applying them to the inverted harmonic oscillator. Our result above (65)
finds that the complexity is equal to the time-dependent squeezing parameter
$r(t)=2\Omega t$, growing linearly with time, while the complexity for the
Gaussian wavefunction approach _saturates_ at late times ${\mathcal{C}}_{\rm
Gauss}\approx\pi/4$ [24]. These differences may simply be due to the fact that
the pure operator complexity, as we have found here, is independent of the
specific forms of the reference and target states, and is thus the complexity
of the quantum circuit as it would be applied to an _arbitrary_ reference
state. A more specific reference state, such as a vacuum state as considered
in [24], perhaps allows one to find additional shortcuts by using the Gaussian
form of the position-space wavefunction. In this way, our results for the
operator complexity (72) may serve as a universal upper bound on the
complexity needed to construct the target operator as applied to an arbitrary
reference state. Specific choices of reference and target states can then
allow one to find shortcuts in the construction of the operator with smaller
circuit depth.

The complexity of a quantum circuit can also be calculated by characterizing
the circuit through a covariance matrix [42]. Applying this approach to a
squeezed state leads to a complexity proportional to the squeezing
${\mathcal{C}}_{\rm cov}=r(t)$ [24, 43], similar to our results above
(although the results [24, 43] appear again to be insensitive to the rotation
angle $\theta$). However, both the covariance matrix and Gaussian wavefunction
approaches suffer from similar limitations: they can only be applied to
Gaussian reference and target states. In contrast, the operator complexity
approach we have outlined here is independent of the reference and target
states, and is potentially generalizable to other groups that are not
quadratic in the raising and lowering operators.

##  5 Scalar Field Complexity

In the previous sections, we studied the operator complexity of the
displacement operator as well as the free and and squeezed quantum harmonic
oscillators. Now we extend the formalism and techniques of operator complexity
from [9, 10] to a quantum scalar field, first in the case of a free massive
scalar field, then to quantum scalar cosmological perturbations.

###  5.1 Free Scalar Field

To begin, let us consider a $(d+1)$-dimensional free scalar field of mass $m$
in a box of size $L$ with periodic boundary conditions (we will take
$L\rightarrow\infty$ at the end). Expanding the field in Fourier modes

$\displaystyle\hat{\phi}(x)=\sum_{\vec{n}}^{\infty}\frac{1}{\sqrt{2E_{\vec{n}}}}\left(\hat{a}_{\vec{n}}\,e^{i\vec{p}_{\vec{n}}\cdot\vec{x}}+\hat{a}_{\vec{n}}^{\dagger}\,e^{-i\vec{p}_{\vec{n}}\cdot\vec{x}}\right)$
(74)

for canonical creation and annihilation operators
$\left[\hat{a}_{\vec{n}},\hat{a}_{\vec{m}}^{\dagger}\right]=\delta_{\vec{n},\vec{m}}$,
where $\vec{p}_{\vec{n}}=\vec{n}\pi/L$ and
$E_{\vec{n}}=\sqrt{\vec{p}_{\vec{n}}^{2}+m^{2}}$ and $\vec{n}$ is a
$d$-dimensional vector of integers. The Hamiltonian becomes a sum over modes

$\displaystyle\hat{H}=\frac{1}{2}\sum_{n_{1}=1}^{\infty}\sum_{n_{2}=1}^{\infty}...\sum_{n_{d}=1}^{\infty}E_{\vec{n}}\left(\hat{a}_{\vec{n}}^{\dagger}\hat{a}_{\vec{n}}+\hat{a}_{\vec{n}}\hat{a}_{\vec{n}}^{\dagger}\right)\,.$
(75)

We will cutoff the infinite sums of modes at the UV scale $\Lambda=N_{\rm
max}\pi/L$ for $N_{\rm max}\gg 1$, allowing us to study how the complexity
diverges as a function of the UV cutoff $\Lambda$. As a result, our target
unitary becomes the product

$\displaystyle\hat{\mathcal{U}}_{\rm
target}=e^{-i\hat{H}t}=\prod_{n_{1}=1}^{N_{\rm max}}\prod_{n_{2}=1}^{N_{\rm
max}}...\prod_{n_{d}=1}^{N_{\rm
max}}e^{-i\frac{1}{2}E_{\vec{n}}\left(\hat{a}_{\vec{n}}^{\dagger}\hat{a}_{\vec{n}}+\hat{a}_{\vec{n}}\hat{a}_{\vec{n}}^{\dagger}\right)}\,.$
(76)

For each mode $\vec{n}$ we can choose our set of fundamental gates to be the
corresponding $\mathfrak{su(1,1)}$ generators from Section 4, e.g.
$\hat{e}_{3}^{\vec{n}}=(\hat{c}_{\vec{n}}^{\dagger}\hat{c}_{\vec{n}}+\hat{c}_{\vec{n}}\hat{c}_{\vec{n}}^{\dagger})/4$.
The corresponding geometry on the space of operators becomes the $(N_{\rm
max})^{d}$ dimensional direct product of copies of the $\mathfrak{su(1,1)}$
geometry (52)

$\displaystyle ds^{2}=\sum_{n_{1}=1}^{N_{\rm max}}\sum_{n_{2}=1}^{N_{\rm
max}}...\sum_{n_{d}=1}^{N_{\rm
max}}G_{IJ}^{\vec{n}}V^{I}_{\vec{n}}V^{J}_{\vec{n}}\,.$ (77)

For any single mode $\vec{n}$ of the target unitary (76), the minimal path,
and corresponding complexity, is given by that of the free harmonic oscillator
of Section 4.1,

$\displaystyle{\mathcal{C}}_{\rm
free}^{\vec{n}}=|v_{3}^{\vec{n}}|=\begin{cases}2(E_{\vec{n}}\,t-2\pi
m)&\mbox{for }2\pi m<E_{\vec{n}}\,t<\pi(2m+1)\cr 2(2\pi
m-E_{\vec{n}}\,t)&\mbox{for }\pi(2m-1)<E_{\vec{n}}\,t<2\pi m\end{cases}\mbox{
for some integer }m$ (78)

which oscillates between $0\leq{\mathcal{C}}_{\rm free}^{\vec{n}}\leq 2\pi$.
Taking into account all of the modes, the total complexity for the
(regularized) free scalar field is

$\displaystyle{\mathcal{C}}_{\phi}=\sqrt{\sum_{\\{n_{i}\\}=1}^{N_{\rm
max}}\left(v_{3}^{\vec{n}}\right)^{2}}\,,$ (79)

where $v_{3}^{\vec{n}}$ is given by (78) Taking the continuum limit
$L\rightarrow\infty$, the sums become integrals $\sum_{\\{n_{i}\\}=1}^{N_{\rm
max}}\rightarrow L^{d}/\pi^{d}\int^{\Lambda}d^{d}p$, and the complexity (79)
becomes

$\displaystyle{\mathcal{C}_{\phi}}=\frac{L^{d/2}}{\pi^{d/2}}\sqrt{\int^{\Lambda}v_{3}(p)^{2}\,d^{d}p}\,$
(80)

where $v_{3}(p)$ is the continuum version of (78) with
$E_{p}=\sqrt{\vec{p}^{2}+m^{2}}$. Even before performing the integral, we see
that the scalar field complexity (80) diverges as the square root of the
volume, similar to other field theory complexity calculations using a
geometric cost function [4].

At very early times $\pi/t\gg\Lambda$ much shorter than the UV scale, none of
the modes in (80) has yet reached its first oscillation in $v_{3}^{\vec{n}}$.
Thus, we have $v_{3}(p)=2E_{p}\,t$ in these early times, and the complexity
becomes

$\displaystyle{\mathcal{C}}_{\phi}\sim\frac{L^{d/2}}{\pi^{d/2}}\sqrt{\int^{\Lambda}4(p^{2}+m^{2})\,t^{2}\,d^{d}p}\sim\frac{L^{d/2}}{\pi^{d/2}}\left(\mbox{vol}(d)\right)^{1/2}\Lambda^{(d+2)/2}t\sim
L^{d/2}\Lambda^{d/2}\Lambda t$ (81)

where we took the integral to be dominated by the high-energy modes $p^{2}\gg
m^{2}$, and $\mbox{vol}(d)$ is the volume of a $d$-dimensional unit sphere.
The UV divergence at these early times scales as $\Lambda^{(d+2)/2}$, leading
to a rapid growth of complexity over a very short time scale.

(a)

(b)

Figure 3: The operator complexity (80) of the time evolution operator for a
free scalar field of mass $m$ with UV cutoff $\Lambda$ (here chosen as
$m=\Lambda/10$) shows a sharp growth ${\mathcal{C}}_{\phi}\sim 2\Lambda t$ at
early times, some transient behavior at intermediate times
$\pi/\Lambda<t<\pi/m$, and a saturation (with damped oscillations) at late
times $t>\pi/m$. (a) The transient intermediate behavior is shown for $d=1$
for $m\neq 0$ (solid); when $m=0$ (dashed) there is no transient period, and
the operator complexity saturates after its initial growth. (b) The growth and
saturation of the operator complexity is similar across different dimensions.

At intermediate times $\pi/\Lambda\ll t\ll\pi/m$, the complexity for the UV
modes $p^{2}\gg m^{2}$ are rapidly oscillating, but the complexity for the IR
modes $p^{2}<m^{2}$ are still increasing with time, $v_{3}^{IR}\sim 2mt$.
Since the UV modes dominate the integral, this leads to a transient period of
mild time dependence, as seen for $d=1$ in Figure 3(a). Finally, at late times
$t\gg\pi/m$, the complexity of each mode is rapidly oscillating with an
${\mathcal{O}}(1)$ value, so the total complexity of the scalar field roughly
saturates (up to oscillations that will damp out over time) to an average
value set by the UV scale,

$\displaystyle{\mathcal{C}}_{\phi}\sim\frac{L^{d/2}}{\pi^{d/2}}\sqrt{\int^{\Lambda}d^{d}p}\sim
L^{d/2}\Lambda^{d/2}\,.$ (82)

Putting this behavior together, we get a plot of complexity versus time Figure
3 that has a sharp linear rise followed by a saturation with damped
oscillations, consistent across multiple values of the dimension $d$ as shown
in Figure 3(b). Taking the scalar field to be massless $m\rightarrow 0$
removes the intermediate transient behavior, so that the complexity simply
saturates after the sharp rise at early times, as seen in Figure 3(a).

The volume and UV divergences in the saturated operator complexity (82) of the
free scalar field Hamiltonian evolution operator are similar to the
divergences found in [4] for the state complexity of the ground state of a
free scalar field in $d$-dimensions. Naturally, the power of $1/2$ in (82) is
coming from the square root in the geometric length of the circuit depth (6).
The Euler-Arnold minimization procedure simultaneously minimizes the “squared
length” circuit depth

$\displaystyle\tilde{\mathcal{C}}=\min_{\\{V^{I}\\}}\tilde{\mathcal{D}}\left[V^{I}\right]=\min_{\\{V^{I}\\}}\int_{0}^{1}G_{IJ}V^{I}V^{J}\
ds\,,$ (83)

so the corresponding saturated complexity for the scalar field would scale
with the volume and UV cutoff instead as

$\displaystyle\tilde{\mathcal{C}}_{\phi}\sim L^{d}\Lambda^{d}\,.$ (84)

This divergence structure compares more favorably with that of common
proposals of complexity from holography [44, 45], as well as other studies of
scalar field complexity [4].

###  5.2 Quantum Scalar Cosmological Perturbations

Another interesting application of operator complexity to field theory is its
application to the unitary evolution of quantum cosmological perturbations in
an expanding background. We will briefly review the description of quantum
cosmological perturbations as squeezed states; see [39, 20, 21, 22, 23] for
details. We will work in $(3+1)$-dimensions and take a spatially flat
Friedmann-Lemaitre-Robertson-Walker (FLRW) metric

$\displaystyle
ds^{2}=-dt^{2}+a(t)^{2}d\vec{x}^{2}=a(\eta)^{2}\left(-d\eta^{2}+d\vec{x}^{2}\right)$
(85)

where $\eta$ is known as the conformal time. The Hubble expansion rate of this
background is characterized by the time derivative of the scale factor
$H=\dot{a}/a=a^{\prime}/a^{2}$, where a dot denotes a derivative with respect
to cosmic time $t$ and a prime denotes a derivative with respect to conformal
time $\eta$. On this background, we will consider fluctuations of a light
scalar field $\phi=\phi_{0}(t)+\delta\phi(x,t)$; the fluctuations $\delta\phi$
combine with linearized fluctuations of the metric to form gauge-invariant
perturbations, such as the curvature perturbation ${\mathcal{R}}$. When
written in terms of the Mukhanov-Sasaki variable $v\equiv z{\mathcal{R}}$,
where $z\equiv a\sqrt{2\epsilon}$ with $\epsilon=-\dot{H}/H^{2}$, the action
expanded to quadratic order becomes

$\displaystyle S=\frac{1}{2}\int d\eta\,d^{3}x\left[v^{\prime
2}-(\vec{\nabla}v)^{2}+\left(\frac{z^{\prime}}{z}\right)v^{2}-2\frac{z^{\prime}}{z}\,v^{\prime}v\right]\,.$
(86)

The action (86) represents a massless scalar field coupled to an external
time-dependent source due to the expanding cosmological background.

According to the inflationary model of the early universe, the observed
classical perturbations in the early universe began as quantum cosmological
perturbations stretched to large scales by the rapid expansion of inflation
[39]. More generally, quantum fields on time-dependent and curved backgrounds
can lead to interesting effects such as particle production and entanglement.
We will promote our scalar cosmological perturbation to a quantum field and
expand in (continuous) Fourier modes leading to the Hamiltonian

$\displaystyle\hat{H}_{\rm cosmo}=\int
d^{3}k\,\hat{\mathcal{H}}_{\vec{k}}=\int
d^{3}k\left[k\left(\hat{a}^{\dagger}_{\vec{k}}\hat{a}_{\vec{k}}+\hat{a}_{-\vec{k}}\hat{a}^{\dagger}_{-\vec{k}}\right)-i\frac{z^{\prime}}{z}\left(\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}-\hat{a}^{\dagger}_{\vec{k}}\hat{a}^{\dagger}_{-\vec{k}}\right)\right]$
(87)

where we wrote the creation and annihilation operators with explicit factors
of $\vec{k}_{\vec{n}}=\vec{n}\pi/L$ to illustrate that the first term
represents the free Hamiltonian evolution of the modes $\pm\vec{k}_{\vec{n}}$,
while the second term represents particle creation from the time-dependent
cosmological background, entangling modes of opposite momenta.

Let us first consider the unitary evolution associated with the evolution of a
single $(\vec{k},-\vec{k})$ pair of modes; we will then return to considering
the total evolution of all of the modes. The unitary time-evolution operator

$\displaystyle\hat{U}_{\vec{k}}(\eta)={\mathcal{P}}\,e^{-i\int_{0}^{\eta}\hat{\mathcal{H}}_{\vec{k}}(\tilde{\eta})d\tilde{\eta}}$
(88)

can be rewritten in terms of two-mode squeezing and rotation operators

$\displaystyle\hat{U}_{\vec{k}}(\eta)=\hat{\mathcal{S}}_{\vec{k}}\left(r_{k},\phi_{k}\right)\hat{\mathcal{R}}_{\vec{k}}\left(\theta_{k}\right)$
(89)

where

$\displaystyle\hat{\mathcal{S}}_{\vec{k}}$ $\displaystyle=$
$\displaystyle\mbox{exp}\left[\frac{r_{k}(\eta)}{2}\left(e^{-2i\phi_{k}(\eta)}\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}-e^{2i\phi_{k}}\hat{a}^{\dagger}_{\vec{k}}\hat{a}^{\dagger}_{-\vec{k}}\right)\right]$
(90) $\displaystyle\hat{\mathcal{R}}_{\vec{k}}$ $\displaystyle=$
$\displaystyle\mbox{exp}\left[-i\theta_{k}(\eta)\left(\hat{a}_{\vec{k}}\hat{a}^{\dagger}_{\vec{k}}+\hat{a}^{\dagger}_{-\vec{k}}\hat{a}_{-\vec{k}}\right)\right]$
(91)

with $k=|\vec{k}|$, are similar to the single-mode squeezing and rotation
operators (36) of Section 4. Indeed, the two-mode operators (90,91) can be
built from the fundamental set of operators

$\displaystyle\hat{e}_{1}^{\vec{k}}=\frac{\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}+\hat{a}^{\dagger}_{\vec{k}}\hat{a}^{\dagger}_{-\vec{k}}}{2},\hskip
14.45377pt\hat{e}_{2}^{\vec{k}}=\frac{i}{2}\left(\hat{a}_{\vec{k}}\hat{a}_{-\vec{k}}-\hat{a}^{\dagger}_{\vec{k}}\hat{a}^{\dagger}_{-\vec{k}}\right),\hskip
14.45377pt\hat{e}_{3}^{\vec{k}}=\frac{\hat{a}_{\vec{k}}\hat{a}^{\dagger}_{\vec{k}}+\hat{a}^{\dagger}_{-\vec{k}}\hat{a}_{-\vec{k}}}{2}$
(92)

which satisfy the $\mathfrak{su(1,1)}$ algebra (40). This allows us to use the
calculations of Section 4.3 so that the complexity for the individual pair of
modes $(\vec{k},-\vec{k})$ is

$\displaystyle{\mathcal{C}}_{\vec{k}}(\eta)=\sqrt{4r_{k}(\eta)^{2}+v_{3}(\eta)^{2}}$
(93)

where $v_{3}(\eta)$ is given in terms of $\theta_{k}(\eta)$ by (70).

Before using the time-dependence of the squeeze and rotation parameters to
determine the resulting time-dependence of the complexity (93), we immediately
notice that by construction, the complexity (93) does not include any
information about the reference state. In particular, we did not need to
assume that the quantum state is in its ground state in the asymptotic past,
as is typically done to study the time-evolution of the quantum state of
cosmological perturbations. Our complexity of the time-evolution operator of
the cosmological background (93) is thus insensitive to details such as the
initial state of the cosmological perturbations, in contrast to state-
dependent methods such as [24, 25, 43, 28]. Thus, it is potentially useful as
a measure of the complexity of the background, independent of the details of
state preparation.

(a)

(b)

(c)

Figure 4: (a) Squeezing parameter $r_{k}$, (b) squeezing angle $\phi_{k}$, and
(c) rotation angle $\theta_{k}$, each as a function of the scale factor
$a=a(\eta)$ for a de Sitter background and a fixed co-moving wavenumber $k$.
While the mode is inside the horizon $kH_{dS}\gg a$, the squeezing is small
$r_{k}\ll 1$ and the squeezing and rotation angles vary with time. When the
mode exits the horizon, the squeezing begins to grow $r_{k}\sim\ln a$ and the
squeezing angle and rotation angles “freeze-out” to fixed values.

The time-dependence for the squeezing parameter $r_{k}(\eta)$ squeezing angle
$\phi_{k}(\eta)$ and rotation angle $\theta_{k}(\eta)$ arises from the
Heisenberg equations of motion with the Hamiltonian
$\hat{\mathcal{H}}_{\vec{k}_{\vec{n}}}$ (e.g. see [21, 22, 23])

$\displaystyle\frac{dr_{k}}{d\eta}$ $\displaystyle=$
$\displaystyle-\frac{z^{\prime}}{z}\cos(2\phi_{k})\,;$ (94)
$\displaystyle\frac{d\phi_{k}}{d\eta}$ $\displaystyle=$ $\displaystyle
k+\frac{z^{\prime}}{z}\coth(2r_{k})\sin(2\phi_{k})\,;$ (95)
$\displaystyle\frac{d\theta_{k}}{d\eta}$ $\displaystyle=$ $\displaystyle
k-\frac{z^{\prime}}{z}\tanh(r_{k})\sin(2\phi_{k})\,.$ (96)

An exact solution of particular interest is known for de Sitter space,
$a(\eta)=-1/(H_{dS}\eta)$ for $\eta<0$

$\displaystyle r_{k}$ $\displaystyle=$
$\displaystyle\sinh^{-1}\left(\frac{1}{2|k\eta|}\right)=\sinh^{-1}\left(\frac{a}{2kH_{dS}}\right)$
(97) $\displaystyle\phi_{k}$ $\displaystyle=$
$\displaystyle-\frac{\pi}{4}-\frac{1}{2}\tan^{-1}\left(\frac{1}{2|k\eta|}\right)=-\frac{\pi}{4}-\frac{1}{2}\tan^{-1}\left(\frac{a}{2kH_{dS}}\right)$
(98) $\displaystyle\theta_{k}$ $\displaystyle=$
$\displaystyle|k\eta|-\tan^{-1}\left(\frac{1}{2|k\eta|}\right)=\frac{kH_{dS}}{a}-\tan^{-1}\left(\frac{a}{2kH_{dS}}\right)$
(99)

At early times $\eta\rightarrow-\infty$ (or $kH_{dS}\gg a$), the mode is
inside the horizon and the squeezing $r_{k}\ll 1$ is small, while
$\phi_{k},\theta_{k}$ are varying with time. As the mode exits the horizon at
late times $|k\eta|\ll 1$ ($a\gg kH_{dS}$), the squeezing begins to grow
$r_{k}\sim\ln a$ and the squeezing and rotation angles “freeze out” to their
superhorizon values. This behavior is shown numerically in Figures 4 and 4(c).

For a fixed time, small wavelength modes $|k\eta|\gg 1$ have small squeezing
$r_{k}\ll 1$ so the the complexity (93) is dominated by the rapidly
oscillating contribution from the phase $\theta_{k}$ through $v_{3}$, and the
complexity is ${\mathcal{C}}_{\vec{k}}\sim{\mathcal{O}}(1)$. For long-
wavelength modes $|k\eta|\ll 1$, the mode is outside the Hubble horizon and
the complexity becomes dominated by the squeezing parameter

$\displaystyle{\mathcal{C}}_{\vec{k}}(\eta)\sim 2r_{k}\sim
2\ln\left(\frac{1}{2|k\eta|}\right)\sim 2\ln\left(a(\eta)/a_{e}\right)\sim
2N_{e}$ (100)

where $a_{e}=-1/(H_{dS}k)$ is the scale factor at horizon exit, and in the
last step we wrote the result in terms of the number of e-folds of expansion
since the horizon exit of the mode. A plot of the full numerical solution of
the single-mode complexity (93) as a function of the scale factor of the
universe illustrating this behavior can be found in Figure 5.

Figure 5: The complexity of a single $(\vec{k},-\vec{k})$ mode pair of
cosmological perturbations in a de Sitter background for several different
measures of complexity as a function of the scale factor. Operator complexity
${\mathcal{C}}_{\vec{k}}$ (93) (black), studied in this paper, is sensitive to
the oscillating phase $\theta_{k}$ at early times. The covariance matrix
complexity of the corresponding state ${\mathcal{C}}_{\rm cov}$ (102) (blue)
is directly proportional to the squeezing parameter $r_{k}$, which is
insensitive to the phase. The wavefunction-method complexity of the
corresponding state ${\mathcal{C}}_{\rm wavefcn}$ (101) (red) is also
insensitive to the phase, though its sensitivity to the squeezing angle gives
very slight differences in intermediate growth. All of these different
measures of complexity grow linearly with the number of e-folds $N_{e}\sim\ln
a$ at late times (with slightly different numerical coefficients due to
differing conventions).

It is interesting to compare the operator complexity (93) for the cosmological
perturbation $(\vec{k},-\vec{k})$ mode pair to the corresponding state
complexity. There are several different techniques for calculating state
complexity777For the purposes of comparison, each of these techniques assumes
a geometric cost function and a Bunch-Davies vacuum state as a reference
state., including the wavefunction method (see [4, 24, 25]) with corresponding
complexity

$\displaystyle{\mathcal{C}}_{\rm
wavefcn}=\frac{1}{\sqrt{2}}\sqrt{\left|\ln\left|\frac{1+e^{-2i\phi_{k}}\,\tanh
r_{k}}{1-e^{-2i\phi_{k}}\,\tanh
r_{k}}\right|\right|^{2}+\rm{arctan}\left(2\sin 2\phi_{k}\,\sinh r_{k}\,\cosh
r_{k}\right)^{2}}\,,$ (101)

and the covariance matrix method (for examples, see the appendix of [24] as
well as [43]), with corresponding complexity,

$\displaystyle{\mathcal{C}}_{\rm cov}=r_{k}\,.$ (102)

In the long-wavelength (superhorizon) limit $kH_{dS}\ll a$, all of the
measures of complexity (93,101,102) grow in a similar way as the number of
e-folds888The wavefunction complexity is smaller by a factor of $2\sqrt{2}$
due to the conventions of (101); the covariance matrix method complexity is
similarly different by a factor of 2 due to conventions.

$\displaystyle{\mathcal{C}}_{\vec{k}}\sim 2\sqrt{2}\,{\mathcal{C}}_{\rm
wavefcn}\sim 2{\mathcal{C}}_{\rm cov}\sim 2\ln\left(a(\eta)/a_{e}\right)\sim
2N_{e}\,.$ (103)

The difference between the measures of complexity is primarily apparent at
small wavelengths $a\ll kH_{dS}$ when the mode is within the horizon: the
operator complexity (93) rapidly oscillates between $0$ and $2\pi$, while both
of the state complexities (101), (102) smoothly go to zero in this UV limit.
This differing behavior of the complexities can be seen in Figure 5 in the
limit of small scale factor $a$, which shows each of these complexities for a
fixed wavelength $k$.

Now that we have examined the operator complexity for a single
$(\vec{k},-\vec{k})$ mode of cosmological perturbations, let us now consider
its extension to field theory by including an integral over the Fourier modes
of the Hamiltonian (87). Analogously to (79), the total cosmological
complexity (93) now becomes

$\displaystyle{\mathcal{C}}_{\rm op}^{\rm tot}\rightarrow
L^{3/2}\sqrt{\int^{\Lambda}\left(4r_{k}(\eta)^{2}+v_{3}(\eta)^{2}\right)d^{3}k}\,.$
(104)

As noted above, in the UV limit $|k\eta|\gg 1$ the integrand is dominated by
the rapid oscillations of the phase about an ${\mathcal{O}}(1)$ value. In the
IR limit $|k\eta|\ll 1$, the integrand is dominated by the squeezing parameter
$r_{k}(\eta)\sim\ln\left(\frac{1}{|k\eta|}\right)\gg 1$. The integral in (104)
thus roughly splits into two regimes, which combine as

$\displaystyle{\mathcal{C}}_{\rm op}^{\rm tot}$ $\displaystyle\sim$
$\displaystyle
L^{3/2}\sqrt{\int_{IR}4r_{k}(\eta)^{2}\,d^{3}k+\int_{UV}v_{3}(\eta)^{2}\,d^{3}k}\sim
L^{3/2}\sqrt{\int_{0}^{|\eta|^{-1}}k^{2}\ln^{2}\left(\frac{1}{|k\eta|}\right)dk+\int_{|\eta|^{-1}}^{\Lambda}k^{2}dk}$
(105) $\displaystyle\sim$ $\displaystyle
L^{3/2}\sqrt{\frac{\alpha_{1}}{|\eta|^{3}}+\alpha_{2}\Lambda^{3}}\,,$

up to ${\mathcal{O}}(1)$ factors absorbed into the $\alpha_{i}$ constants. As
expected from Section 5.1, the UV part of the integral diverges as
$\Lambda^{3}$ and is independent of the de Sitter background. Surprisingly,
this competes with the first term, which represents growth in complexity in
the IR due to the squeezing and is proportional to the growth in volume of de
Sitter space $1/|\eta|^{3}\sim H_{dS}^{3}a(\eta)^{3}\sim H_{dS}^{3}e^{N_{e}}$.
In fact, for a UV cutoff near the Planck scale $\Lambda\sim 10^{19}$ GeV and
60 e-folds of expansion, the squeezing contribution to the complexity
dominates over that from the UV modes for Hubble scales down to approximately
$H_{dS}>1$ keV!

How does this result for the total operator complexity of quantum cosmological
perturbations (105) compare to the total state complexities (102),(101) in the
field theory limit? The generalization of the covariance matrix-method state
complexity (102) to include multiple modes in the continuum limit should be
the geometric sum

$\displaystyle{\mathcal{C}}_{\rm cov}^{\rm tot}\sim
L^{3/2}\sqrt{\int^{\Lambda}r_{k}^{2}\,d^{3}k}\,.$ (106)

The IR contribution to the integral (106) is the same as for (105), while in
the UV we have $r_{k}\approx 1/(2k|\eta|)\ll 1$ so (106) becomes

$\displaystyle{\mathcal{C}}_{\rm cov}^{\rm tot}\sim
L^{3/2}\sqrt{\int_{IR}r_{k}^{2}\,d^{3}k+\int_{UV}r_{k}^{2}\,d^{3}k}\sim
L^{3/2}\sqrt{\frac{\beta_{1}}{|\eta|^{3}}+\beta_{2}\frac{\Lambda}{|\eta|^{2}}}\,,$
(107)

where $\beta_{i}\sim{\mathcal{O}}(1)$. Interestingly, the UV divergence is
weaker here, depends on the de Sitter scale through
$\Lambda/|\eta|^{2}\sim\Lambda H_{dS}^{2}a(\eta)^{2}$, and scales with the
square of the scale factor.

Finally, let’s take the continuum field theory limit of the wavefunction-
method complexity (101)

$\displaystyle{\mathcal{C}}_{\rm wavefcn}^{\rm tot}$ $\displaystyle\sim$
$\displaystyle\frac{L^{3/2}}{\sqrt{2}}\sqrt{\int^{\Lambda}\left(\left|\ln\left|\frac{1+e^{-2i\phi_{k}}\,\tanh
r_{k}}{1-e^{-2i\phi_{k}}\,\tanh
r_{k}}\right|\right|^{2}+\rm{arctan}\left(2\sin 2\phi_{k}\,\sinh r_{k}\,\cosh
r_{k}\right)^{2}\right)d^{3}k}$ (108)

Again, we can approximate this expression by dividing the integral into IR
$|k\eta|\ll 1$ and UV $|k\eta|\gg 1$ parts,
$\int^{\Lambda}\approx\int_{IR}+\int_{UV}$. In the IR, we take the approximate
solutions (see [25]) $r_{k}\approx\ln(1/|k\eta|)\gg 1,\phi_{k}\approx-\pi/2$
and the IR integral is dominated by the first term of the integrand

$\displaystyle\int_{0}^{|\eta|^{-1}}\left|\ln\left|\frac{1+e^{-2i\phi_{k}}\,\tanh
r_{k}}{1-e^{-2i\phi_{k}}\,\tanh
r_{k}}\right|\right|^{2}\,k^{2}\,dk\sim\int_{0}^{|\eta|^{-1}}k^{2}\ln^{2}\left(\frac{1}{|k\eta|}\right)\,dk\sim\lambda_{1}\frac{1}{|\eta|^{3}}$
(109)

where $\lambda_{1}\sim{\mathcal{O}}(1)$ is an order one constant. As with the
IR contributions of the other measures of complexity, this scales as the
volume growth of de Sitter, exponentially with the number of e-folds
$1/|\eta|^{3}=H_{dS}^{3}a(\eta)^{3}\sim H_{dS}^{3}e^{3N_{e}}$. For the UV
$|k\eta|\gg 1$, the squeezing parameter and angle have the approximate
solutions [25] $r_{k}\approx 1/(2|k\eta|)\ll
1,\phi_{k}\approx-\pi/4-1/(4|k\eta|)$, where we kept the subleading dependence
in $\phi_{k}$. Inserting these solutions into the integrand of (108), the
leading behavior in the UV $|\eta|^{-1}\leq k\leq\Lambda$ is

$\displaystyle\int_{|\eta|^{-1}}^{\Lambda}k^{2}\left(\left|\ln\left|\frac{1+e^{-2i\phi_{k}}\,\tanh
r_{k}}{1-e^{-2i\phi_{k}}\,\tanh
r_{k}}\right|\right|^{2}+\rm{arctan}\left(2\sin 2\phi_{k}\,\sinh r_{k}\,\cosh
r_{k}\right)^{2}\right)\,dk$ $\displaystyle\approx$
$\displaystyle\int_{|\eta|^{-1}}^{\Lambda}k^{2}\left(\frac{8}{(k|\eta|)^{4}}\right)\,dk\approx\frac{\lambda_{2}}{|\eta|^{4}}\left(|\eta|-\frac{1}{\Lambda}\right)\rightarrow\frac{\lambda_{2}}{|\eta|^{3}}\,,$
(110)

with $\lambda_{2}\sim{\mathcal{O}}(1)$. Remarkably, the UV integral (110) does
not diverge, and we can take $\Lambda\rightarrow 0$ to obtain a UV-finite
result, as we did in the last step above. We then find that the UV
contribution (110) also scales as $1/|\eta|^{3}$, similar to the IR part of
the integral. Combining the IR (109) and UV (110) contributions to the
integrand of (108), then, we obtain a result for the total field theory state
complexity (wavefunction method) of cosmological perturbations in de Sitter
space as

$\displaystyle{\mathcal{C}}_{\rm wavefcn}^{\rm tot}\sim\gamma
L^{3/2}\frac{1}{|\eta|^{3/2}}\sim\gamma L^{3/2}H_{dS}^{3/2}a(\eta)^{3/2}\,,$
(111)

where again $\gamma$ is some other ${\mathcal{O}}(1)$ constant.

Summarizing our results for the three different complexities (105),(107),(111)
of cosmological perturbations in the field theory limit,

$\displaystyle{\mathcal{C}}_{\rm op}^{\rm tot}\sim$ $\displaystyle
L^{3/2}\sqrt{\frac{\alpha_{1}}{|\eta|^{3}}+\alpha_{2}\Lambda^{3}}\hskip
7.22743pt$ Operator Complexity (112) $\displaystyle{\mathcal{C}}_{\rm
cov}^{\rm tot}\sim$ $\displaystyle
L^{3/2}\sqrt{\frac{\beta_{1}}{|\eta|^{3}}+\beta_{2}\frac{\Lambda}{|\eta|^{2}}}$
Covariance Matrix-Method State Complexity (113)
$\displaystyle{\mathcal{C}}_{\rm wavefcn}^{\rm tot}\sim$ $\displaystyle\gamma
L^{3/2}\frac{1}{|\eta|^{3/2}}$ Wavefunction-Method State Complexity (114)

The wavefunction-method complexity (114) simply scales as the square root of
the volume, and is UV finite in contrast to the operator complexity (112) or
covariance matrix-method complexity (113). Nevertheless, as we argued above,
even for a Planck-scale UV cutoff the IR terms dominate for around 60 e-folds
of expansion when $H_{dS}>1$ keV. Assuming the IR term dominates in all of the
complexities above, the scaling ${\mathcal{C}}^{\rm tot}\sim a(\eta)^{3/2}$
means that during inflation, the total complexity of the universe due to the
cosmological perturbations grows by a factor,

$\displaystyle\frac{{\mathcal{C}}^{\rm tot}(\eta_{f})}{{\mathcal{C}}^{\rm
tot}(\eta_{i})}\sim\left(\frac{a_{f}}{a_{i}}\right)^{3/2}\sim e^{3N_{e}/2}\,,$
(115)

which is a factor of $\sim e^{90}\sim 10^{39}$ for $N_{e}\sim 60$ e-folds of
inflation.

##  6 Discussion

As a step towards a better understanding of the complexity of operators in
scalar field quantum field theories, we analyzed the operator complexity
associated with the displacement, squeeze, and rotation operators of a quantum
harmonic oscillator. Applying the approach of [9, 10], we define the
complexity of a target operator as a minimal length geodesic between two
points, identified with the identity and the target operator, in a geometry
associated with the group algebra of the fundamental operators that generate
the operator. As discussed in the Introduction, this notion of operator
complexity is independent of the choice of reference and target states.

We focused here on two sets of unitary operators of the quantum harmonic
oscillator – the displacement operator, and the squeeze and rotation operators
– which together characterize any unitary operation that is at most quadratic
in the creation and annihilation operators. The displacement operator can be
constructed with generators of the Heisenberg group, and the resulting
operator-space geometry is 3-dimensional hyperbolic space. The corresponding
complexity of a (time-dependent) displacement operator is proportional to the
magnitude of the coherent state parameter, independent of time. This result is
similar to some previous results for the complexity of coherent states
obtained using state-based complexity techniques [5, 40], but is in contrast
to other results for the complexity of the displacement operator in which a
non-trivial dependence on time was found [34, 41]. It would be interesting to
study the reasons for these differences.

We also considered the squeeze and rotation operators, which are elements of
the group SU(1,1) (isomorphic to ${\rm SL}(2,\mathbb{R})$), finding again a
non-compact, negatively curved group manifold. As two simple examples, we
confirmed that the complexity for time evolution by a free harmonic oscillator
is bounded by $2\pi$ and oscillates with a frequency given by the quantum
revival time, while the complexity for an inverted harmonic oscillator grows
linearly with time without bound, reflecting the instability of the inverted
oscillator. More generally, a generic quadratic Hamiltonian can be written as
the product of squeeze and rotation operators, and the corresponding operator
grows linearly in the squeezing parameter for large squeezing, and is
independent of the squeezing angle. This stands in contrast with the
wavefunction-method complexity of a squeezed state [24, 25], in which the
squeezing angle plays an important role in determining the time-dependent
behavior of complexity. It would be interesting to study further the
relationships and qualitative differences between the wavefunction-based and
operator-based methods for calculating complexity.

Our analysis of the complexity of the displacement, squeezing and rotation
operators may have interesting applications. For example, coherent and
squeezed states of light are essential building blocks of quantum optics [46,
47, 18], and these states can play an important role in continuous variable
quantum computation, see e.g. [15, 16, 17]. For example, in one such algorithm
[19] the squeezing $r$ is inversely proportional to the precision $\Delta_{E}$
of phase estimation (for fixed computational time), $r\sim 1/\ln(\Delta_{E})$.
Since we found that the complexity is proportional to the squeezing (73), this
implies that the resulting precision for a measurement constructed from a
squeezed state with complexity ${\mathcal{C}}_{*}$ scales exponentially with
the complexity $\Delta_{E}\sim e^{-{\mathcal{C}}_{*}}$, suggesting that
complexity itself might be an exploitable resource for quantum computation
algorithms. Alternatively, the average energy (or particle number) of a
squeezed vacuum state – another useful quantum information resource – scales
with the squeezing parameter as

$\displaystyle\langle E\rangle\sim\bar{N}=\langle
0|\hat{S}^{\dagger}(r,\phi)\hat{N}\hat{S}(r,\phi)0\rangle=\sinh^{2}r\,.$ (116)

For small squeezing $r\ll 1$, the average energy scales as the square of the
squeezing parameter $\langle E\rangle\sim r^{2}$ so that the complexity scales
as the square root of the average energy ${\mathcal{C}}\sim\sqrt{\langle
E\rangle}$. Interestingly, this dependence of the complexity on the average
energy is identical to that of the coherent state (31). For large squeezing,
however, the average energy scales exponentially with the squeezing $\langle
E\rangle\sim e^{2r}$. The resulting complexity therefore scales logrithmically
with the average energy ${\mathcal{C}}\sim\ln\sqrt{\langle E\rangle}$. We see
here that a benefit to such large-squeezing states is that the quantum circuit
complexity needed to build a squeezed state with some average energy $\langle
E\rangle_{*}$ scales slower with $\langle E\rangle_{*}$ than its coherent
state counterpart. It would be interesting to explore whether this flattening
of the dependence of complexity on average energy occurs for other resources,
and whether complexity itself can serve as a quantum information resource.

In Section 5, we used our results for the operator complexity of the quantum
harmonic oscillator to study the complexity of a free massive scalar field. In
the continuum limit with a UV cutoff $\Lambda$, we found that the complexity
rapidly grows linearly with time followed by saturation at a value
${\mathcal{C}}_{\phi}\sim L^{d/2}\Lambda^{d/2}$ that depends on the UV cutoff
and the number of spatial dimensions. As an application of our quantum
mechanical and field theory results, we studied the operator complexity of
quantum cosmological perturbations in a de Sitter background. The time
evolution of Fourier modes of cosmological perturbations can be characterized
by two-mode squeezing and rotation operators with time-dependent squeezing and
rotation parameters. Restricted to pairs of Fourier modes, we find that while
the operator complexity oscillates rapidly at early times (corresponding to
modes deep within the Hubble horizon), at late times the squeezing dominates
and the operator complexity grows linearly with the number of e-folds of
expansion ${\mathcal{C}}_{\vec{k}}\sim 2N_{e}\sim 2\ln a$. Up to the overall
numerical coefficient, this agrees with previous results on the complexity of
cosmological perturbations at late times obtained using the wavefunction-
method complexity and covariance matrix-method complexity [24, 25].
Integrating over all Fourier modes, the operator complexity roughly splits
into the geometric sum of two regimes: a UV part, which diverges as the free
scalar field, and an IR part, which represents growth due to squeezing and
grows exponentially as the volume of de Sitter space $\sim
H_{dS}^{3}e^{N_{e}}$. A similar integration for the covariance matrix-method
complexity yields a similar IR part but a weaker UV divergence, while the
total integrated wavefunction-method complexity is remarkably UV-finite and
scales as the (square root of the) volume of de Sitter space. Together, these
results for the total, integrated complexity of quantum cosmological
perturbations suggest that the complexity of the observable universe grows by
a factor of $e^{3N_{e}/2}\sim e^{90}$ during $N_{e}\sim 60$ e-folds of de
Sitter inflation due to the growth in volume. It would be interesting to
consider whether this rapid growth in complexity is consistent with bounds on
the growth or saturation of complexity expected from more general
considerations (e.g.see [29, 30] for examples), and its relation to the
entanglement entropy of cosmological perturbations (see [48, 49, 50] for some
examples).

Finally, we close with some thoughts on future directions. We have focused
here the operator geometry generated by simple operators based on the creation
and annihilation operators of a quantum harmonic oscillator. It would be
interesting to extend this analysis to more general systems and their
associated groups, including self-interacting and coupled oscillators. It is
also unclear what relationship, if any, there is between the many different
approaches towards computing complexity in the literature, even for the simple
harmonic oscillator system considered here. It would also be interesting to
consider whether there are physical constraints in which some measures of
complexity are better suited to characterizing the “circuit depth” cost of
construction than others.

## Acknowledgments

We would like to thank Arpan Bhattacharyya and Sayura Das for helpful
conversations and discussions. S.H. would like to thank the University of Cape
Town for funding this project.

##  Appendix A Alternative Matrix Representations

In this appendix, we will choose an alternative matrix representation for the
Heisenberg group, establishing representation-independence of the result.

We begin with the Heisenberg group by noting that the solutions for the
operator tangent vectors from (17) are independent of the matrix
representation of the generators of the Heisenberg group

$\displaystyle V^{1}(s)$ $\displaystyle=$ $\displaystyle v_{1}\,;$
$\displaystyle V^{2}(s)$ $\displaystyle=$ $\displaystyle v_{2}\,;$ (117)
$\displaystyle V^{3}(s)$ $\displaystyle=$ $\displaystyle 0\,,$

with a corresponding expression for the complexity (21)

$\displaystyle{\mathcal{C}}_{\rm target}=\sqrt{v_{1}^{2}+v_{2}^{2}}\,.$ (118)

We will determine the $v_{1},v_{2}$ constants by matching them to our target
operator, which is a solution to

$\displaystyle\frac{d\hat{U}(s)}{ds}=-iV^{I}(s)\ \hat{e}_{I}\ \hat{U}(s)\,,$
(119)

subject to the boundary conditions $\hat{U}(0)=\hat{\mathds{1}}$,
$\hat{U}(1)=\hat{\mathcal{U}}_{\rm target}=\hat{\mathcal{D}}(\alpha)$. In
order to find an explicit solution, we need to use a matrix representation of
the Heisenberg group. In the main text, we used a standard 3 x 3 upper-
triangular matrix representation of the Heisenberg group generators. In this
appendix, we will instead use a 4 x 4 matrix representation, which takes the
form

$\displaystyle\hat{e}_{1}=\begin{pmatrix}0&0&0&0\cr 0&0&0&0\cr
0&-i/2&0&0\cr-i/2&0&0&0\end{pmatrix}\,,\hskip
14.45377pt\hat{e}_{2}=\begin{pmatrix}0&i/2&0&0\cr 0&0&0&0\cr 0&0&0&0\cr
0&0&-i/2&0\end{pmatrix}\,,\hskip
14.45377pt\hat{e}_{3}=\begin{pmatrix}0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr
0&i/2&0&0\end{pmatrix}\,.$ (120)

Within this representation, a general element of this 4 x 4 matrix
representation of the Heisenberg group becomes

$\displaystyle\hat{U}(s)=\begin{pmatrix}1&b(s)/2&0&0\cr 0&1&0&0\cr
0&-a/2&1&0\cr-a/2&c/2&-b/2&0\end{pmatrix}\,.$ (121)

Using (120) and (121) in (119) and imposing the boundary conditions, the
parameterizations $a(s),b(s),c(s)$ of the group generators have the solutions

$\displaystyle a(s)$ $\displaystyle=$ $\displaystyle v_{1}s=-\sqrt{2}{\rm
Im}\left[\alpha(t)\right]\,;$ (122) $\displaystyle b(s)$ $\displaystyle=$
$\displaystyle v_{2}s=-\sqrt{2}{\rm Re}\left[\alpha(t)\right]\,;$ (123)
$\displaystyle c(s)$ $\displaystyle=$ $\displaystyle 0\,.$ (124)

The resulting complexity (118)

$\displaystyle{\mathcal{C}}_{\rm Heis}=\sqrt{2}\ |\alpha|\,,$ (125)

is identical to the result (29) obtained using the 3 x 3 matrix
representation. We conclude that the physical result – the circuit depth –
appears to be independent of the matrix representation, as expected.

## References

  * [1] M. A. Nielsen, A geometric approach to quantum circuit lower bounds, Science 311 (2006), no. 4, 92 [0502070].
  * [2] M. A. Nielsen, M. R. Dowling, M. Gu and A. C. Doherty, Quantum computation as geometry, Science 311 (Feb., 2006) 1133–1135.
  * [3] M. R. Nielsen, M. A.and Dowling, The geometry of quantum computation, Science 311 (2006), no. 4, 1133–1135 [0701004].
  * [4] R. Jefferson and R. C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017) 107 [1707.08570].
  * [5] M. Guo, J. Hernandez, R. C. Myers and S.-M. Ruan, Circuit Complexity for Coherent States, JHEP 10 (2018) 011 [1807.07677].
  * [6] E. Caceres, S. Chapman, J. D. Couch, J. P. Hernandez, R. C. Myers and S.-M. Ruan, Complexity of Mixed States in QFT and Holography, JHEP 03 (2020) 012 [1909.10557].
  * [7] T. Ali, A. Bhattacharyya, S. Shajidul Haque, E. H. Kim and N. Moynihan, Time Evolution of Complexity: A Critique of Three Methods, JHEP 04 (2019) 087 [1810.02734].
  * [8] T. Ali, A. Bhattacharyya, S. S. Haque, E. H. Kim, N. Moynihan and J. Murugan, Chaos and Complexity in Quantum Mechanics, Phys. Rev. D 101 (2020), no. 2, 026021 [1905.13534].
  * [9] V. Balasubramanian, M. Decross, A. Kar and O. Parrikar, Quantum Complexity of Time Evolution with Chaotic Hamiltonians, JHEP 01 (2020) 134 [1905.05765].
  * [10] V. Balasubramanian, M. DeCross, A. Kar, Y. C. Li and O. Parrikar, Complexity growth in integrable and chaotic models, JHEP 07 (2021) 011 [2101.02209].
  * [11] C. Bai, W.-H. Li and X.-H. Ge, Towards the non-equilibrium thermodynamics of the complexity and the Jarzynski identity, 2107.08608.
  * [12] P. Basteiro, J. Erdmenger, P. Fries, F. Goth, I. Matthaiakakis and R. Meyer, Quantum Complexity as Hydrodynamics, 2109.01152.
  * [13] R. Auzzi, S. Baiguera, G. B. De Luca, A. Legramandi, G. Nardelli and N. Zenoni, Geometry of quantum complexity, Phys. Rev. D 103 (2021), no. 10, 106021 [2011.07601].
  * [14] V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Annales de l’Institut Fourier 16 (1966), no. 1, 319–361.
  * [15] S. L. Braunstein and P. van Loock, Quantum information with continuous variables, Rev. Mod. Phys. 77 (2005) 513–577 [quant-ph/0410100].
  * [16] N. Liu, J. Thompson, C. Weedbrook, S. Lloyd, V. Vedral, M. Gu and K. Modi, Power of one qumode for quantum computation, Physical Review A 93 (May, 2016).
  * [17] Q. Zhuang, T. Schuster, B. Yoshida and N. Y. Yao, Scrambling and Complexity in Phase Space, Phys. Rev. A 99 (2019), no. 6, 062334 [1902.04076].
  * [18] B. L. Schumaker, Quantum mechanical pure states with gaussian wave functions, ”Phys. Rept.” 135 (Apr., 1986) 317–408.
  * [19] N. Liu, J. Thompson, C. Weedbrook, S. Lloyd, V. Vedral, M. Gu and K. Modi, Power of one qumode for quantum computation, Physical Review A 93 (May, 2016).
  * [20] L. P. Grishchuk and Y. V. Sidorov, Squeezed quantum states of relic gravitons and primordial density fluctuations, Phys. Rev. D 42 (Nov, 1990) 3413–3421.
  * [21] A. Albrecht, P. Ferreira, M. Joyce and T. Prokopec, Inflation and squeezed quantum states, Phys. Rev. D50 (1994) 4807–4820 [astro-ph/9303001].
  * [22] J. Martin, Inflationary perturbations: The Cosmological Schwinger effect, Lect. Notes Phys. 738 (2008) 193–241 [0704.3540].
  * [23] J. Martin, Cosmic Inflation, Quantum Information and the Pioneering Role of John S Bell in Cosmology, Universe 5 (2019), no. 4, 92 [1904.00083].
  * [24] A. Bhattacharyya, S. Das, S. S. Haque and B. Underwood, Cosmological complexity, Physical Review D 101 (May, 2020).
  * [25] A. Bhattacharyya, S. Das, S. S. Haque and B. Underwood, Rise of cosmological complexity: Saturation of growth and chaos, Physical Review Research 2 (Aug, 2020).
  * [26] S. S. Haque and B. Underwood, Squeezed out-of-time-order correlator and cosmology, Phys. Rev. D 103 (2021), no. 2, 023533 [2010.08629].
  * [27] S. S. Haque, C. Jana and B. Underwood, Saturation of thermal complexity of purification, JHEP 01 (2022) 159 [2107.08969].
  * [28] K. Adhikari, S. Choudhury, H. N. Pandya and R. Srivastava, PGW Circuit Complexity, 2108.10334.
  * [29] J. Maldacena, S. H. Shenker and D. Stanford, A bound on chaos, Journal of High Energy Physics 2016 (Aug., 2016) 106.
  * [30] A. R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018), no. 8, 086015 [1701.01107].
  * [31] H. A. Camargo, P. Caputa, D. Das, M. P. Heller and R. Jefferson, Complexity as a Novel Probe of Quantum Quenches: Universal Scalings and Purifications, Physical Review Letters 122 (Feb, 2019).
  * [32] A. Bhattacharyya, W. Chemissany, S. Shajidul Haque and B. Yan, Towards the web of quantum chaos diagnostics, Eur. Phys. J. C 82 (2022), no. 1, 87 [1909.01894].
  * [33] J. Kudler-Flam, L. Nie and S. Ryu, Conformal field theory and the web of quantum chaos diagnostics, JHEP 01 (2020) 175 [1910.14575].
  * [34] A. Bhattacharyya, W. Chemissany, S. S. Haque, J. Murugan and B. Yan, The Multi-faceted Inverted Harmonic Oscillator: Chaos and Complexity, SciPost Phys. Core 4 (2021) 002 [2007.01232].
  * [35] A. Bhattacharyya, S. S. Haque and E. H. Kim, Complexity from the reduced density matrix: a new diagnostic for chaos, JHEP 10 (2021) 028 [2011.04705].
  * [36] T. Ali, A. Bhattacharyya, S. Shajidul Haque, E. H. Kim and N. Moynihan, Post-Quench Evolution of Complexity and Entanglement in a Topological System, 1811.05985.
  * [37] F. Liu, S. Whitsitt, J. B. Curtis, R. Lundgren, P. Titum, Z.-C. Yang, J. R. Garrison and A. V. Gorshkov, Circuit complexity across a topological phase transition, Phys. Rev. Research 2 (Mar, 2020) 013323.
  * [38] Z. Xiong, D.-X. Yao and Z. Yan, Nonanalyticity of circuit complexity across topological phase transitions, Phys. Rev. B 101 (2020), no. 17, 174305 [1906.11279].
  * [39] V. F. Mukhanov, H. Feldman and R. H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203–333.
  * [40] M. Guo, Z.-Y. Fan, J. Jiang, X. Liu and B. Chen, Circuit complexity for generalized coherent states in thermal field dynamics, Phys. Rev. D101 (2020), no. 12, 126007 [2004.00344].
  * [41] P. Caputa, J. M. Magan and D. Patramanis, Geometry of Krylov Complexity, 2109.03824.
  * [42] L. Hackl and R. C. Myers, Circuit complexity for free fermions, JHEP 07 (2018) 139 [1803.10638].
  * [43] J.-L. Lehners and J. Quintin, Quantum circuit complexity of primordial perturbations, Physical Review D 103 (Mar, 2021).
  * [44] A. Reynolds and S. F. Ross, Divergences in Holographic Complexity, Class. Quant. Grav. 34 (2017), no. 10, 105004 [1612.05439].
  * [45] D. Carmi, R. C. Myers and P. Rath, Comments on Holographic Complexity, JHEP 03 (2017) 118 [1612.00433].
  * [46] B. L. Schumaker and C. M. Caves, New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation, Phys. Rev. A 31 (May, 1985) 3093–3111.
  * [47] A. Perelomov, Generalized coherent states and their applications.  Springer, Berlin, 1986.
  * [48] R. Brandenberger, V. Mukhanov and T. Prokopec, Entropy of a classical stochastic field and cosmological perturbations, Physical Review Letters 69 (Dec, 1992) 3606–3609.
  * [49] R. Brandenberger, V. Mukhanov and T. Prokopec, Entropy of the gravitational field, Physical Review D 48 (Sep, 1993) 2443–2455.
  * [50] S. Brahma, O. Alaryani and R. Brandenberger, Entanglement entropy of cosmological perturbations, Phys. Rev. D 102 (2020), no. 4, 043529 [2005.09688].