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Published for SISSA by Springer

Received: April 12, 2018

Revised: May 25, 2018

Accepted: June 3, 2018

Published: June 5, 2018

Evolution of complexity following a quantum quench

in free field theory

Daniel W.F. Alvesa,b and Giancarlo Camiloc

aSão Paulo State University (UNESP), Institute for Theoretical Physics (IFT),

R. Dr. Bento T. Ferraz 271, Bl. II, 01140-070, Sao Paulo, SP, Brazil
bCenter for Quantum Mathematics and Physics, Department of Physics, University of California,

Davis, CA 95616, U.S.A.
cInternational Institute of Physics, Federal University of Rio Grande do Norte,

Campus Universitário, Lagoa Nova, 59078-970, Natal, RN, Brazil

E-mail: dwfalves@gmail.com, gcamilo@iip.ufrn.br

Abstract: Using a recent proposal of circuit complexity in quantum field theories intro-

duced by Jefferson and Myers, we compute the time evolution of the complexity following

a smooth mass quench characterized by a time scale δt in a free scalar field theory. We

show that the dynamics has two distinct phases, namely an early regime of approximately

linear evolution followed by a saturation phase characterized by oscillations around a mean

value. The behavior is similar to previous conjectures for the complexity growth in chaotic

and holographic systems, although here we have found that the complexity may grow or

decrease depending on whether the quench increases or decreases the mass, and also that

the time scale for saturation of the complexity is of order δt (not parametrically larger).

Keywords: Field Theories in Higher Dimensions, Gauge-gravity correspondence

ArXiv ePrint: 1804.00107

Open Access, c© The Authors.

Article funded by SCOAP3.
https://doi.org/10.1007/JHEP06(2018)029

mailto:dwfalves@gmail.com
mailto:gcamilo@iip.ufrn.br
https://arxiv.org/abs/1804.00107
https://doi.org/10.1007/JHEP06(2018)029


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Contents

1 Introduction 1

2 Mass quenches in free scalar fields 3

3 Complexity in quantum field theory 5

4 Introducing the circuit 6

4.1 Reference and target states 6

4.2 Choosing the gates 8

4.3 Matrix representation 10

4.4 Boundary conditions for the geodesics 12

4.5 Phase factor ambiguity 13

5 Time evolution of the complexity 13

5.1 Geometrizing the problem 13

5.2 Symmetries and Killing vectors 15

5.3 Geodesics and the complexity 16

5.3.1 Static limit: the vacuum complexity 18

5.3.2 Sudden quench limit 19

5.3.3 Small and large reference scale M 19

5.3.4 UV behavior 20

5.3.5 Numerical analysis 20

5.4 Higher dimensions 22

6 Discussion and conclusion 22

A Relation between Schrödinger and Heisenberg picture circuits 24

1 Introduction

Entanglement, the most distinct trait of quantum mechanics, has proven to be a key concept

in many different areas of physics. In particular, it has recently become a standard term in

the high-energy physics vocabulary due mainly to the AdS/CFT correspondence, the Ryu-

Takayanagi formula [1] and its generalization [2]. Many previous works have explored these

connections (see, e.g., [3] for a review) with the aim of elucidating the basic mechanism of

the AdS/CFT correspondence and space-time emergence.

Some questions remain unanswered though, in particular how the behavior of the

interior of black holes (which is a region often inaccessible to Ryu-Takayanagi surfaces) in

asymptotically AdS space-times is encoded in the dual CFT. In particular, by a classical

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gravity computation, it is possible to show that the volume of the Einstein-Rosen bridge

connecting the two sides of an eternal AdS black hole grows linearly in time, which raises

the question of what quantity this corresponds to in the dual CFT [4].

As we will explain below, it has been proposed that the computational complexity in

the field theory, or complexity for short, could be the quantity whose growth is the dual to

that of the Einstein-Rosen bridge in the gravity side. Complexity is a concept originally

introduced in the field of classical computation [5], though it is also extensively studied

in quantum information. The basic idea is to consider some reference state |Ω〉 and some

target state |ψ〉, as well as a set of quantum operators or gates, OI . The complexity of |ψ〉
is then the length of the minimal circuit connecting it to |Ω〉 by successive applications of

the operators OI . To properly define “length” and “minimal”, we need a notion of distance,

or cost, associated to a given circuit, that in principle we are free to define. The definitions

of cost and the choice of OI should be made as to capture the intuition that the gates are

easy to apply. For instance, the operators should act on only a few degrees of freedom at

a time. It is also possible to study the complexity of quantum operators themselves [6]

instead of quantum states, but in this paper we focus on the latter.

Two conjectures which aim to connect the complexity of the boundary field theory to

geometric constructs in the bulk space-time have been proposed, the so called “complexity

= action” [7] and “complexity = volume” [8] proposals. In these works, it was argued that

these objects in the bulk indeed grow as expected. The bulk computation of the evolution

of holographic complexity was put on firmer grounds by [9], and it has also been subject

of many recent works [10–19]. For instance, in [20], the holographic complexity following

a quantum quench was calculated by modeling the quench with an AdS-Vaidya spacetime

and it was found that the complexity evolves approximately linearly.

To truly check the proposed correspondence, it is necessary to have a precise definition

of complexity in quantum field theory in the first place. That was only put forward very

recently [21–25], where the authors defined and calculated the ground state complexity of

free quantum field theories. Their approach was inspired by previous work by Nielsen [26]

where the task of finding the optimal circuit connecting two quantum states was turned

into a geometric problem of finding geodesics in a curved space. Some similarities with the

general relativistic computation of holographic complexity were reported in [21], and [22],

even though the theories with classical gravity duals are known to be very different, i.e.,

strongly coupled and with a large number of degrees of freedom.

The study of complexity in quantum field theory might also prove useful for subjects

unrelated to holography. For instance, in [27], a lower bound on the complexity of an

ensemble of operators in lattice systems was obtained in terms of out-of-time order corre-

lators. So, in this sense quantum chaos and complexity seem to be quantitatively related

and it would be interesting to check if something similar happens in continuum QFT’s. It

has also been conjectured [28] that a universal second law of complexity should hold for

operators in quantum chaotic systems (analogous to the second law of thermodynamics).

In the present work, we use the definition of [21] and, following a construction similar

to the one done in [29] and [30] in the context of cMERA, we calculate the complexity of

circuits preparing the instantaneous state after a quantum quench and study its time evo-

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lution. Our basic aim here is to explore the definition [21] in a dynamical, non-equilibrium,

setting, to compare our results with holographic calculations, and to use them as a bench-

mark for future studies on interacting theories. The calculation becomes particularly simple

if the so called F2 complexity is used, since in this case the problem of finding the optimal

circuit geometrizes to that of computing geodesics in a Riemannian curved space. We

shall deal with mass quenches in free bosonic theory. By choosing to work with free field

quenches, we keep the problem computationally tractable while still allowing for non-trivial

dynamics. In this case, one should not expect thermalization in the usual sense of Gibbs

ensemble, since the free scalar field is an integrable system. Nevertheless, it was shown

in [31] that many such integrable systems do thermalize in a broader sense, approaching

the so called Generalized Gibbs Ensemble (GGE). In fact, it has been shown in [32] that

after the quench the entanglement entropy in momentum-space agrees with the prediction

from a GGE, with the density matrix after the quench being explicitly computed in terms

of the quench data.

The paper is organized as follows. We begin with a brief review of smooth mass

quenches in free scalar theories in section 2. We then discuss the definition of circuit

complexity in QFT as proposed in [21] in section 3. In section 4, we introduce our circuit

by defining the reference and target states, as well as the elementary gates. The complexity

is then calculated in section 5, the final result being given by (5.24). The final remarks

appear in section 6. In appendix A, we prove the equivalence of Schrödinger and Heisenberg

picture calculations of the complexity, which is an important step of the calculation.

Note added. After the paper appeared online we have been informed of an upcoming

paper [33] containing a very similar analysis.

2 Mass quenches in free scalar fields

We begin with a quick review of the approach introduced in [34] (the reader is referred

to [32] for details) to study mass quenches in a free bosonic quantum field theory. The

action describing the system is

I =
1

2

∫
ddx

(
ηµν∂µφ∂νφ−m(t)2φ2) , (2.1)

where the mass profile m(t) asymptotes to constants min ≡ m(−∞) and mout ≡ m(+∞)

at early and late times, respectively, so that asymptotic states exist before and after the

quench. We shall focus on d = 2 for simplicity. This problem is equivalent to a standard

quantum scalar field of constant mass m0 placed in a cosmological background and therefore

can be understood using intuition from quantum field theory in curved spacetimes [32]. We

focus on the particular choice

m(t)2 =
1

2

(
mout

2 +min
2
)

+
1

2

(
mout

2 −min
2
)

tanh
t

δt
, (2.2)

a smoothened version of a step function connecting the initial and final masses where the

nontrivial time dependence happens roughly within the scale δt, since in this case exact

solutions are known for the classical mode functions.

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There are two types of mode functions, the so-called in- and out-modes φ
in/out
k (t, x) =

(2π)−1/2ei kxχ
in/out
k (t), depending on how we choose to fix the boundary conditions. They

are given by

χin
k (t) =

1√
2ωin

e−iω+t−iω−δt log(2 cosh t
δt

)
2F1

[
1 + iω−δt; iω−δt; 1− iωinδt;

1 + tanh t
δt

2

]
(2.3a)

χout
k (t) =

1√
2ωout

e−iω+t−iω−δt log(2 cosh t
δt

)
2F1

[
1 + iω−δt; iω−δt; 1 + iωoutδt;

1− tanh t
δt

2

]
,

(2.3b)

where 2F1[...] is the hypergeometric function and

ωin =
√
k2 +min

2 ωout =
√
k2 +mout

2 ω± =
1

2
(ωout ± ωin) . (2.4)

The in-mode satisfies limt→−∞ χ
in
k (t) ∼ e−iωint and therefore is adapted to observers at

early times (at t = −∞, before the quench starts), while the out-mode similarly satisfies

limt→+∞ χ
out
k (t) ∼ e−iωoutt and is adapted to post-quench observers at t = +∞.

Canonical quantization of the field φ then proceeds in the usual way. Both in and

out-modes are equally good and the field can be expanded using either of them, i.e.,

φ(t, x) =

∫
dk

2π
ei kx

[
a

in/out
k χ

in/out
k (t) + a

† in/out
−k χ

∗ in/out
−k (t)

]
, (2.5)

To each set of modes (in/out) one can associate a corresponding vacuum state |0in/out〉,
namely,

a
in/out
k |0in/out〉 = 0 , (2.6)

as well as particle excitations by acting with a
† in/out
k . They are of course different, being

adapted either to early or late-time measurements only, though both correct. In fact, the

two definitions are connected by a Bogoliubov transformation that is block-diagonal with

respect to k (i.e., it mixes only opposite momentum modes) of the type

ain
k = Ak a

out
k +Bk a

†out
−k , (2.7)

where

Ak =

(√
ωout

ωin

Γ(1− iωinδt)Γ(−iωoutδt)

Γ(−iω+δt)Γ(1− iω+δt)

)∗
(2.8a)

Bk =

(
−
√
ωout

ωin

Γ(1− iωinδt) Γ(iωoutδt)

Γ(iω−δt) Γ(1 + iω−δt)

)∗
(2.8b)

and |Ak|2 − |Bk|2 = 1. From this it is straightforward to check that the in and out vacua

are indeed physically different: |0in〉, which contains no particles for an initial observer, is

populated by |Bk|2 particles of the out type (and vice-versa), which is the phenomenon of

particle production by the quench.

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3 Complexity in quantum field theory

We start by reviewing the definition of circuit complexity in quantum field theory as pro-

posed in [21]. As mentioned in the Introduction, to properly define complexity one first

needs to specify a reference state |Ω〉, a target state |ψ〉, and a set of elementary gates

(quantum operators) {OI} to be used to build a quantum circuit connecting the them.

The circuit then consists of a continuum set of operations parametrized by some parame-

ter s (we take s ∈ [0, 1] with no loss of generality) as

|ψ〉 = P e−
∫ 1
0 Y

I(s)OI ds |Ω〉 , (3.1)

where the set of complex functions Y I(s) characterize a particular circuit roughly by de-

ciding whether or not the gate OI is applied at “level s” of the circuit. As we shall see,

in general there will be many distinct such functions preparing the same final state. The

path-ordering symbol P here ensures that the circuit is built in the direction of growing s,

i.e., that operators are applied sequentially from s = 0 to s = 1. In the following it will be

convenient to introduce

U(s) = P e−
∫ s
0 Y

I(s′)OI ds′ , U(0) = 1 (3.2)

such that the circuit becomes

|ψ〉 = U(1)|Ω〉 . (3.3)

Next, we need a notion of circuit depth that will allow us to quantify the costs of

applying different circuits to accomplish the same task of connecting a pair of reference and

target states. The computational complexity of this task (or, equivalently, the complexity

of the state |ψ〉 relative to |Ω〉) will then be defined as the cost of the optimal circuit

amongst all, i.e., the one with minimal depth. Given a circuit, or set of functions {Y I(s)},
its depth can be quantified by any functional of the form

D [U ] =

∫ 1

0
F
(
Y I(s), ∂sY

I(s)
)
ds (3.4)

with the cost function F satisfying a list of reasonable physical requirements such as

smoothness, positivity, and the triangle inequality (see [26] for details). It remains un-

clear whether there is a preferred choice of F and, for the sake of simplicity, we shall

follow [21] and focus here on the so called F2 distance defined by F =
√
δIJ Y I (Y J)∗, even

though the strategy is exactly the same for any other distance measure. This means that,

for our purposes, the circuit depth is defined by

DF2 [U ] =

∫ 1

0
ds

√∑
I

|Y I(s)|2 (3.5)

and the corresponding complexity is simply

CF2 = DF2 [Uoptimal] = min
{Y I}
DF2 [U ] . (3.6)

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The main advantage of this approach, to be discussed in detail below, relies on the

fact that the problem of minimizing DF2 [U ] to calculate the complexity can in principle be

rephrased as a geometric problem of finding geodesics in a Riemannian manifold. Namely,

for specific choices of reference and target states the set of gates OI needed to implement

the task closes a Lie algebra and hence the circuits U are elements of the corresponding

Lie group. One can then cover the group manifold with coordinates x in such a way

that quantum circuits become curves in this manifold; the curves are defined by Y I(s) =

Y I (x(s), ∂sx(s)) and finding the optimal circuit amounts simply to finding the minimal

length curve connecting the reference and target states. In particular, using the F2 cost

function above reduces the problem to that of calculating a geodesic in a Riemannian

space1 whose metric is given by

gµν
dxµ

ds

dxν

ds
=
∑
I

∣∣Y I(s)
∣∣2 . (3.7)

In the following we shall design quantum circuits preparing the time-dependent state

following a quantum quench from a product state having no entanglement over spatial

subregions. We will see that an infinite number of elementary gates is needed, each acting

on a particular momentum mode k of the quantum field.2 These gates will be denoted by

OI,k and the circuit by a set of functions Y I
k (s). Thus, we will write the circuit as

U(s) = P ei
∫ s
0 ds′

∑
k Y

I
k (s′)OI,k . (3.8)

The sum runs over discrete values of k for the field in the compact space and must be

replaced by V
∫
dk in the continuum limit, where V is the spatial volume factor needed to

make the exponent dimensionless. In our case, when it is time to take the continuum limit

we shall ignore this factor and set V = 1 for simplicity.

4 Introducing the circuit

4.1 Reference and target states

In this section, we specify the target and reference states that define our circuit. We shall

choose our reference |Ω〉 to be a product state, characterized by the absence of entanglement

in any spatial subregion. A nice way to impose this condition (see [21, 29, 30] for similar

work in the context of cMERA tensor networks) is by demanding the state to satisfy

Reference:

(√
M φ(x) +

i√
M
π(x)

)
|Ω〉 = 0 (4.1)

for some arbitrarily chosen mass scale M . It is straightforward to check that this definition

implies [29]

〈φ(x)φ(x′)〉 = δ(x− x′) (4.2a)

〈π(x)π(x′)〉 = δ(x− x′) , (4.2b)

which indeed indicates that the state have no spatial entanglement.

1For more general cost functions F we end up with Finsler geometries instead [21].
2If we work in a non-compact space, the k’s will form a continuous set, meaning that we would have an

infinite and uncountable set of elementary gates. We shall work in a compact space, where the momentum

label takes values on a discrete set and things are well-defined.

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Our target state, on the other hand, will be time-dependent and shall be denoted

|ψ(t)〉. We choose it to be the instantaneous state of the system at time t following a mass

quench that starts from the vacuum of the pre-quench Hamiltonian at t = −∞, i.e.,

Target: |ψ(t)〉 = E(t,−∞) |0in〉 (4.3)

where E is the time evolution operator.

Notice that the states above are defined in the Schrödinger picture, which is why

no time dependence shows up in the fields in (4.1) and the whole time dependence is

contained in the target state |ψ(t)〉. Indeed, in the field representation |{φ(x)}〉 (where

π(x) = −i δ
δφ(x)) equation (4.1) can be seen as a functional differential equation defining

the wave functional Ω[φ(x)] ≡ 〈{φ(x)}|Ω〉, whose solution is a simple Gaussian function

with width controlled by M . By discretizing the field into a chain of coupled harmonic

oscillators this becomes precisely the Gaussian product state used by the authors of [21]

as the reference state with respect to which the free scalar field complexity is evaluated.

Similarly, in the absence of a quench, i.e. the limit min = mout , the time evolution in (4.3)

is trivial and the target state reduces simply to the vacuum of a free massive scalar field,

which also corresponds to the choice of target state used in [21].3

Nevertheless, since the analysis of the exactly solvable quantum quench model of pre-

vious section is done in the Heisenberg picture, it will prove more convenient to redefine

our whole circuit and work in the Heisenberg picture instead. The physical intuition be-

comes slightly less obvious in this case though, since the roles are interchanged: the time

dependence due to the quench now appears in the constraint defining the reference state

(since the fields themselves are now time dependent), whereas the target state becomes

constant. Anyway, the Heisenberg picture reference and target states are defined by

Reference:

(√
M φ (x, t) +

i√
M

π (x, t)

)
|Ω (t)〉 = 0 (4.4a)

Target: |ψ〉 = |ψ(−∞)〉 = |0in〉 . (4.4b)

or equivalently, using (2.5), (2.6) and (2.7), by the constraints

[
αk(t) a

out
k + βk(t) a

†out
−k

]
|Ω(t)〉 = 0 (4.5a)

(Ak a
out
k +Bk a

†out
−k ) |ψ〉 = 0 , (4.5b)

3Actually even in the presence of the quench (min 6= mout) our target state is still a Gaussian state of the

same type used in [21], since the pre-quench vacuum is Gaussian and the time evolution operator (despite

of its nontrivial time-dependence) is basically quadratic in the fields. However, in this case a different set

of fundamental gates than the ones used in [21] is needed to construct the time-dependent circuit in their

setup (one including also the φ2 and π2 generators, since they are the ones responsible for the nontrivial

time evolution in the quench Hamiltonian and our target state is defined by time evolution). We thank

Rob Myers for pointing this out.

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where A and B are the Bogoliubov coefficients (2.8) and we have introduced

αk(t) ≡
1

2

(√
M χout

k (t) +
i√
M

χ̇out
k (t)

)
(4.6a)

βk(t) ≡
1

2

(√
M χout ∗

−k (t) +
i√
M

χ̇out ∗
−k (t)

)
. (4.6b)

Presenting the circuit in terms of the out modes is interesting since we are interested in

studying time evolution of the complexity and, in particular, its late-time behavior after

the quench has finished (a regime where the out modes are the most natural ones to

use). For simplicity of notation, hereinafter we drop the superscript out and convention

that mode functions χk and creation/annihilation operators ak, a
†
k carrying no superscript

always refer to out.

In principle it is not quite obvious that the circuit complexity will be the same in both

Schrödinger and Heisenberg pictures, so in appendix A we show explicitly that this is the

case. Also, as discussed above, in the limit where min = mout (no quench) our setup is

equivalent to the one studied in [21], so as consistency check we shall be able to recover

their results using Heisenberg picture calculations.

Having defined the target and reference states, the next task is choosing a set of gates

connecting them. From now on it is always implicit that we are working in the Heisenberg

picture.

4.2 Choosing the gates

The choice of gates used to build a circuit connecting the reference and target states is

ambiguous, since there are not many requirements to constrain the options. In addition to

being able to implement the desired task in a unitary way, we only require that the gates

should act on the smallest possible number of degrees of freedom at each step, so that they

constitute the most basic possible set of operations needed to construct the final state.

This captures the intuition that the gates are easy to apply, and that the complexity then

measures how difficult it is to complete the task in the sense of counting the minimum

number of elementary operations needed. In a lattice system, for instance, this can be

achieved by using gates that only act on neighboring lattice sites, i.e., gates comprising

a notion of real space locality. But this is not the only option, and indeed the gates we

shall introduce are different: they are local in momentum space, in the sense that they

only act on specific fixed-momentum subspaces of the full Fock space (recall that the Fock

space of free theories factorizes into subspaces of fixed-momentum modes [32]). Clearly

these are non-local operations from the point of view of real space, but the important

thing is that they also allow us to break the task of connecting two states in the full Fock

space into a sequence of operations that act only fixed-momentum subspaces (the minimal

number of which defines the complexity). This is, for instance, the way in which cMERA

quantum circuits are constructed for free fields [29], where the procedure of acting with

the gates has the nice interpretation of introducing entanglement at different energy scales.

Ultimately, the definition of the gates is subjective and we often have to resort to choices

of practical convenience, but the underlying idea here is to search for universal features of

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the complexity that do not depend on this specific choice. We shall see in the following

that this is precisely what happens in the present work, namely, the results of [21] for

the free scalar field complexity (obtained using a different set of gates based on real-space

intuition) are exactly recovered using our set of gates in the limit of no-quench.

The target state |ψ(t)〉 that we want to construct is built from the initial vacuum using

essentially the quench Hamiltonian (through time evolution). So, to motivate our choice of

gates, it is instructive to write the Hamiltonian associated with our model (2.1) in terms

of creation and annihilation operators [32],4

H =
1

2

∫
dd−1k

[
2
(
|χ̇k|2 + ω2

k|χk|2
)
a†kak +

(
χ̇2
k + ω2

kχ
2
k

)
aka−k +

(
χ̇∗k

2 + ω2
kχ
∗
k

2
)
a†ka
†
−k

]
+E0.

(4.7)

We see that the Hamiltonian is written as a linear combination of operators OI,k which

mix only k and −k modes (here I = 1, 2, 3 or, equivalently, I = +,−, 3), namely5

O+,k = O1,k + iO2,k ≡ a†ka
†
−k (4.8a)

O−,k = O1,k − iO2,k ≡ aka−k (4.8b)

O3,k ≡
1

2
(a†kak + a−ka

†
−k) . (4.8c)

It is straightforward to check that (for each k) these operators close a SU(1, 1) Lie algebra,6

[O3,k,O±,k] = ±O±,k, [O+,k,O−,k] = −2O3,k ,

or

[O1,k,O2,k] = −iO3,k, [O2,k,O3,k] = iO1,k, [O3,k,O1,k] = iO2,k .

As we shall see below, using them as fundamental gates is enough to build a quantum

circuit connecting the target and reference states introduced in the last section. Hence,

they constitute a natural set of operators to construct our circuit, and will be our choice.

A similar choice of gates has been used recently in [22] to study a different notion of

complexity for quantum field theory states. For future convenience we shall adopt the set

of gates {O1,k,O2,k,O3,k}, so from now on whenever a summation over I appears it refers

to I = 1, 2, 3.

The quantum circuit is written as

U(s) = P ei
∫ s
0 ds′

∫
dk Y Ik (s′)OI,k (4.9)

where the functions Y I
k (s) define the circuit in the same spirit as in [21]. We emphasize

here the abuse of language when writing
∫
dk, which is to be understood as

∑
k for the field

4When min = mout (no quench) this reduces to the standard particle number piece ∼
∫
dd−1kωk a

†
kak.

5Here for convenience we have added an extra piece ∼ a−ka
†
−k in the definition of O3,k that did not

seem to be present in (4.7). Actually it was hidden in the constant term at the end and reduces to first

piece after using the commutation relation to convert a−ka
†
−k into a†−ka−k and renaming the integration

variable k → −k.
6When writing this we are implicitly assuming that the field is defined is a compact space so that k

is a discrete index, otherwise a delta function divergence appears at conciding momenta. There is also a

subtlety at the zero modes (when k = −k) which we shall ignore here since it plays no role.

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in the compact space or as V
∫
dk in the continuum limit (V is the spatial volume, which

we ignore by taking V ≡ 1). Unitarity of U is guaranteed by taking the functions to be

real,
(
Y I
k

)∗
= Y I

k , since the OI,k are all Hermitian. Our task is then to find the functions

Y I
k (s) that minimize the circuit depth (3.5) while satisfying the boundary conditions

U(0) = 1 (4.10a)

|ψ〉 = U(1)|Ω〉 . (4.10b)

Note that differentiation of (4.9) with respect to s yields∫ Λ

−Λ
dk Y Ik (s)OIk = ∂sU.U

−1 . (4.11)

The main task at this point is to isolate the Y I
k to find the line element (3.7), but this is

not possible in the present form. A similar problem was faced by the authors of [21] when

dealing with circuits that build the ground state of free scalar fields. Their trick in that

case was to reformulate the problem in matrix language and focus on how the circuit acted

on 2 × 2 matrices defining the ground state, which allowed them to solve for the Y I
k . We

shall follow the same steps in the next sub-section.

4.3 Matrix representation

Let us begin by taking a look at how the transformation U(s) acts on the creation and

annihilation operators. We first define their transformed versions by

ak(s) = U(s)−1 ak U(s) (4.12a)

a†−k(s) = U(s)−1 a†−k U(s) (4.12b)

and differentiate with respect to s, i.e.,

d

ds

(
ak(s)

a†−k(s)

)
=
dU(s)−1

ds

(
ak
a†−k

)
U(s) + U(s)−1

(
ak
a†−k

)
dU(s)

ds
(4.13)

Now we use (4.9) and notice that, due to the path-ordering symbol P (which puts higher

values of s always to the left), the derivative of U yields a factor on the left,

dU(s)

ds
=

(
i

∫ Λ

−Λ
dk Y Ik(s)OIk

)
U(s) . (4.14)

For U(s)−1 (defined with anti path-ordering P̄) we have the opposite and the operator

appears on the right,

dU(s)−1

ds
= U(s)−1

(
−i

∫ Λ

−Λ
dk Y Ik (s)OIk

)
, (4.15)

such that

d

ds

(
ak(s)

a†−k(s)

)
= U(s)−1

[(
−i

∫ Λ

−Λ
dk′ Y Ik′(s)OIk′

)
,

(
ak
a†−k

)]
U(s) . (4.16)

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The commutator is straightforwardly calculated using the canonical commutation relations

for ak, a
†
k and yields a linear combination of ak and a†k themselves. These become ak(s)

and a†k(s) after taking into account the U−1 and U in equation (4.16), so that one gets

d

ds

(
ak(s)

a†−k(s)

)
= −Gk(s)

(
ak(s)

a†−k(s)

)
(4.17)

where Gk(s) is the matrix

Gk(s) =

(
−iY 3

k (s) −iY 1
k (s)− Y 2

k (s)

+iY 1
k (s)− Y 2

k (s) +iY 3
k (s)

)
. (4.18)

We immediately recognize Gk (s) as an element of the SU(1, 1) algebra in the 2× 2 matrix

representation, i.e.,

Gk(s) = 2
[
iY 2

k (s)Σ1 − iY 1
k (s)Σ2 − iY 3

k (s)Σ3

]
≡ XI

k(s)ΣI , (4.19)

where

Σ1 =
1

2

(
0 i

i 0

)
, Σ2 =

1

2

(
0 1

−1 0

)
, Σ3 =

1

2

(
1 0

0 −1

)
(4.20)

satisfy the SU(1, 1) algebra

[Σ1,Σ2] = −i Σ3, [Σ2,Σ3] = i Σ1, [Σ3,Σ1] = i Σ2 .

To simplify the notation, in the last step we have introduced

X1
k(s) ≡ 2iY 2

k (s), X2
k(s) ≡ −2iY 1

k (s), X3
k(s) ≡ −2iY 3

k (s) . (4.21)

Equation (4.17) is formally solved by(
ak(s)

a†−k(s)

)
= Mk(s)

(
ak
a†−k

)
(4.22)

with

Mk(s) = P e−
∫ s
0 ds

′Gk(s′) = P e−
∫ s
0 ds

′XI
k(s′)ΣI . (4.23)

The crucial point is that Mk(s), being the exponential of the ΣI , is itself nothing but

a SU(1, 1) transformation. In other words, U(s) acts on the creation and annihilation

operators as

U(s)−1

(
ak
a†−k

)
U(s) = Mk(s)

(
ak
a†−k

)
, (4.24)

where Mk(s) is a generic SU(1, 1) transformation that we can parametrize as

Mk(s) =

(
pk(s) −qk(s)
−q∗k(s) p∗k(s)

)
(4.25)

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in terms of some pk, qk satisfying |pk|2 − |qk|2 = 1. Hence our whole circuit can be under-

stood as a sequence of SU(1, 1) transformations Mk(s).

This fact is what allows us to express the functions Y I
k (s) defining a particular circuit

in terms of a particular sequence of SU(1, 1) transformations, solving the problem raised

at the end of the previous subsection. Using the fact that the generators ΣI satisfy the

orthogonality relation

Tr
(

ΣIΣ
†
J

)
=

1

2
δIJ , (4.26)

equation (4.23) can be easily solved for the XI
k(s) through differentiation with respect to

s, namely,

XI
k(s) = 2 Tr

(
∂sMk(s)Mk(s)

−1 Σ†I

)
. (4.27)

This determines the Y I
k (s) associated with a given SU(1, 1) transformation Mk(s) via (4.21).

4.4 Boundary conditions for the geodesics

Having isolated the Y I
k (s) we now proceed to the second issue raised on the last section,

namely, to express the boundary conditions (4.10) on U(s) as boundary conditions for the

corresponding trajectories in the SU(1, 1) manifold.

Notice that the condition (2.7) can be written using (4.10b) as

(Ak Bk)

(
ak
a†−k

)
U(1)|Ω〉 = 0 . (4.28)

Applying U(1)−1 on both sides and using (4.24) we have

(Ak Bk)Mk(1)

(
ak
a†−k

)
|Ω〉 = 0 , (4.29)

and by comparing with the defining constraint (4.5a) for the reference state |Ω(t)〉 we

immediately identify

(Ak Bk)Mk(1) = Nk (αk βk) , (4.30)

where Nk is a proportionality constant.

This is a pair of equations that can be solved for pk(1) and qk(1) (the matrix compo-

nents of Mk(1)) in terms of Ak, Bk, αk, βk (the coefficients characterizing the reference and

target states). The boundary conditions (4.10) are then rephrased as

p∗k(1) = N∗kα
∗
kAk −NkβkB

∗
k (4.31a)

qk(1) = N∗kα
∗
kBk −NkβkA

∗
k (4.31b)

p∗k(0) = 1 (4.31c)

qk(0) = 0 , (4.31d)

where we included also the last two which correspond simply to the statement that Mk(0) =

1. The proportionality constant is fixed by requiring compatibility with the SU(1, 1) nor-

malization |pk(1)|2 − |qk(1)|2 = 1, which leads to

|Nk| =
1√

|αk|2 − |βk|2
. (4.32)

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We see that there is still a phase factor freedom in Nk. This is related to an ambiguity

in the definition of our reference and target states which we shall discuss in detail in the

next section, so for now one can simply take Nk to be real without loss of generality and

postpone the discussion of the phases to the next section.

4.5 Phase factor ambiguity

It is important to notice (see [29] for a similar discussion in the context of cMERA) that

there is an ambiguity in the definitions (4.5) of our reference and target states. If one

multiplies any of the constraints by a time-dependent phase factor we still get the same

physical constraint, i.e.,

ei ηk(t)
(
αk ak + βk a

†
−k

)
|Ω〉 = 0

ei η̃k(t)
(
Ak ak +Bk a

†
−k

)
|ψ〉 = 0 .

Equivalently, this means that there is the ambiguity of replacing (αk, βk) → ei ηk(αk, βk)

and (Ak, Bk) → ei η̃k(Ak, Bk).
7 Since they appear in a mixed way in (4.31), there is

a resulting relative phase ambiguity between the two terms that is nonvanishing. Even

though the initial and final states are obviously not affected by this choice of phase, the

circuit connecting them in general is and, as a result, the circuit depth associated to a

given path can depend on it. In practice, this means that the boundary conditions (4.31)

for pk, qk have to be replaced by (here ξk ≡ η̃k − ηk)

p∗k(1) =
ei ξkα∗k Ak − e−i ξkβk B

∗
k√

|αk|2 − |βk|2
(4.34a)

qk(1) =
ei ξkα∗k Bk − e−i ξkβk A

∗
k√

|αk|2 − |βk|2
(4.34b)

p∗k(0) = 1 (4.34c)

qk(0) = 0 , (4.34d)

and, when seeking for the optimal circuit to find the complexity, in addition to finding the

geodesics associated with the metric (5.7) we also need to find the choice of relative phase

ξk(t) that minimizes the circuit depth. We will show explicitly that such a choice of ξk is

not the simplest one (ξk = 0).

5 Time evolution of the complexity

5.1 Geometrizing the problem

We are now ready to geometrize and solve the complexity problem. The first step is to

parametrize the manifold of SU(1, 1) matrices using the set of coordinates {θk, φk, χk}
defined by

p∗k(s) ≡ cosh θk(s) e
iφk(s) qk(s) ≡ sinh θk(s) e

−iχk(s) , (5.1)

7One could in principle consider including also a real number rk(t) in front of the phase factors. However,

for the case of Ak, Bk the condition |Ak|2 − |Bk|2 = 1 forces rk = 1 while in the case of αk, βk this number

gets cancelled from the boundary conditions (4.31) by the normalization factor and hence plays no role.

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which clearly satisfy |pk(s)|2 − |qk(s)|2 = 1. The boundary conditions (4.34) then can be

translated as

φk(0) = 0 (5.2a)

θk(0) = 0 (5.2b)

χk(0) = χ0 (5.2c)

φk(1) = Arg (p∗k(1)) (5.2d)

θk(1) = arccosh (|p∗k(1)|) = arcsinh (|qk(1)|) (5.2e)

χk(1) = −Arg (qk(1)) (5.2f)

with χ0 an arbitrary constant. Here, p∗k(1) and qk(1) are to be understood as shorthand

for the expressions (4.34a), (4.34b).

The functions Y I
k (s) characterizing our circuit can be expressed in terms of the coor-

dinates using equations (4.21), (4.25), (4.27) and (5.1). One finds

Y 1
k = − sin(φk − χk) θ′k +

1

2
cos(φk − χk) sinh(2θk)

(
φ′k + χ′k

)
(5.3a)

Y 2
k = − cos(φk − χk) θ′k −

1

2
sin(φk − χk) sinh(2θk)

(
φ′k + χ′k

)
(5.3b)

Y 3
k = − cosh2(θk)φ

′
k − sinh2(θk)χ

′
k , (5.3c)

where a prime is used here to denote d
ds .

The circuit depth (3.5) associated with a given curve Y I
k (s) ≡ {θk(s), φk(s), χk(s)} in

the SU(1, 1) manifold is then given by

DF2(U) =

∫ 1

0
ds

√∫ Λ

−Λ
dk
(
|Y 1
k (s)|2 + |Y 2

k (s)|2 + |Y 3
k (s)|2

)
=

∫ 1

0
ds

√∫ Λ

−Λ
dk

[
θ′k

2 +
1

4
v′k

2 +
1

4
cosh(4θk)u

′
k

2 +
1

2
cosh(2θk)u

′
k v
′
k

]
, (5.4)

where we defined the new variables

uk = φk + χk (5.5a)

vk = φk − χk . (5.5b)

The problem now becomes a geometrical one by noticing that this can be seen as the length

of a particle’s trajectory in a curved Riemannian manifold with metric

ds2 =

∫ Λ

−Λ
dk

[
dθk

2 +
1

4
dv2
k +

1

4
cosh(4θk) du

2
k +

1

2
cosh(2θk) duk dvk

]
(5.6)

(again, we stress that there is an abuse of notation here: the way to make sense of this is

to formulate the problem in a compact space so that
∫
dk is replaced by a discrete sum

over k). The optimal path is simply a geodesic in this geometry, the length of which defines

the circuit complexity.

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One can see that the metric is block diagonal with respect to the momentum-mode

label k, i.e., ds2 =
∫ Λ
−Λ dk ds

2
k. It is also important to notice that each block k depends only

on the coordinates {θk, ψk, φk} associated with that same momentum k (there is no mode

mixing). This means that we can find the curves minimizing the length in each of these

momentum subspaces and add them together to find the geodesic in the whole manifold,

which can also be checked by simply looking at the geodesic equation. Therefore the main

goal now is to calculate the geodesic associated with the metric

ds2
k = dθk

2 +
1

4
dv2
k +

1

4
cosh(4θk) du

2
k +

1

2
cosh(2θk) duk dvk (5.7)

subject to the boundary conditions (5.2).

Before delving into the calculations to find the geodesics, a comment is in order here.

The metric above is diffeomorphic to the one found in [21] restricted to a hypersurface

y = const. This should not come as a surprise since our choice of gates {O1,k,O2,k,O3,k}
leads to geodesics in the space of SU(1, 1) transformations, while the gates used by the

authors of [21] lead to geodesics in the space of GL(2,R) transformations. It is well-known

that SU(1, 1) is isomorphic to SL(2,R), a subgroup of GL(2,R), so at the end of the day

we should expect to deal with a submanifold of the one appearing in [21].

5.2 Symmetries and Killing vectors

The task of finding the geodesics associated with the metric (5.7) can be simplified by taking

advantage of symmetries and their corresponding conserved quantities. By inspection one

can check that the metric (5.7) possesses a trivial Killing vector, k0 ≡ ∂vk , but the question

remains whether there are others. As shown in [21], to each generator of the symmetry

group there must be a corresponding Killing vector. This is because of the way the metric

is defined, via the functions Y I
k (s) in (4.27): if we redefine Mk(s)→Mk(s)M̃ , where M̃ is

a constant SU(1, 1) matrix, the Y I
k (and consequently the metric) remain unchanged. So

in our case we should expect at least 3 Killing vectors (one for each SU(1, 1) generator). It

turns out that none of them happens to be the k0 above, so there are actually 4 in total.8

Following [21], let us take M̃ = eεIΣI with εI (I = 1, 2, 3) constants. To first order in

εI , the statement is that the metric is invariant under

δMk = Mk ΣIεI . (5.8)

To find the corresponding Killing vectors, we assume this change in Mk is coming from a

diffeomorphism δxj , i.e,

δMk = ∂jMk δx
j , (5.9)

and use (4.26) to find

εI = 2 Tr
(
M−1
k ∂jMk Σ†I

)
δxj . (5.10)

This can be inverted in the (j, I) indices to yield

δxj = (kI)j ε
I (5.11)

8k0 can be seen as an “accidental” symmetry reminiscent of our choice of GIJ = δIJ as the F2-metric

in (3.5) [21].

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with the Killing vectors (kI)j given by

(kI)j =
1

2

[
Tr
(
M−1
k ∂jMk Σ†I

)]−1
. (5.12)

Indeed, we see that there are as many of them as there are SU(1, 1) generators ΣI .

In terms of the coordinates {θk(s), φk(s), χk(s)}, the Killing vectors above read

k1 =
i

2
[cos(uk)∂θk − sin(uk) (tanh(θk) + coth(θk)) ∂u − sin(uk) (tanh(θk)− coth(θk)) ∂vk ]

(5.13a)

k2 =
i

2
[sin(uk)∂θk + cos(uk) (tanh(θk) + coth(θk)) ∂u + cos(uk) (tanh(θk)− coth(θk)) ∂vk ]

(5.13b)

k3 = −i ∂θk , (5.13c)

which together with k0 = ∂vk discussed before consist of all the isometries of the met-

ric (5.7).

To each of these Killing vectors one can associate conserved momenta cI = (kI)
i gijx

′j

which remain constant along the geodesics and can be used to simplify the task of inte-

grating the geodesic equations. Their explicit expressions (apart from overall numerical

factors) are the following

c0 = v′k + cosh(2θk)u
′
k (5.14a)

c1 = cos(uk) θ
′
k −

1

2
sinh(4θk) sin(uk)u

′
k −

1

2
sinh(2θk) sin(uk) v

′
k (5.14b)

c2 = sin(uk) θ
′
k +

1

2
sinh(4θk) cos(uk)u

′
k +

1

2
sinh(2θk) cos(uk) v

′
k (5.14c)

c3 = cosh(4θk)u
′
k + cosh(2θk) v

′
k . (5.14d)

We now proceed to use these equations to find the geodesics.

5.3 Geodesics and the complexity

From (5.14a) and (5.14d) it follows that

u′k =
c3 − c0 cosh(2θk)

sinh2(2θk)
and v′k =

c0 cosh(4θk)− c3 cosh(2θk)

sinh2(2θk)
. (5.15)

Now, the boundary conditions at s = 0 (see (5.2)) demand that c0 = c3 = 0, otherwise

u′k, v
′
k (and hence the length of the curve) would be divergent at s = 0. This implies

u′k = 0, v′k = 0, i.e.,

uk(s) = uk(0) = χ0 (5.16a)

vk(s) = vk(0) = −χ0 , (5.16b)

where in the last step we have expressed the result in terms of the boundary condition for

χk using the definition (5.5) of uk, vk in terms of the original variables φk, χk. By plugging

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these results into (5.14b) and (5.14c) and imposing the boundary condition θk(0) = 0 we

get simply

θk(s) =
c1

cos(χ0)
s and

c2

c1
= tan(χ0) . (5.17)

What remains is to determine the constant c1 in addition to fixing the global phase

factor ξk using the boundary conditions at s = 1. These can be rewritten (using (5.16)

and (5.17)) as

φk(1) = Arg (p∗k(1)) = arctan

(
Im p∗k(1)

Re p∗k(1)

)
= 0 (5.18a)

θk(1) = arccosh (|p∗k(1)|) = arccosh

(√(
Re p∗k(1)

)2
+
(
Im p∗k(1)

)2)
=

c1

cos(χ0)
(5.18b)

χk(1) = −Arg (qk(1)) = − arctan

(
Im qk(1)

Re qk(1)

)
= χ0 (5.18c)

with pk(1), qk(1) given in (4.34). Equation (5.18a) allows us to fix the phase ambiguity,

namely we have to solve

Im p∗k(1) =
1√

|αk|2 − |βk|2
Im
(
α∗k e

i ξkAk − βk e−i ξkB∗k

)
= 0 (5.19)

for ξk. The solution is easily found to be

ξk = arctan

(
Im(βk B

∗
k − α∗k Ak)

Re(βk B
∗
k + α∗k Ak)

)
, (5.20)

which then implies

Re p∗k(1) =
1√

|αk|2 − |βk|2
|αk|2 |Ak|2 − |βk|2 |Bk|2√[

Re(α∗k Ak + βk B
∗
k)
]2

+
[
Im(α∗k Ak − βk B∗k)

]2 . (5.21)

This completely fixes θk(1) in (5.18b) in terms of the target and reference state data, i.e.,

θk(1) = arccosh (|Re p∗k(1)|) . (5.22)

The remaining constants χ0, c1 are also uniquely determined in terms of these data by

equations (5.18b) and (5.18c), but the expressions will not be needed here.

Having obtained the geodesic {θk(s) = θk(1) s, uk(s) = χ0, vk(s) = −χ0}, the com-

plexity is then readily found by substituting it into expression (5.4) to find its length

(the minimal circuit depth). Since uk and vk have vanishing derivatives, we see that the

complexity is fully determined by the boundary value of θk(s), namely,

CF2(U) = min (DF2(U)) =

√∫ Λ

−Λ
dk θk(1)2 . (5.23)

Using (5.22) and (5.21) one can write this explicitly as

CF2(U)=

√√√√√∫ Λ

−Λ
dk arccosh2


∣∣∣|αk|2 |Ak|2 − |βk|2 |Bk|2∣∣∣ /√|αk|2 − |βk|2√[
Re(α∗k Ak + βk B

∗
k)
]2

+
[
Im(α∗k Ak − βk B∗k)

]2
 . (5.24)

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This analytic expression for the complexity in terms of the reference and target state data

(Ak, Bk, αk(t), βk(t)) given in (2.8) and (4.6), with the out-modes given in (2.3) is the main

result of the present paper.

The time evolution due to the quench is encoded in the coefficients αk(t), βk(t) and is

highly nontrivial, via hypergeometric functions. For this reason the integral over k cannot

be carried out analytically, but it is straightforward to evaluate it numerically for specific

choices of the parameters characterizing the quench if we impose a finite cutoff Λ. We shall

do that in a moment, but first let us analyze some interesting limits.

5.3.1 Static limit: the vacuum complexity

As a consistency check, we can particularize to the static or no quench case that corresponds

to mout = min. Equivalently, this can be seen as the early time limit t→ −∞ of the time

evolution following the mass quench. In this limit, the construction above gives simply the

ground state complexity of a massive scalar field (with mass min) previously studied in [21]

(remember that our choices of reference and target states in this limit become exactly the

same as those of [21]).

The coefficients Ak, Bk, αk, βk take the very simple form

Ak = 1

Bk = 0

αk(t) =
M + ωin

2
√

2Mωin
e−iωint

βk(t) =
M − ωin

2
√

2Mωin
eiωint ,

and the complexity (5.24) reduces to

CF2(U) =

√∫ Λ

−Λ
dk arccosh2

(
M + ωin√

4Mωin

)
. (5.26)

Using the identity

arccosh(x) = log
(
x+

√
x2 − 1

)
(5.27)

we get

Cvacuum
F2

=
1

2

√∫ Λ

−Λ
dk
(

log
ωin

M

)2
, (5.28)

which is exactly (the continuum version of) the expression for the complexity in a dis-

cretized massive scalar field computed in [21]. It is interesting to stress that the choice of

elementary gates used in [21] differs significantly from the one used here, yet both results

for the complexities agree. This indicates some degree of universality of the complexity

with respect to the choice of elementary gates.

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5.3.2 Sudden quench limit

Another interesting limit to look at is the infinitely fast or sudden quench limit δt → 0,

where the mass profile (2.2) reduces to a step function. For t < 0 there is essentially no

dynamics since m = min is held fixed and we get simply the vacuum result above Cvacuum
F2

for the complexity. For t > 0, on the other hand, we have

Ak =
ω+√
ωinωout

Bk = − ω−√
ωinωout

αk(t) =
M + ωout

2
√

2Mωout
e−iωoutt

βk(t) =
M − ωout

2
√

2Mωout
eiωoutt ,

and the complexity (5.24) reduces to CF2 =
√∫ Λ
−Λ dk arccosh2 (fk(t)) with

fk(t) =
|(M + ωout)

2ω2
+ − (M − ωout)

2ω2
−|/
√

4Mωin ω2
out√

(M + ωout)2ω2
+ + (M − ωout)2ω2

− − 2(M2 − ω2
out)ω+ω−(cosωoutt− sinωoutt)

.

(5.30)

In other words, in this limit the complexity jumps from the constant value (5.28) at t < 0

to an oscillatory behavior at t > 0 that persists forever. The frequency of oscillation is

not easy to guess though, since ωout depends on the momentum variable k which is being

integrated over. As we shall see below, this behavior is compatible with the smooth quench

results, where the same late time oscillations appear for t & δt.

5.3.3 Small and large reference scale M

One can also play with the free scale M that characterizes the reference state and obtain

reasonably simple exact expressions for the complexity (5.24) in the extreme limits where

M → 0 and M → ∞. Since M appears only in the functions αk(t), βk(t), taking these

limits is straightforward using (4.6), namely

M →∞ : αk(t) =

√
M

2
χk(t) , βk(t) =

√
M

2
χ∗−k(t)

M → 0 : αk(t) =
i

2
√
M
χ̇k(t) , βk(t) =

i

2
√
M
χ̇∗−k(t)

while the difference |αk|2 − |βk|2 = Im (χk χ̇
∗
k) is independent of M . This leads to

CF2 =

√√√√√∫ Λ

−Λ
dk log2

√M |χk|2/
√

Im(χk χ̇
∗
k)√[

Re(χ∗k (Ak +B∗k)
]2

+
[
Im(χ∗k (Ak −B∗k))

]2
 (5.32)

for M →∞ and

CF2 =

√√√√√∫ Λ

−Λ
dk log2

 1√
M

|χ̇k|2/
√

Im(χk χ̇
∗
k)√[

Re(χ̇∗k (Ak +B∗k)
]2

+
[
Im(χ̇∗k (Ak −B∗k))

]2
 (5.33)

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for M → 0, where we have used arccosh(x) ≈ log(2x) (x → ∞) to convert the integrands

into logarithms. The main difference between the two limits appears in the second ratio

inside the argument of the log, being the time dependence dictated by χ̇k(t) as M → 0

and by χk(t) as M → ∞ as already anticipated by (5.31). Intermediate scales of M are

therefore characterized by an interplay between these two behaviors.

5.3.4 UV behavior

It is important to notice that the ground state complexity (5.24) is UV divergent. This is

easily seen by analyzing the large k behavior of the integrand, namely

CF2 =

√√√√∫ Λ

−Λ
dk arccosh2

(√
|k|
4M

)
+ · · · ∼ Λ

(
log

Λ

M

)2

+ · · · . (5.34)

However, the leading UV divergence happens to be exactly the same as the one of the

ground state complexity (5.26), since ωin ∼ |k| for large k. It is tempting then to subtract

this constant divergent piece and introduce the “renormalized” complexity (i.e., the relative

complexity of the the instantaneous state |ψ(t)〉 with respect to the ground state one)

∆CF2(t) ≡ CF2(t)− Cvacuum
F2

, (5.35)

which is guaranteed to be UV finite during the whole time evolution. By definition, this

quantity vanishes at early times since the pre-quench state is nothing but the vacuum of

the initial Hamiltonian. In the following we shall study it numerically for different choices

of quench parameters as well as reference state data.

5.3.5 Numerical analysis

Having looked at special solvable limits of interest, we now analyze the renormalized com-

plexity ∆CF2(t) introduced in (5.35) in its full generality using numerical methods. As

discussed above, this object is UV finite unlike the complexity (5.24) itself. In practice, the

analysis shall be carried for a fixed and finite (but large) value for the momentum cutoff

Λ. We emphasize once again that the integrand appearing in (5.24) is known exactly in

terms of the reference and target state data, but its k-dependence is highly nontrivial such

that doing the integration analytically becomes hopeless. That is why we need to resort to

numerical integration, at this very last step of the analysis.

The goal is to understand how the dynamics of ∆CF2(t) is affected by the quench

parameters δt,min,mout (or δm ≡ mout −min), which characterize the target state |ψ(t)〉,
as well as by the scale M that characterizes the reference state |Ω〉. Notice that, with the

mass taken to depend on time as defined in (2.2), the quench effectively starts at t ≈ −δt
and ends at t ≈ δt, so the plots below will always focus on this interval (−δt, δt) where

a nontrivial time evolution is expected. The relevant results are shown in figure 1. For

numerical convenience we have fixed Λ = 100 and all the plots were made using specific

numerical values for the parameters chosen for presentation purposes only, although the

qualitative behavior illustrated in each one has been checked to hold generically.

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δm=5

δm=4

δm=3

δm=2

δm=1

-4 -2 2 4
t

0.05

0.10

0.15

ΔC(t)

(a) Mass-increasing quenches with fixed δt,M .

δm=-1

δm=-2

δm=-3

δm=-4

δm=-5

-4 -2 2 4
t

-0.15

-0.10

-0.05

ΔC(t)

(b) Mass-decreasing quenches with fixed δt,M .

δt=0

δt=0.3

δt=0.6

δt=1

δt=2

δt=3

-4 -2 2 4
t

0.005

0.010

0.015

0.020

0.025

ΔC(t)

(c) Different quench rates with fixed δm,M .

M=0.01

M=0.1

M=1

M=3

M=10

M=20

M=50

-10 -5 5 10
t

-0.3

-0.2

-0.1

0.1
ΔC(t)

(d) Different reference scales with fixed δt, δm.

Figure 1. Time dependence of the complexity for many combinations of the mass difference

δm = mout−min, the quench rate δt, and the scaleM characterizing the reference state. All the plots

were made with specific numerical values of the parameters chosen for presentation purposes, but the

qualitative behavior above was checked to hold generically. Figure (a) has δt = 1,M = 2,min = 1

and mout is changed as indicated by the inset; (b) has δt = 1,M = 2,min = 6 and varying mout;

(c) has min = 1,mout = 2,M = 2 and varying δt; and (d) was made with min = 1,mout = 2, δt = 1

while M was changed.

Regardless of the choice of parameters one can see two distinct regimes for the time

dependence of the complexity that are illustrated in figures (a) through (d): an approxi-

mate linear evolution for −δt . t . δt, followed by a saturation phase at t & δt, where

the complexity approaches a constant with eventual fluctuations around this value. As

illustrated in (a) and (b), the slope of the linear phase can be positive or negative (i.e.,

the complexity can grow or shrink) depending on the sign of δm. Namely, for quenches

that increase the mass (δm > 0) the slope is always positive, while for mass-decreasing

quenches it is negative. The amplitude of the final state oscillations is influenced not only

by the mass difference δm, but also by the initial value min as one can see by comparing

(a) and (b): the former has min = 1 and the oscillations become more pronounced as δm

is increased, while the latter with min = 6 and decreasing δm shows almost imperceptible

oscillations.

The quench rate δt also has significant influence in the time evolution as indicated in

(c). First, the slope of the linear phase increases the faster the quench is done. Second, the

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late time oscillations of the complexity are also sensitive to δt and become more pronounced

as δt decreases. This is compatible with the infinitely fast quench limit δt → 0 discussed

analytically in section 5.3.2, which is also shown in the plot, where the complexity jumps

(hence having an infinite slope) at t = 0 from a constant value to an oscillatory behavior

for t > 0.

In (d) we analyze also the dependence of the complexity of the energy scale M that

defines the reference state. To do this, we keep the quench data min,mout, δt (i.e., the target

state) fixed. Interestingly, we see that the renormalized complexity growth rate decreases

and becomes even negative as the scale M is increased. In particular, there is a particular

scale (M ≈ 3 for the set of parameters used in the plot) for which there is a transition

from growth to decrease of the renormalized complexity. At this scale, ∆CF2(t) remains

approximately zero during the whole time evolution, i.e., the quench barely increases the

complexity of the initial state. It is important to keep in mind that varying M affects not

only the instantaneous complexity CF2(t), but also the vacuum contribution Cvacuum
F2

that

is being subtracted to yield ∆CF2(t).

5.4 Higher dimensions

The analysis of the complexity above is easily extended to an arbitrary number of spacetime

dimensions. Namely, the gate operators now carry a vector index k, i.e. OI,k, but still close

the same SU(1, 1) algebra. The reference and target states remain unchanged, since the out

modes χout
k (t) are the same in higher dimensions (see e.g. [34]) and therefore the defining

constraints (4.5) do not care about the number of dimensions. The circuit is thus essentially

the same and the calculation of geodesics proceeds in a similar way by focusing on a single

k to later join them all together. As a result, the complexity in d = D+1 dimensions takes

the same form as in (5.24) but with∫ Λ

−Λ
dk →

∫
dDk = SD−1

∫ Λ

0
dk kD−1 ,

where SD−1 is the area of the (D − 1)-sphere.9

6 Discussion and conclusion

In this work, we have studied the time evolution of the computational complexity as the

result of a smooth mass quench in a free bosonic field theory. This model at late times

is known to thermalize in the sense of generalized Gibbs ensemble [31, 32], so this can

be seen as a toy model for studying the evolution of complexity during a thermalization

process. The reference state was chosen to be one with no spatial entanglement, while

the target state was a time-dependent one obtained from the pre-quench vacuum through

time evolution. As elementary gates, we chose the ones appearing in the momentum-space

Hamiltonian (4.7) of the model. Since they close a SU(1, 1) algebra, one can follow the

9Of course there is also the spatial volume factor V in front of the momentum integrals that we are here

ignoring and setting to 1.

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procedure of Jefferson and Myers in [21] to geometrize the problem of finding the complexity

as the one of finding a geodesic in the manifold of SU(1, 1) transformations.

The main result of the paper is formula (5.24) for the time evolution of the complexity

CF2(t) in terms of the data (Ak, Bk, αk(t), βk(t)) characterizing the reference and target

states. In the limit where there is no quench and we have simply a massive scalar field in the

vacuum state, the time dependence cancels out and we recover the same expression (5.28)

for the complexity obtained in [21]. This is interesting since the set of elementary gates

chosen here is quite different from the ones used in that case.

The complexity itself is well-known to be UV divergent, since there are infinitely many

momentum modes (gates) k contributing to the circuit. However, we have shown that the

UV divergent piece happens to be time-independent and exactly the same as the one from

the initial state complexity Cvacuum
F2

, such that the difference ∆CF2(t) ≡ CF2(t)− Cvacuum
F2

(the “renormalized” complexity) is by construction UV finite. We then studied the time

dependence of this object as a function of the quench data min,mout, δt as well as the free

scale M characterizing the reference state.

We found that the time evolution of the renormalized complexity is marked by two

distinct phases, separated by a time scale set by the quench rate δt (or, equivalently, the

thermalization scale for the model). For −δt . t . δt, the complexity grows (or decays,

depending on the sign of δm ≡ mout−min) approximately in a linear way, with a slope dC
dt

inversely proportional to δt and directly proportional to δm. Then, at a time of order δt

the complexity saturates and starts oscillating around a mean value. We checked that the

amplitude of the late time oscillations is sensitive not only to δt and the mass difference

δm, but also to the starting point of the quench (min) and the reference scale M . Also, for

fixed quench data the renormalized complexity shifts from growing to decreasing in time

as the scale M is increased, and in particular there is an intermediate scale for which the

change in complexity due to the quench is nearly vanishing.

As mentioned before, it has been conjectured that the complexity growth might be dual

to the growth of Einstein-Rosen bridges in eternal AdS black hole geometries. Classical

gravity computations show that observables characterizing this growth [9] evolve linearly

at late times. The holographic complexity was observed to grow linearly as well in AdS-

Vaidya model of a quantum quench [20]. Also, a linear growth followed by saturation

with oscillations around a mean value had been previously conjectured to be the universal

behavior of the complexity for chaotic systems [28, 35]. The time scale for saturation in

these systems is believed to be parametrically larger than the thermalization scale. Our

results show some similarities with these conjectures and calculations even though we dealt

with a free (hence non-chaotic) field theory, with the important difference that the time

scale for saturation here is of the order of the thermalization scale δt. For this reason,

it seems to be an interesting future prospect to extend the calculations presented here to

interacting theories, which are the ones most likely to have a simple classical gravitational

dual, and check if the early linear behavior shown here can be pushed to last up to larger

time scales. Another interesting prospect in this direction would be to use the κ = 1

measure discussed in [21] instead of the F2-distance used here, which has been argued to

be more appropriate to get closer to holographic results.

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Acknowledgments

We would like to thank Rob Myers and Veronika Hubeny for valuable comments on the

draft, and also Ying Zhao, and Mukund Rangamani for discussions. D.W.F.A. would like

to acknowledge hospitality at the QMAP center of UC Davis, where part of this work was

developed. D.W.F.A is supported by the PDSE Program scholarship of CAPES – Brazilian

Federal Agency for Support and Evaluation of Graduate Education within the Ministry of

Education of Brazil, and by the CNPq grant 146086/2015-5. G.C. acknowledges financial

support from the Brazilian ministries MCTI and MEC.

A Relation between Schrödinger and Heisenberg picture circuits

As introduced in section 4, in the Schrödinger picture the target state is time-dependent

while the reference state is not. Namely, the target state is |ψ (t)〉 obtained by evolving on

time the initial state |0in〉 while the reference state |Ω〉 is constrained to satisfy(√
M φ (x, 0) +

i√
M

π (x, 0)

)
|Ω〉 = 0 . (A.1)

where we have chosen, with no loss of generality, the time slice t = 0 for the Schrödinger

picture operators. The optimal circuit is

|ψ (t)〉 = USch
optimal (s = 1, t) |Ω〉 ≡ P ei

∫ 1
0 ds

′ Y I(t,s′)OI |Ω〉 , (A.2)

which can also be written as

|ψ (0)〉 = E−1 (t, 0) P ei
∫ 1
0 ds

′ Y I(t,s′)OI |Ω〉 , (A.3)

where E (t, 0) is the time evolution operator. In the Schrödinger picture, the task is there-

fore to find a circuit as above satisfying the boundary condition

USch
optimal (s = 0, t) = 1 , (A.4)

which amounts to finding the optimal choice of Y I (t, s) that minimizes
∫
ds
√
δIJY IY J∗.

In the Heisenberg picture, on the other hand, the target state is time-independent

while the reference state is not. The target reads

|ψ (0)〉 ≡ E−1 (t, 0) |ψ (t)〉 (A.5)

while the reference state is given by

|Ω (t)〉 ≡ E−1 (t, 0) |Ω〉 . (A.6)

By acting with E−1(t, 0) on (A.1) it is straightforward to see that such a reference state

satisfies a similar relation for the Heisenberg picture quantities, namely(√
M φ (x, t) +

i√
M

π (x, t)

)
|Ω (t)〉 = 0 . (A.7)

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One can write the optimal circuit as

|ψ (0)〉 = UHei
optimal (s = 1, t) |Ω (t)〉

= P ei
∫ 1
0 ds

′XI(t,s′)OI |Ω (t)〉

= P ei
∫ 1
0 dsX

I(t,s)OIE−1 (t, 0) |Ω〉 . (A.8)

and the task is to find the set of functions XI that satisfy the boundary condition

UHei
optimal (s = 0, t) = 1 (A.9)

and minimize
∫
ds
√
δIJXIXJ∗.

The question then is how to relate the two apparently distinct ways to calculate the

complexity. To answer that, let us multiply the last condition by E−1E = 1 on the left to

obtain

|ψ (0)〉 = E−1 (t, 0)
(
E (t, 0)P ei

∫ 1
0 dsX

I(t,s)OIE−1 (t, 0)
)
|Ω〉 . (A.10)

Since E does not depend on s (the circuit “level”), it can go inside the path-ordering

such that

|ψ (0)〉 = E−1 (t, 0)
(
P ei

∫ 1
0 dsX

I(t,s)[E(t,0)OIE−1(t,0)]
)
|Ω〉 . (A.11)

Now, consider the Heisenberg picture version of the operator OI , namely,

OI (t) ≡ E (t, 0)−1OI E (t, 0) , (A.12)

which satisfies

∂OI (t)

∂t
=

∂

∂t

(
T ei

∫ t
0 dt
′H(t′)OI T e−i

∫ t
0 dt
′H(t′)

)
= iE (t, 0)−1 [H (t) , OI ]E (t, 0) . (A.13)

Recall that the Hamiltonian for our problem is a combination of creation and annihilation

operators that close a SU(1, 1) algebra, and we have chosen the {OI} to be exactly that

set of operators, that is,

H (t) = fJ (t)OJ (A.14)

for some functions fJ(t) that can be seen from (4.7). This means that

∂OI (t)

∂t
= i fJ (t) cJIKOK(t) , (A.15)

where cJIK are the structure constants of SU(1, 1). The solution is

OI (t) = M J
I (t)OJ (A.16)

where MI is an element of SU(1, 1) in the adjoint representation.

We can now return to (A.11) and write

|ψ (0)〉 = E (t, 0)−1
(
P ei

∫ 1
0 dsX

I(t,s)(M−1)
IK
MKL[E(t,0)OLE(t,0)−1]

)
|Ω〉 . (A.17)

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Defining X̃K ≡ XI
(
M−1

)
IK

and using (A.16) together with (A.12) to identify the

Schrödinger picture operator OI in the exponent, one can now rephrase the Heisenberg

problem as that of finding the circuit

|ψ (0)〉 = E (t, 0)−1
(
P ei

∫ 1
0 ds X̃

k(t,s)OK
)
|Ω〉 (A.18)

that minimizes∫
ds
√
δIJXIXJ∗ =

∫
ds

√
(M)IKX̃

KδIJ(M)J∗L X̃
L∗ =

∫
ds

√
(M)IKX̃

K(M)I∗L X̃
L∗ (A.19)

Finally, by noticing that (M)IK (M)I∗L = (MM †)KL = δKL since M ∈ SU(1, 1) we see that

this becomes exactly the Schrödinger picture problem, i.e., that of finding the circuit

|ψ (0)〉 = E (t, 0)−1
(
P ei

∫ 1
0 ds X̃

k(t,s)Ok
)
|Ω〉 (A.20)

that minimizes ∫
ds

√
δKLX̃KX̃∗L . (A.21)

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

References

[1] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT,

Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].

[2] V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement

entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].

[3] M. Van Raamsdonk, Lectures on Gravity and Entanglement, in Proceedings of Theoretical

Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and

Strings (TASI 2015), Boulder U.S.A. (2015), pg. 297 [arXiv:1609.00026] [INSPIRE].

[4] L. Susskind, Entanglement is not enough, Fortsch. Phys. 64 (2016) 49 [arXiv:1411.0690]

[INSPIRE].

[5] S. Arora and B. Barak, Computational Complexity: A Modern Approach, first edition,

Cambridge University Press, New York U.S.A. (2009).

[6] R.-Q. Yang, Y.-S. An, C. Niu, C.-Y. Zhang and K.-Y. Kim, Axiomatic complexity in

quantum field theory and its applications, arXiv:1803.01797 [INSPIRE].

[7] A.R. Brown, D.A. Roberts, L. Susskind, B. Swingle and Y. Zhao, Holographic Complexity

Equals Bulk Action?, Phys. Rev. Lett. 116 (2016) 191301 [arXiv:1509.07876] [INSPIRE].

[8] D. Stanford and L. Susskind, Complexity and Shock Wave Geometries, Phys. Rev. D 90

(2014) 126007 [arXiv:1406.2678] [INSPIRE].

[9] D. Carmi, S. Chapman, H. Marrochio, R.C. Myers and S. Sugishita, On the Time

Dependence of Holographic Complexity, JHEP 11 (2017) 188 [arXiv:1709.10184] [INSPIRE].

[10] B. Swingle and Y. Wang, Holographic Complexity of Einstein-Maxwell-Dilaton Gravity,

arXiv:1712.09826 [INSPIRE].

– 26 –

https://creativecommons.org/licenses/by/4.0/
http://dx.doi.org/10.1103/PhysRevLett.96.181602
https://arxiv.org/abs/hep-th/0603001
https://inspirehep.net/search?p=find+EPRINT+hep-th/0603001
https://doi.org/10.1088/1126-6708/2007/07/062
https://arxiv.org/abs/0705.0016
https://inspirehep.net/search?p=find+EPRINT+arXiv:0705.0016
https://arxiv.org/abs/1609.00026
https://inspirehep.net/search?p=find+EPRINT+arXiv:1609.00026
http://dx.doi.org/10.1002/prop.201500095
https://arxiv.org/abs/1411.0690
https://inspirehep.net/search?p=find+EPRINT+arXiv:1411.0690
https://arxiv.org/abs/1803.01797
https://inspirehep.net/search?p=find+EPRINT+arXiv:1803.01797
http://dx.doi.org/10.1103/PhysRevLett.116.191301
https://arxiv.org/abs/1509.07876
https://inspirehep.net/search?p=find+EPRINT+arXiv:1509.07876
http://dx.doi.org/10.1103/PhysRevD.90.126007
http://dx.doi.org/10.1103/PhysRevD.90.126007
https://arxiv.org/abs/1406.2678
https://inspirehep.net/search?p=find+EPRINT+arXiv:1406.2678
https://doi.org/10.1007/JHEP11(2017)188
https://arxiv.org/abs/1709.10184
https://inspirehep.net/search?p=find+EPRINT+arXiv:1709.10184
https://arxiv.org/abs/1712.09826
https://inspirehep.net/search?p=find+EPRINT+arXiv:1712.09826


J
H
E
P
0
6
(
2
0
1
8
)
0
2
9

[11] M. Alishahiha, A. Faraji Astaneh, M.R. Mohammadi Mozaffar and A. Mollabashi,

Complexity Growth with Lifshitz Scaling and Hyperscaling Violation, arXiv:1802.06740

[INSPIRE].

[12] M. Alishahiha, A. Faraji Astaneh, A. Naseh and M.H. Vahidinia, On complexity for F(R)

and critical gravity, JHEP 05 (2017) 009 [arXiv:1702.06796] [INSPIRE].

[13] P.A. Cano, R.A. Hennigar and H. Marrochio, Complexity Growth Rate in Lovelock Gravity,

arXiv:1803.02795 [INSPIRE].

[14] S. Chapman, H. Marrochio and R.C. Myers, Complexity of Formation in Holography, JHEP

01 (2017) 062 [arXiv:1610.08063] [INSPIRE].

[15] J. Couch, S. Eccles, W. Fischler and M.-L. Xiao, Holographic complexity and

noncommutative gauge theory, JHEP 03 (2018) 108 [arXiv:1710.07833] [INSPIRE].

[16] Z. Fu, A. Maloney, D. Marolf, H. Maxfield and Z. Wang, Holographic complexity is nonlocal,

JHEP 02 (2018) 072 [arXiv:1801.01137] [INSPIRE].

[17] M. Ghodrati, Complexity growth in massive gravity theories, the effects of chirality and more,

Phys. Rev. D 96 (2017) 106020 [arXiv:1708.07981] [INSPIRE].

[18] Y.-S. An and R.-H. Peng, Effect of the dilaton on holographic complexity growth, Phys. Rev.

D 97 (2018) 066022 [arXiv:1801.03638] [INSPIRE].

[19] R.-Q. Yang, C. Niu, C.-Y. Zhang and K.-Y. Kim, Comparison of holographic and field

theoretic complexities for time dependent thermofield double states, JHEP 02 (2018) 082

[arXiv:1710.00600] [INSPIRE].

[20] M. Moosa, Evolution of Complexity Following a Global Quench, JHEP 03 (2018) 031

[arXiv:1711.02668] [INSPIRE].

[21] R. Jefferson and R.C. Myers, Circuit complexity in quantum field theory, JHEP 10 (2017)

107 [arXiv:1707.08570] [INSPIRE].

[22] S. Chapman, M.P. Heller, H. Marrochio and F. Pastawski, Toward a Definition of

Complexity for Quantum Field Theory States, Phys. Rev. Lett. 120 (2018) 121602

[arXiv:1707.08582] [INSPIRE].

[23] R. Khan, C. Krishnan and S. Sharma, Circuit Complexity in Fermionic Field Theory,

arXiv:1801.07620 [INSPIRE].

[24] K. Hashimoto, N. Iizuka and S. Sugishita, Time evolution of complexity in Abelian gauge

theories, Phys. Rev. D 96 (2017) 126001 [arXiv:1707.03840] [INSPIRE].

[25] R.-Q. Yang, Complexity for quantum field theory states and applications to thermofield

double states, Phys. Rev. D 97 (2018) 066004 [arXiv:1709.00921] [INSPIRE].

[26] M.A. Nielsen, M.R. Dowling, M. Gu and A.C. Doherty, Quantum Computation as Geometry,

Science 311 (2006) 1133 [quant-ph/0603161].

[27] D.A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 04 (2017) 121

[arXiv:1610.04903] [INSPIRE].

[28] A.R. Brown and L. Susskind, Second law of quantum complexity, Phys. Rev. D 97 (2018)

086015 [arXiv:1701.01107] [INSPIRE].

[29] M. Nozaki, S. Ryu and T. Takayanagi, Holographic Geometry of Entanglement

Renormalization in Quantum Field Theories, JHEP 10 (2012) 193 [arXiv:1208.3469]

[INSPIRE].

– 27 –

https://arxiv.org/abs/1802.06740
https://inspirehep.net/search?p=find+EPRINT+arXiv:1802.06740
https://doi.org/10.1007/JHEP05(2017)009
https://arxiv.org/abs/1702.06796
https://inspirehep.net/search?p=find+EPRINT+arXiv:1702.06796
https://arxiv.org/abs/1803.02795
https://inspirehep.net/search?p=find+EPRINT+arXiv:1803.02795
https://doi.org/10.1007/JHEP01(2017)062
https://doi.org/10.1007/JHEP01(2017)062
https://arxiv.org/abs/1610.08063
https://inspirehep.net/search?p=find+EPRINT+arXiv:1610.08063
https://doi.org/10.1007/JHEP03(2018)108
https://arxiv.org/abs/1710.07833
https://inspirehep.net/search?p=find+EPRINT+arXiv:1710.07833
https://doi.org/10.1007/JHEP02(2018)072
https://arxiv.org/abs/1801.01137
https://inspirehep.net/search?p=find+EPRINT+arXiv:1801.01137
http://dx.doi.org/10.1103/PhysRevD.96.106020
https://arxiv.org/abs/1708.07981
https://inspirehep.net/search?p=find+EPRINT+arXiv:1708.07981
http://dx.doi.org/10.1103/PhysRevD.97.066022
http://dx.doi.org/10.1103/PhysRevD.97.066022
https://arxiv.org/abs/1801.03638
https://inspirehep.net/search?p=find+EPRINT+arXiv:1801.03638
https://doi.org/10.1007/JHEP02(2018)082
https://arxiv.org/abs/1710.00600
https://inspirehep.net/search?p=find+EPRINT+arXiv:1710.00600
https://doi.org/10.1007/JHEP03(2018)031
https://arxiv.org/abs/1711.02668
https://inspirehep.net/search?p=find+EPRINT+arXiv:1711.02668
https://doi.org/10.1007/JHEP10(2017)107
https://doi.org/10.1007/JHEP10(2017)107
https://arxiv.org/abs/1707.08570
https://inspirehep.net/search?p=find+EPRINT+arXiv:1707.08570
http://dx.doi.org/10.1103/PhysRevLett.120.121602
https://arxiv.org/abs/1707.08582
https://inspirehep.net/search?p=find+EPRINT+arXiv:1707.08582
https://arxiv.org/abs/1801.07620
https://inspirehep.net/search?p=find+EPRINT+arXiv:1801.07620
http://dx.doi.org/10.1103/PhysRevD.96.126001
https://arxiv.org/abs/1707.03840
https://inspirehep.net/search?p=find+EPRINT+arXiv:1707.03840
http://dx.doi.org/10.1103/PhysRevD.97.066004
https://arxiv.org/abs/1709.00921
https://inspirehep.net/search?p=find+EPRINT+arXiv:1709.00921
http://dx.doi.org/10.1126/science.1121541
https://arxiv.org/abs/quant-ph/0603161
https://doi.org/10.1007/JHEP04(2017)121
https://arxiv.org/abs/1610.04903
https://inspirehep.net/search?p=find+EPRINT+arXiv:1610.04903
http://dx.doi.org/10.1103/PhysRevD.97.086015
http://dx.doi.org/10.1103/PhysRevD.97.086015
https://arxiv.org/abs/1701.01107
https://inspirehep.net/search?p=find+EPRINT+arXiv:1701.01107
https://doi.org/10.1007/JHEP10(2012)193
https://arxiv.org/abs/1208.3469
https://inspirehep.net/search?p=find+EPRINT+arXiv:1208.3469


J
H
E
P
0
6
(
2
0
1
8
)
0
2
9

[30] A. Mollabashi, M. Nozaki, S. Ryu and T. Takayanagi, Holographic Geometry of cMERA for

Quantum Quenches and Finite Temperature, JHEP 03 (2014) 098 [arXiv:1311.6095]

[INSPIRE].

[31] S. Sotiriadis and P. Calabrese, Validity of the GGE for quantum quenches from interacting to

noninteracting models, J. Stat. Mech. 1407 (2014) P07024 [arXiv:1403.7431] [INSPIRE].

[32] D.W.F. Alves and G. Camilo, Momentum-space entanglement after smooth quenches,

arXiv:1712.01400 [INSPIRE].

[33] H. Camargo, P. Caputa, D. Das, M. Heller and R. Jefferson, to appear.

[34] S.R. Das, D.A. Galante and R.C. Myers, Universality in fast quantum quenches, JHEP 02

(2015) 167 [arXiv:1411.7710] [INSPIRE].

[35] A.R. Brown, L. Susskind and Y. Zhao, Quantum Complexity and Negative Curvature, Phys.

Rev. D 95 (2017) 045010 [arXiv:1608.02612] [INSPIRE].

– 28 –

https://doi.org/10.1007/JHEP03(2014)098
https://arxiv.org/abs/1311.6095
https://inspirehep.net/search?p=find+EPRINT+arXiv:1311.6095
http://dx.doi.org/10.1088/1742-5468/2014/07/P07024
https://arxiv.org/abs/1403.7431
https://inspirehep.net/search?p=find+EPRINT+arXiv:1403.7431
https://arxiv.org/abs/1712.01400
https://inspirehep.net/search?p=find+EPRINT+arXiv:1712.01400
https://doi.org/10.1007/JHEP02(2015)167
https://doi.org/10.1007/JHEP02(2015)167
https://arxiv.org/abs/1411.7710
https://inspirehep.net/search?p=find+EPRINT+arXiv:1411.7710
http://dx.doi.org/10.1103/PhysRevD.95.045010
http://dx.doi.org/10.1103/PhysRevD.95.045010
https://arxiv.org/abs/1608.02612
https://inspirehep.net/search?p=find+EPRINT+arXiv:1608.02612

	Introduction
	Mass quenches in free scalar fields
	Complexity in quantum field theory
	Introducing the circuit
	Reference and target states
	Choosing the gates
	Matrix representation
	Boundary conditions for the geodesics
	Phase factor ambiguity

	Time evolution of the complexity
	Geometrizing the problem
	Symmetries and Killing vectors
	Geodesics and the complexity
	Static limit: the vacuum complexity
	Sudden quench limit
	Small and large reference scale M
	UV behavior
	Numerical analysis

	Higher dimensions

	Discussion and conclusion
	Relation between Schrödinger and Heisenberg picture circuits