# Nature and origin of the operators entering the master equation of an open
quantum system

Giovanni Spaventa
Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-
Allee 11 D-89081, Ulm (Germany) &<EMAIL_ADDRESS>
Paola Verrucchi
Istituto dei Sistemi Complessi CNR, and Dipartimento di Fisica, Università di
Firenze, and INFN sezione di Firenze, via G.Sansone 1, 50019 Sesto Fiorentino
(Italy)
&<EMAIL_ADDRESS>
###### Abstract

By exploiting the peculiarities of a recently introduced formalism for
describing open quantum systems (the Parametric Representation with
Environmental Coherent States) we derive an equation of motion for the reduced
density operator of an open quantum system that has the same structure of the
celebrated
Gorini–Kossakowski–Sudarshan–Lindblad equation, but holds regardless of
markovianity being assumed. The operators in our result have explicit
expressions in terms of the Hamiltonian describing the interactions with the
environment, and can be computed once a specific model is considered. We find
that, instead of a single set of Lindblad operators, in the general (non-
markovian) case there one set of Lindblad-like operators for each and every
point of a symplectic manifold associated to the environment. This intricacy
disappears under some assumptions (which are related to markovianity and the
classical limit of the environment), under which it is possible to recover the
usual master-equation formalism. Finally, we find such Lindblad-like operators
for two different models of a qubit in a bosonic environment, and show that in
the classical limit of the environment their renown master equations are
recovered.

Prologue

I am not amongst the lucky persons who have been knowing Prof. Kossakowski for
decades. I knew his extraordinary works, of course, but I only met him in
person 10 years ago, during the 44th Symposium on Mathematical Physics. It was
my first time in Torun and I was nervous: I knew I would have presented my
work to some of the most distinguished experts in OQS of the world, Prof.
Kossakowski in the first place.

On the morning of the second day Andrzej gave his talk and my tension melt
away like snow: he was authoritative and friendly, rigorous and talkative,
focused and serene. He was a real master. Suddenly I could not wait to tell
him about my work.

When I gave the talk, the day after, he was sitting in the first row of the
lecture-hall and I simply felt proud and happy, because he was there. And this
is how we got to know each other. Listening to Andrzej, looking him writing at
the blackboard, following his extraordinary thoughts has been a pleasure.
Talking with him a true honour.

Paola

##  1\. Introduction

Quantum information science, with its rapid growth over the past years, has
been crucial in providing insights in the foundations of quantum mechanics,
while allowing the development of quantum technologies. In fact, controlling a
quantum device would be an impossible task without a deep understanding of the
interplay between systems and their surrounding environments [1, 2, 3, 4].
This is indeed one of the main goals in the analysis of Open Quantum Systems
(OQS): to study the dynamics of quantum systems interacting with their equally
quantum environments [5, 6, 7, 8]

One of the main features OQS, at variance with their isolated counterparts, is
that they inherently exhibit memory effects, due to the dynamical generation
of entanglement between the principal system and its environment. This means
that their time evolution (in particular the time evolution of their density
operator $\rho(t)$) is generally non-markovian, and therefore not described by
differential equations. However, there are some cases in which entanglement
between the two subsystems does not severely alter their dynamics, the best
example being that of macroscopic environments, when their behaviour can be
effectively described by classical dynamics [9]. In such situations, time
evolution in terms of a differential equation for $\rho(t)$ can be recovered,
in the form of a so-called master equation, $\dot{\rho}=\mathcal{L}[\rho(t)]$,
where $\mathcal{L}$ is dubbed generator of the master equation itself [6, 8].

Thank to the work of Gorini, Kossakowski, Sudarshan [10] and Lindblad [11], it
is known that $\mathcal{L}$ must be of a certain form if it has to describe
markovian dynamics, thus defining the so-called GKSL equation [12]. The
operators entering such general expression (the Lindblad operators or
lindbladians) are not derived from the microscopic details of the theory, and
this makes it difficult to guess their form, and gives the GKSL result a
phenomenological character it should not actually have. Moreover, the
relevance of OQS dynamics in controlling quantum devices, such as the
processors of quantum computers, makes it necessary to explicitly determine
the lindbladians for the most diverse systems and environments [13, 14, 15,
16, 17, 18].

In this work we want to derive a GKSL-like master equation in a way such that
the Lindblad operators emerge in terms of the actual interaction entering the
model under analysis. To this aim we use the formalism of Generalized Coherent
States (GCS) [19, 20, 21], which is one of the best tools for describing the
quantum-to-classical crossover, i.e. the way a quantum system may feature a
classical behaviour when becoming macroscopic [22, 23]. In fact, referring to
a recently introduced [24] method to study OQS, namely the Parametric
Representation with Environmental Coherent States (PRECS) [24], we exploit the
peculiarities of GCS and describe the open system under analysis in terms of
an ensemble of normalized pure states, parametrically dependent on the
environmental configurations. This ensemble defines a representation of
$\rho(t)$ for which we obtain a master equation that can be cast in the GKSL-
like form, once an ansatz on the nature of some specific mathematical objects
entering its derivation is made. In this equation we recognize the operators
playing the role of the lindbladians, and get their explicit form in terms of
the original hamiltonian $H$ of the total system, so that $\dot{\rho}(t)$ can
be written once $H$ is known or, alternatively, information on $H$ is obtained
if $\rho(t)$ can be phenomenologically deduced, but $H$ cannot.

The structure of the paper is as follows: in Sec. 2. we outline the main
features of the PRECS, which will serve as a basis for Sec. 3., where we
perform the time-derivative of the parametric representation of $\rho(t)$.
Exploiting the known results about the dynamics of GCS, in Sec. 3.1. we show
that the resulting equation is GKSL-like, provided that the nature of some
mathematical objects appearing in its derivation is properly interpreted. The
classical limit for the environment is considered in Sec. 3.2., while in Sec.
4. we consider two models of qubits in bosonic environments, namely the pure-
dephasing and the Jaynes-Cummings model, in order to see how our results
compare with what is already known. Finally, in Sec. 5. we discuss about the
additional information one can possibly obtain by our approach, and draw some
conclusions.

##  2\. Parametric Representation with environmental coherent states

The description of OQS in terms of reduced density operators, stems from an
exact procedure that preserves the quantum character of the environment,
namely that of tracing out the environmental degrees of freedom from the
density operator of the total system.

In fact, there exists another approach for dealing with OQS, based on the idea
of considering them as if they were closed, i.e. described by an effective
hamiltonian, whose time-dependent parameters (such as oscillating fields or
fluctuating couplings) account for environmental effects. This approach
implicitly assumes that the environment behaves classically, since the
environmental operators are replaced by time-dependent functions. This fact
has two consequences: firstly, entanglement between the principal system and
its environment has no place in this description (there cannot be entanglement
between a quantum system and a classical one); secondly, by choosing ad-hoc
time dependencies for the parameters in the effective hamiltonian, one
completely neglects the environmental dynamics due to the interaction with the
principal system, often dubbed back-action [25, 26].

In what follows we will rather adopt the recently introduced [24, 27]
Parametric Representation with Environmental Coherent States (PRECS), which
provides a formally exact way to study the OQS dynamics in terms of a
collection of pure states, each labelled by a set of parameters which are in
one-to-one correspondence with (coherent) quantum states of the environment.
In fact, the use of Generalized Coherent States (GCS) [20, 21, 19] for
describing the environment, naturally emerges in the study of the quantum-to-
classical crossover [22, 9], that is, the limit in which a quantum system
displays an effectively classical behaviour due to its being macroscopic. This
implies that the PRECS provides an ideal formalism for studying the twilight
zone where an environment retains some of its quantum features, and yet starts
being quite properly described by a classical-like theory [23, 28].

Let us briefly summarize the PRECS formalism, by considering a bipartite
quantum system $\Psi=\Gamma\cup\Xi$, where $\Gamma$ and $\Xi$ play the roles
of principal system and environment, respectively. The GCS are constructed for
the environment $\Xi$, following the group-theoretical procedure of Ref. [19],
which we briefly outline below.

Starting from the hamiltonian of the environment, $H_{\Xi}$, one identifies
the so called dynamical group $G$, i.e, the group of propagators that
determine the evolution of $\Xi$. The procedure futher requires the choice of
an arbitrary normalized element $\ket{R}$, called reference state, in the
Hilbert space $\mathcal{H}_{\Xi}$ of the environment, from which the
identification of the maximum stability subgroup of $G$ follows: this is the
subgroup $F$ whose elements leave $\ket{R}$ unchanged, up to an irrelevant
phase factor. From $F$, the coset $G/F$ and its associated symplectic manifold
${\cal M}$, whose points are in one-to-one correspondence with the elements of
$G/F$ [29], are defined, and each element of $G/F$ defines a state
$\ket{\Omega}\in\mathcal{H}_{\Xi}$. By construction, hence, GCS are in one-to-
one correspondence with points on ${\cal M}$. GCS for the environment will be
hereafter dubbed environmental coherent states (ECS). These states are
normalized but non-orthogonal, and they form an overcomplete set on
$\mathcal{H}_{\Xi}$, i.e.

$\int_{\mathcal{M}}d\mu(\Omega)\ket{\Omega}\bra{\Omega}=\mathbb{1}_{\Xi}\,,$
(1)

where $d\mu(\Omega)$ is a measure on $\mathcal{M}$ that is invariant under the
action of elements in $G/F$.

Introducing the bases $\\{\ket{k}\\}$ and $\\{\ket{\xi}\\}$ of the Hilbert
spaces $\mathcal{H}_{\Gamma}$ and $\mathcal{H}_{\Xi}$ respectively, any state
of $\Psi$

$\ket{\psi}=\sum_{k\xi}c_{k\xi}\ket{k}\otimes\ket{\xi}\,,$ (2)

can be expressed in terms of ECS thanks to the resolution of the identity (1):

$\ket{\psi}=\sum_{k\xi}c_{k\xi}\ket{k}\otimes\int_{\mathcal{M}}d\mu(\Omega)\ket{\Omega}\bra{\Omega}\ket{\xi}=\int_{\mathcal{M}}d\mu(\Omega)\chi(\Omega)\ket{\phi(\Omega)}\otimes\ket{\Omega}\,,$
(3)

where

$\begin{split}\ket{\phi(\Omega)}=\frac{1}{\chi(\Omega)}\sum_{k}a_{k}(\Omega)\ket{k}~{}~{}&,~{}~{}a_{k}(\Omega)=\sum_{\xi}c_{k\xi}\bra{\Omega}\ket{\xi}\,,\\\
\chi(\Omega)=&\sqrt{\sum_{k}|a_{k}(\Omega)|^{2}}\,.\end{split}$ (4)

The positive function $\chi^{2}(\Omega)$ is normalized on $\mathcal{M}$, i.e.

$\int_{\mathcal{M}}d\mu(\Omega)\chi^{2}(\Omega)=1~{},$ (5)

and can be interpreted as a probability distribution for $\Xi$ to be in the
state $\ket{\Omega}$ when the global state is $\ket{\psi}$.

The density operator for the principal system, $\rho_{\Gamma}$, can be cast
[24] into the form

$\rho_{\Gamma}=\int_{\mathcal{M}}d\mu(\Omega)|\chi(\Omega)|^{2}\ket{\phi(\Omega)}\bra{\phi(\Omega)}\,,$
(6)

which means that the PRECS allows us to describe an open system $\Gamma$ in
terms of normalized pure states $\ket{\phi(\Omega)}$, parametrically dependent
on the environmental parameters $\Omega$. Each $\Omega$, on the other hand, is
in one-to-one correspondence with ECS $\ket{\Omega}$, the probability of whose
occurrence is given by $\chi^{2}(\Omega)$.

##  3\. A time derivative

The above expression (6) above provides the reduced density operator of an OQS
in terms of a probability distribution $\chi^{2}(\Omega)$ on the manifold
$\mathcal{M}$. It is clear that, under the Markovian approximation, such
density operator must satisfy a GKSL equation

$\dot{\rho}_{\Gamma}=-i[H_{\rm
eff},\rho_{\Gamma}]+\sum_{k}\gamma_{k}\Big{(}L_{k}\rho_{\Gamma}L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho_{\Gamma}\\}\Big{)}$
(7)

for some $H_{\rm eff},\gamma_{k}$ and set of lindbladians $L_{k}$.

In this section we explicitly perform the time derivative of Eq.(6), with the
goal of casting the resulting expression into a form that might allow the
identification of the Lindblad operators in the Markovian limit.

For the sake of clarity, fron now on we will adopt the following conventions:
the $\Omega$-dependencies, as well as the time ones, will be understood
whenever possible, and restored when leading to a better understanding;
moreover, in this section, and this one only, operators acting on the whole
composite system, i.e. on ${\cal H}_{\Gamma}\otimes{\cal H}_{\Xi}$, will
appear in bold, operators acting on ${\cal H}_{\Gamma}$ will be denoted by a
hat, $\widehat{\cdot}$, while operators acting on ${\cal H}_{\Xi}$ will have
no distinct sign; hats will not be used for density operators and their time-
derivatives.

Starting from Eq.(6), we consider the time derivative of $\rho_{\Gamma}$:

$\dot{\rho}_{\Gamma}=\int
d\mu\bigg{[}\widehat{K}+\chi^{2}(\Omega_{t})\frac{d}{dt}\Big{(}\ket{\phi}\bra{\phi}\Big{)}\bigg{]},$
(8)

where we have defined

$\begin{split}\widehat{K}:=\dot{\chi}\ket{\phi}\bra{\phi}\chi^{*}+\chi\ket{\phi}\bra{\phi}\dot{\chi}^{*}\,.\end{split}$
(9)

The time derivative of $\chi(\Omega_{t})$ is

$\displaystyle\dot{\chi}=$
$\displaystyle\frac{1}{2\chi}\frac{d}{dt}\sum_{k}|\sum_{\xi}c_{k\xi}|\langle{\Omega_{t}|\xi}\rangle|^{2}$
(10) $\displaystyle=$
$\displaystyle\frac{1}{2\chi}\sum_{k\xi\xi^{\prime}}\Big{[}\Big{(}\dot{c}_{k\xi}c_{k\xi^{\prime}}^{*}+c_{k\xi}\dot{c}_{k\xi^{\prime}}^{*}\Big{)}\langle{\Omega_{t}|\xi}\rangle\bra{\xi^{\prime}}\ket{\Omega}$
$\displaystyle+$ $\displaystyle
c_{k\xi}c_{k\xi^{\prime}}^{*}\Big{(}\bra{\xi^{\prime}}\ket{\Omega_{t}}\frac{d}{dt}\langle{\Omega_{t}|\xi}\rangle+\langle{\Omega_{t}|\xi}\rangle\frac{d}{dt}\bra{\xi^{\prime}}\ket{\Omega_{t}}\Big{)}\Big{]}\,;$

the quantities $\frac{d}{dt}\langle{\Omega_{t}|\xi}\rangle$ require some care
in being dealt with: they contain the dynamics of the parameter $\Omega$ on
the manifold that, if $\Xi$ were isolated, would be given by the Hamilton-like
equations of motion (see for instance Ref. [19]), with the function
$H_{\Xi}(\Omega):=\langle{\Omega|H_{\Xi}|\Omega}\rangle$ playing the role of
the classical hamiltonian. On the other hand, $\Xi$ is not isolated, and the
hamiltonian $\bm{H}$ with which we will be dealing acts upon ${\cal
H}_{\Gamma}\otimes{\cal H}_{\Xi}$: therefore,
$\langle{\Omega|\bm{H}|\Omega}\rangle$ is an operator acting on
$\mathcal{H}_{\Gamma}$, and so is
$\frac{d}{dt}\langle{\Omega_{t}|\xi}\rangle$. This statement is not crystal
clear from a mathematical viepoint, and we will get back to it in the
subsection below; however, we anticipate that whenever an expression for the
above derivatives is explicitly available, it is indeed in the form of an
operator for $\Gamma$.

Having said that, we define

$\displaystyle a_{k}=\sum_{\xi}c_{k\xi}\langle{\Omega_{t}\xi}\rangle$
$\displaystyle\,,\quad
b_{k}=\sum_{\xi}\dot{c}_{k\xi}\langle{\Omega_{t}|\xi}\rangle$ (11)
$\displaystyle\text{and }\quad\widehat{F}_{k}=$
$\displaystyle\sum_{\xi}c_{k\xi}\frac{d}{dt}\langle{\Omega_{t}|\xi}\rangle,$

and get

$\displaystyle\dot{\chi}=\frac{1}{2\chi}\sum_{k}\Big{[}a_{k}^{*}b_{k}+a_{k}b_{k}^{*}+a_{k}^{*}\widehat{F}_{k}+a_{k}\widehat{F}_{k}^{\dagger}\Big{]}$
(12)
$\displaystyle=\frac{1}{2\chi}\sum_{k}\Big{[}r_{k}+a_{k}^{*}\widehat{F}_{k}+a_{k}\widehat{F}_{k}^{\dagger}\Big{]}\,,$

with $r_{k}\equiv 2\real(a_{k}^{*}b_{k})=a_{k}^{*}b_{k}+a_{k}b_{k}^{*}\,$.

Introducing the operators

$\widehat{L}_{k}=\frac{1}{a_{k}}(\widehat{\mathbb{1}}-a_{k}^{*}\widehat{F}_{k})~{},$
(13)

it can be shown that $\widehat{K}$ in Eq.(9) reads

$\displaystyle\widehat{K}=$
$\displaystyle\sum_{k}\big{(}r_{k}+1\big{)}\ket{\phi}\bra{\phi}$ (14)
$\displaystyle-$
$\displaystyle\Big{\\{}\frac{1}{2}\sum_{k}|a_{k}|^{2}\left(\widehat{L}_{k}^{\dagger}\widehat{L}_{k}-\widehat{F}_{k}^{\dagger}\widehat{F}_{k}\right),\ket{\phi}\bra{\phi}\Big{\\}}\,,$

where $\\{A,B\\}=AB+BA$.
In order to get rid of the operators $\widehat{F}_{k}$, we use

$\tr\rho_{\Gamma}=1\implies\tr\dot{\rho}_{\Gamma}=0\,,$ (15)

implying

$\tr\int
d\mu\bigg{[}\widehat{K}+\chi^{2}\frac{d}{dt}\ket{\phi}\bra{\phi}\bigg{]}=0\,.$
(16)

By the definition of $\widehat{K}$, and by making use of the cyclicity of the
trace, the above condition can be rewritten as

$\displaystyle\tr\int
d\mu\bigg{[}\Big{\\{}\frac{1}{2}\sum_{k}|a_{k}|^{2}\widehat{F}_{k}^{\dagger}\widehat{F}_{k},\ket{\phi}\bra{\phi}\Big{\\}}\bigg{]}$
(17) $\displaystyle=\tr\int
d\mu\bigg{[}-|\chi|^{2}\frac{d}{dt}\ket{\phi}\bra{\phi}+\sum_{k}|a_{k}|^{2}\widehat{L}_{k}\ket{\phi}\bra{\phi}\widehat{L}_{k}^{\dagger}-\sum_{k}(r_{k}+1)\ket{\phi}\bra{\phi}\bigg{]}\,.$

What we have obtained above is an equation of the form $\tr A=\tr B$, which
tells us that the operators $A$ and $B$ are equal up to a traceless operator
$C$ (which is, in fact, simply their difference $C=A-B$).
Now, using this fact into the expression for $\dot{\rho}_{\Gamma}$ we get

$\displaystyle\dot{\rho}_{\Gamma}=\widehat{C}+\int
d\mu\bigg{[}\sum_{k}|a_{k}|^{2}\bigg{(}\widehat{L}_{k}\ket{\phi}\bra{\phi}\widehat{L}_{k}^{\dagger}-\frac{1}{2}\Big{\\{}\widehat{L}_{k}^{\dagger}\widehat{L}_{k},\ket{\phi}\bra{\phi}\Big{\\}}\bigg{)}\bigg{]}\,.$
(18)

Let us now introduce the operator $\widehat{R}(\Omega)$, defined as

$\widehat{R}(\Omega)=\chi^{2}(\Omega)\ket{\phi(\Omega)}\bra{\phi(\Omega)}\,,$
(19)

and such that $\rho_{\Gamma}=\int d\mu(\Omega)\widehat{R}(\Omega)$. In terms
of $\widehat{R}(\Omega)$, we can write

$\displaystyle\dot{\rho}_{\Gamma}=\widehat{C}+\int
d\mu\bigg{[}\sum_{k}\gamma_{k}\bigg{(}\widehat{L}_{k}\widehat{R}\widehat{L}_{k}^{\dagger}-\frac{1}{2}\Big{\\{}\widehat{L}_{k}^{\dagger}\widehat{L}_{k},\widehat{R}\Big{\\}}\bigg{)}\bigg{]}\,,$
(20)

in whose second term we recognize the structure of the dissipator as in the
GKSL equation, with
$\gamma_{k}=\absolutevalue{\frac{a_{k}}{\chi}}^{2}=\absolutevalue{\bra{k}\ket{\phi(\Omega)}}^{2}$.

By comparing the results above with the GKSL equation, it is clear that the
operator $\widehat{C}$ must be related to the unitary term
$-i[\widehat{H}_{\rm eff},\rho_{\Gamma}]$, and this can be further justified
by the fact that $\tr\widehat{C}=0$ and every traceless operator can be
written as a commutator, hence we take $\widehat{H}_{\rm eff}$ such that

$\widehat{C}=-i\Big{[}\widehat{H}_{\rm eff},\rho_{\Gamma}\Big{]}\,.$ (21)

Finally, the equation for $\dot{\rho}_{\Gamma}$ reads

$\dot{\rho}_{\Gamma}=-i\big{[}\widehat{H}_{\rm eff},\rho_{\Gamma}\big{]}+\int
d\mu\sum_{k}\gamma_{k}\bigg{(}\widehat{L}_{k}\widehat{R}\widehat{L}_{k}^{\dagger}-\frac{1}{2}\Big{\\{}\widehat{L}_{k}^{\dagger}\widehat{L}_{k},\widehat{R}\Big{\\}}\bigg{)}\,,$
(22)

which is indeed of the GKSL form with the Lindblad operators as from Eqs.(13),
with their explicit form and actual nature as operators acting on ${\cal
H}_{\Gamma}$ considered in the following subsection.

###  3.1. Time evolution of ECS

Let us get back to one of the key definitions in the above derivation, namely
that of $\frac{d}{dt}\bra{\xi}\ket{\Omega_{t}}$ introduced in Eq.(10). We have
consistently considered these objects as operators, which is ultimately the
reason why the $\Omega$-dependent lindbladians $\widehat{L}_{k}(\Omega)$ are
operators on $\mathcal{H}_{\Gamma}$. We now clarify this claim, resorting to
the well-known dynamical properties of GCS, as found, for instance, in Ref.
[19].

The theory of GCS defines a bijective map between states $\ket{\Omega}$ and
points $\Omega$ on $\mathcal{M}$, whose coordinates dynamically evolve on
${\cal M}$ according to the Hamilton equations of motion

$\dot{\Omega}=-i\frac{\partial
H(\Omega)}{\partial\Omega^{*}}\quad,\qquad\dot{\Omega}^{*}=+i\frac{\partial
H(\Omega)}{\partial\Omega}\,,$ (23)

with $H(\Omega):=\bra{\Omega}H\ket{\Omega}$ if $\Xi$ were isolated; therefore,
it seems reasonable to associate a time evolution to the ECS, exploiting the
fact that there is one defined on the manifold,
$\ket{\Omega_{t}}=\ket{\Omega(t)}$.

In particular, for an isolated system with GCS $\ket{\Omega}$, the inner
product $\bra{\xi}\ket{\Omega}$ is a function $\phi_{\xi}(\Omega)$ on ${\cal
M}$ with time derivatives

$\displaystyle\frac{d\phi_{\xi}}{dt}=\frac{\partial\phi_{\xi}}{\partial\Omega}\dot{\Omega}+\frac{\partial\phi_{\xi}}{\partial\Omega^{*}}\dot{\Omega}^{*}$
(24) $\displaystyle=-i\frac{\partial\phi_{\xi}}{\partial\Omega}\frac{\partial
H(\Omega)}{\partial\Omega^{*}}+i\frac{\partial\phi_{\xi}}{\partial\Omega^{*}}\frac{\partial
H(\Omega)}{\partial\Omega}=-i\\{\phi_{\xi}(\Omega),H(\Omega)\\}\,,$

where

$\\{f,g\\}_{\rm P}=\frac{\partial f}{\partial\Omega}\frac{\partial
g}{\partial\Omega^{*}}-\frac{\partial f}{\partial\Omega^{*}}\frac{\partial
g}{\partial\Omega}$ (25)

is the Poisson bracket on $\mathcal{M}$. Therefore, for an isolated system,
the time evolution of $\phi_{\xi}(\Omega)$ is dictated by the Poisson bracket
of $\phi_{\xi}(\Omega)$ and the function
$H(\Omega)=\bra{\Omega}H\ket{\Omega}$.

The problem with which we are dealing, though, is that of a composite system
$\Psi=\Gamma\cup\Xi$, with a generic hamiltonian

$\bm{H}=\sum_{i}g_{i}\widehat{O}_{i}^{\Gamma}\otimes O_{i}^{\Xi}$ (26)

where $g_{i}$ are the couplings, while $\widehat{O}_{i}^{\Gamma}$ and
$O_{i}^{\Xi}$ are operators on $\mathcal{H}_{\Gamma}$ and $\mathcal{H}_{\Xi}$,
respectively.
This means that, once the ECS are constructed for $\Xi$,
$\bra{\Omega}\bm{H}\ket{\Omega}:=\widehat{H}(\Omega)$ is not a function but
rather an operator on $\mathcal{H}_{\Gamma}$, in fact:

$\widehat{H}(\Omega)=\sum_{i}g_{i}\bra{\Omega}O_{i}^{\Xi}\ket{\Omega}\widehat{O}_{i}^{\Gamma}\equiv\sum_{i}g_{i}\theta_{i}(\Omega)\widehat{O}_{i}^{\Gamma}\,,$
(27)

where we have introduced the expectation values

$\theta_{i}(\Omega)\equiv\bra{\Omega}O_{i}^{\Xi}\ket{\Omega}\,.$ (28)

Following the above reasoning, though, we can still refer to Eq.(23) and
write:

$\displaystyle\widehat{\dot{\phi}}_{\xi}:={\frac{d\phi_{\xi}}{dt}}=-i\bigg{(}\frac{\partial\phi_{\xi}}{\partial\Omega}\frac{\partial\widehat{H}(\Omega)}{\partial\Omega^{*}}-\frac{\partial\phi_{\xi}}{\partial\Omega^{*}}\frac{\partial\widehat{H}(\Omega)}{\partial\Omega}\bigg{)}$
(29)
$\displaystyle=-i\sum_{i}g_{i}\Big{\\{}\phi_{\xi}(\Omega),\theta_{i}(\Omega)\Big{\\}}_{\rm
P}\widehat{O}_{i}^{\Gamma}\,:$

getting back to Eq. (11), we can then explicitly write

$\displaystyle\widehat{F}_{k}(\Omega)=-i\sum_{\xi}\sum_{i}g_{i}c_{k\xi}\Big{\\{}\bra{\Omega}\ket{\xi},\theta_{i}(\Omega)\Big{\\}}_{\rm
P}\widehat{O}_{i}^{\Gamma}$ (30)
$\displaystyle=-i\sum_{i}g_{i}\Big{\\{}\sum_{\xi}c_{k\xi}\bra{\Omega}\ket{\xi},\theta_{i}(\Omega)\Big{\\}}_{\rm
P}O_{i}^{\Gamma}$
$\displaystyle=-i\sum_{i}b_{i}^{(k)}(\Omega)\widehat{O}_{i}^{\Gamma}\,,$

with

$b_{i}^{(k)}(\Omega)=g_{i}\big{\\{}a_{k}(\Omega),\theta_{i}(\Omega)\big{\\}}_{\rm
P}\,.$ (31)

Thanks to the above results one obtains, via Eq. (13), the explicit form of
the operators playing the role of the lindbladians in our GKSL-like equation
(22). We underline that, as mentioned in the Introduction, these operators
depend on $\Omega$, from whom they inherit an essential time dependence.

###  3.2. Remarks on Markovianity and the Classical Limit

The integro-differential equation (22) evidently resembles a GKSL equation.
though it does not embody any markovian approximation. Consistently it cannot
be considered a genuine master equation, as it turns clear after noticing, for
instance, that the integral on its r.h.s. contains the operator
$\widehat{R}=\widehat{R}(\Omega_{t})$ rather than $\rho_{\Gamma}(t)$. However,
reminding that $\widehat{R}$ is proportional to $\chi^{2}(\Omega)$ by
definition (19), one can guess that conditions upon the probability
distribution of ECS on ${\cal M}$, embodied by $\chi^{2}(\Omega)$ itself, can
transform Eq.(22) into some more familiar, purely differential, equation. In
fact, it can be demonstrated [22, 24, 30, 23] that taking the classical limit
for $\Xi$, and $\Xi$ only, implies

$\chi^{2}(\Omega)\to\sum_{i}p_{i}\delta(\Omega-\Omega_{i})~{},$ (32)

with $p_{i}$ positive coefficients such that $\sum_{i}p_{i}=1$, and

$\bra{\Omega_{i}}\ket{\Omega_{j}}\to\delta_{ij}\,\,\forall
t\,\,~{},~{}\,\,\bra{\xi}\ket{\Omega_{i}}\bra{\Omega_{i}}\ket{\xi^{\prime}}\to\delta_{\xi\xi^{\prime}}\,\,\forall
t\forall i\,,$ (33)

where $\to$ indicates the classical limit of the environment. If this is the
case, it is

$\rho_{\Gamma}(t)\to\sum_{i}p_{i}\ket{\phi(\Omega_{i})}\bra{\phi(\Omega_{i})}\,,$
(34)

with $\bra{\phi(\Omega_{i}(t)}\ket{\phi(\Omega_{j}(t))}\to\delta_{ij}$, and
hence, defining

$P_{i}=\ket{\phi(\Omega_{i})}\bra{\phi(\Omega_{i})}\,,$ (35)

one gets the following equation of motion for $\rho_{\Gamma}$:

$\dot{\rho}_{\Gamma}\to-i\Big{[}\widehat{H}_{\rm
eff},\rho_{\Gamma}\Big{]}+\sum_{ki}p_{i}\gamma_{ki}\bigg{(}\widehat{F}_{ki}\widehat{P}_{i}\widehat{F}_{ki}^{\dagger}-\frac{1}{2}\Big{\\{}\widehat{F}^{\dagger}_{ki}\widehat{F}_{ki},\widehat{P}_{i}\Big{\\}}\bigg{)}\,,$
(36)

where $\widehat{F}_{ki}\equiv\widehat{F}_{k}(\Omega_{i}(t))$ and
$\gamma_{ki}\equiv\gamma_{k}(\Omega_{i}(t))$.
Under the condition

$\sqrt{\gamma_{ki}}F_{ki}\equiv\sqrt{\gamma_{k}}F_{k}\quad\forall i\,,$ (37)

the equation becomes

$\dot{\rho}_{\Gamma}\to-i\Big{[}\widehat{H}_{\rm
eff},\rho_{\Gamma}\Big{]}+\sum_{k}\gamma_{k}\bigg{(}\widehat{F}_{k}\rho_{\Gamma}\widehat{F}_{k}^{\dagger}-\frac{1}{2}\Big{\\{}\widehat{F}^{\dagger}_{k}\widehat{F}_{k},\rho_{\Gamma}\Big{\\}}\bigg{)}\,,$
(38)

which is a genuine GKSL equation for $\rho_{\Gamma}$; moreover, if the
parameters $\Omega_{i}$ do not vary in time, so do the operators
$\widehat{F}_{k}$, that are hence recognized as true lindbladians. This result
confirms that there are conditions leading to a markovian master equation for
$\rho_{\Gamma}(t)$: they are are embodied in Eqs. (32) and (33), which are
demonstrated to hold [31], when $\Xi$ becomes macroscopic and can be
effectively described in the classical formalism, i.e. for a classical
environment.

Before moving to the next section, we notice that whether we consider the
operators $\widehat{F}_{k}$ or their siblings $\widehat{L}_{k}$ as
lindbladians does not make much difference, as Eq. 38 can be written using
these or the others by properly redefining $\widetilde{H}_{\rm eff}$ [32].

##  4\. A qubit in a bosonic environment

Having found a GKSL-like equation for the time derivative of the density
operator of an OQS in the parametric representation, we now want to see how
this result works for two specific models and, in particular, derive an
explicit form for the operators $\widehat{F}_{k}$ playing the role of
lindbladians. Both models describe a qubit $\Gamma$ interacting with a bosonic
environment $\Xi$: the first one is the so-called pure-dephasing model, while
the second is the well-known Jaynes-Cummings model.

For the sake of a lighter notation, and given that the formalism should now be
clear, in this section we drop any distinctive signs for operators, no matter
upon which Hilbert space they act.

###  4.1. Pure-Dephasing Model

By pure-dephasing model it is usually meant one in which the operators acting
on $\Gamma$ that enter the total hamiltonian, commute with each other: this
feature reflects into the existence of a preferred basis in
$\mathcal{H}_{\Gamma}$, hereafter indicated by
$\\{\ket{\gamma}\\}_{\mathcal{H}_{\Gamma}}$, which simultaneously diagonalizes
all of the above operators. The model’s dynamics is often referred to as off-
diagonal, since the diagonal elements of $\rho_{\Gamma}(t)$ with respect to
the preferred basis are constant in time.

Introducing the Pauli matrices $\sigma_{x},\sigma_{y},\sigma_{z}$ to describe
the qubit, and the creation,annihilation operators $a^{\dagger},a$ for the
bosonic environment, the total hamiltonian, with $\hbar=1$, is

$\displaystyle H$ $\displaystyle=$ $\displaystyle\mathbb{1}\otimes\omega
a^{\dagger}a+g\sigma_{z}\otimes\big{(}a+a^{\dagger}\big{)}$ (39)
$\displaystyle=$ $\displaystyle H_{\Xi}+H_{int}~{},$ (40)

where

$\quad H_{\Xi}=\omega a^{\dagger}a\,,\quad
H_{int}=g\sigma_{z}\otimes(a+a^{\dagger})\,.$ (41)

In this case the proper ECS are the usual Glauber coherent states

$\ket{\alpha}=D(\alpha)\ket{0},$ (42)

with $\alpha\in\mathbb{C}$ and

$D(\alpha)=e^{\alpha a^{\dagger}-\alpha^{*}a}\,,$ (43)

often called displacement operator. These coherent states are such that

$\displaystyle
a\ket{\alpha}=\alpha\ket{\alpha}\quad,\qquad\ket{\alpha}=e^{-|\alpha|^{2}/2}\sum_{n}\frac{\alpha^{n}}{\sqrt{n!}}\ket{n},$
(44)
$\displaystyle\bra{\beta}\ket{\alpha}=e^{-\frac{1}{2}(|\alpha|^{2}+|\beta|^{2})}e^{\beta^{*}\alpha},$

where $\ket{n}$ are the Fock states (i.e. eigenstates of $a^{\dagger}a$),
while the manifold $\mathcal{M}$ is the complex plane, with the invariant
measure

$d\mu(\alpha)=\frac{d\alpha d\alpha^{*}}{\pi}\,.$ (45)

We now move to the derivation of an explicit form for the operators
$F_{k}(\alpha)$ for this model: we have

$\frac{\partial}{\partial\alpha^{*}}H(\alpha)=\frac{\partial}{\partial\alpha^{*}}\Big{(}\omega|\alpha|^{2}\mathbb{1}+g(\alpha+\alpha^{*})\sigma_{z}\Big{)}=g\sigma_{z}+\omega\alpha\mathbb{1}\,,$
(46)

and

$D\left(\dot{\alpha}\right)=\exp[-i(g\sigma_{z}+\omega\alpha\mathbb{1})\otimes
a^{\dagger}-i(g\sigma_{z}+\omega\alpha^{*}\mathbb{1})\otimes a\Big{]}\,.$ (47)

To compute the explicit action of the operator above on the reference state,
we write it as

$D(\dot{\alpha})=e^{-i(Y+\bar{Y})}$ (48)

with

$Y=g\sigma_{z}+\omega\alpha\mathbb{1}\otimes
a^{\dagger}\,,\quad\bar{Y}=g\sigma_{z}+\omega\alpha^{*}\mathbb{1}\otimes a,$
(49)

and notice that

$\displaystyle[Y,\bar{Y}]=-\Big{(}g^{2}+\omega^{2}|\alpha|^{2}\Big{)}\mathbb{1}\otimes\mathbb{1}-g\omega\Big{(}\alpha+\alpha^{*}\Big{)}\sigma_{z}\otimes\mathbb{1}\,,$
(50)
$\displaystyle\big{[}Y,\big{[}Y,\bar{Y}\big{]}\big{]}=\big{[}\bar{Y},\big{[}Y,\bar{Y}\big{]}\big{]}=0\,,$

implying that the Baker-Campbell-Hausdorff formula provides

$e^{-i(Y+\bar{Y})}=e^{-iY}e^{-i\bar{Y}}e^{-\frac{1}{2}[Y,\bar{Y}]}\,.$ (51)

The expression for $F_{\pm}(\alpha)$ consequently reads

$F_{\pm}(\alpha)=d_{\pm}(\alpha)\mathbb{1}+b_{\pm}(\alpha)\sigma_{z}\,,$ (52)

where $d_{\pm}(\alpha)$ and $b_{\pm}(\alpha)$ are functions of $\alpha$ whose
expression is rather involved [26, 32], but gets simplified when the classical
limit for the environment is taken. In particular, in such limit we find
(assuming the initial state is
$\frac{1}{\sqrt{2}}(\ket{+}+\ket{-})\otimes\ket{\Xi}$ for some state
$\ket{\Xi}$ of the environment)

$\displaystyle
d^{C}_{\pm}(\alpha)=\frac{1}{\sqrt{2}}e^{-\frac{1}{2}|\beta|^{2}}e^{\mp
i\omega\beta\alpha^{*}}\cos{g\beta}\,,$ (53) $\displaystyle
b^{C}_{\pm}(\alpha)=\mp\frac{i}{\sqrt{2}}e^{-\frac{1}{2}|\beta|^{2}}e^{\mp
i\omega\beta\alpha^{*}}\sin{g\beta}\,,$

where

$\beta(t)=\frac{g}{\omega}(1-e^{-i\omega t})\,.$ (54)

Consequently, Eq. (22) takes the form

$\dot{\rho}_{\Gamma}=-i[H_{\rm eff},\rho_{\Gamma}]+\int
d\mu\,\Gamma\,\Big{(}\sigma_{z}R\sigma_{z}-R\Big{)}$ (55)

where

$H_{\rm eff}=\int
d\mu\frac{1}{\chi^{2}}\sum_{k=\pm}\imaginary(a_{k}b_{k}+|a_{k}|^{2}d^{*}_{k}b_{k}\Big{)}\sigma_{z}=h\sigma_{z}\,$
(56)

is the free hamiltonian of a qubit in an external time-dependent magnetic
field pointing in the $z$ direction, while

${\rm
T}:=\sum_{k=\pm}\gamma_{k}|b_{k}|^{2}=\frac{1}{2}e^{-|\beta|^{2}}|\sin(g\beta)|^{2}e^{-2\omega\imaginary(\alpha^{*}\beta)}.$
(57)

Since $R=\chi^{2}\ket{\phi}\bra{\phi}$ contains the distribution $\chi^{2}$,
which is significantly nonzero only in a small region around
$\alpha=\pm\beta(t)$ (where $\imaginary(\beta^{*}\alpha)\approx 0$), we can
safely take T out of the integral, use $\int d\mu R=\rho_{\Gamma}$, and write

$\dot{\rho}_{\Gamma}=-i\big{[}H_{\rm
eff},\rho_{\Gamma}\big{]}+\Gamma\Big{(}\sigma_{z}\rho_{\Gamma}\sigma_{z}-\rho_{\Gamma}\Big{)}\,,$
(58)

where

${\rm T}=\frac{1}{2}e^{-|\beta|^{2}}|\sin(g\beta)|^{2}\,.$ (59)

Fig. 1: Plot of ${\rm T}(t)$ for several values of the coupling $g$. The
functions are normalized to $1$.

For small values of $g$ it is

${\rm
T}(t)\sim\frac{1}{2}g^{2}|\beta(t)|^{2}=\frac{g^{4}}{\omega^{2}}(1-\cos(\omega
t))\,;$ (60)

however, as $g\to\infty$, the interval of time $\Delta t$ during which the
function T$(t)$ is approximately zero, grows larger and larger in proportion
to its period. This can be seen in Fig.1 and it means that there are large
intervals of time during which T$(t)\approx 0$ and the system evolves
unitarily under the effective hamiltonian $H_{\rm eff}$, which, we recall, is
the free hamiltonian of a qubit in an external time-dependent magnetic field.

In other words, in the strong coupling regime $g\to\infty$, there is no
decoherence and the qubit evolves as if it were a closed system, under the
influence of an effective hamiltonian containing a time-dependent field.

###  4.2. Jaynes-Cummings Model

Let us now consider another model for a qubit in a bosonic environment, namely
the Jaynes-Cummings model. This model is defined by the hamiltonian

$H=\mathbb{1}\otimes\omega a^{\dagger}a+g(\sigma^{+}\otimes
a+\sigma^{-}\otimes a^{\dagger})=H_{\Xi}+H_{int}~{},$ (61)

where

$\quad H_{\Xi}=\omega a^{\dagger}a\,,\quad H_{int}=g\left(\sigma^{+}\otimes
a+\sigma^{-}\otimes a^{\dagger}\right)~{}.$ (62)

The operators acting on $\Gamma$, $\sigma^{+}$ and $\sigma^{-}$, do not
commute with each others: therefore $H_{int}$ does not describe a pure-
dephasing interaction. The proper ECS are still the Glauber coherent states,
while the operators $H(\alpha)$, acting on $\Gamma$, read

$H(\alpha)=\bra{\alpha}H\ket{\alpha}=\omega|\alpha|^{2}\mathbb{1}+g(\alpha\sigma^{+}+\alpha^{*}\sigma^{-})\,,$
(63)

and we can write

$\frac{\partial
H(\alpha)}{\partial\alpha^{*}}=g\sigma^{-}+\omega\alpha\mathbb{1}\,,\qquad\frac{\partial
H(\alpha)}{\partial\alpha}=g\sigma^{+}+\omega\alpha^{*}\mathbb{1}\,;$ (64)

therefore, the objects $\ket{\smash{\dot{\alpha}}}=D(\dot{\alpha})$ are
obtained as

$\displaystyle
D(\dot{\alpha})=\exp[-i(g\sigma^{-}+\omega\alpha\mathbb{1})\otimes
a^{\dagger}-i(g\sigma^{+}+\omega\alpha^{*}\mathbb{1})\otimes a\Big{]}$ (65)
$\displaystyle=\exp[-ig(\sigma^{+}\otimes a+\sigma^{-}\otimes
a^{\dagger})-i\omega\mathbb{1}\otimes(\alpha^{*}a+\alpha
a^{\dagger})\Big{]}~{}.$

To get the operators $F_{\pm},F_{\pm}^{\dagger}$, we need to explicitly write
the action of $D(\dot{\alpha})$ on $\ket{0}$. On the other hand, the presence
of the non-commuting operators $\sigma^{+}$ and $\sigma^{-}$ (in contrast with
the previous case, where we only had to deal with $\sigma_{z}$), makes it more
difficult to factorize $D(\dot{\alpha})$ as a product of exponentials.
Therefore, at variance with the pure-dephasing case, we here consider the
classical limit of the environment right from the beginning, just to factorize
$D(\dot{\alpha})$. In fact, it can be shown[26, 28] that

$D(\dot{\alpha})\ket{0}\to e^{-ig\sigma^{-}\otimes
a^{\dagger}}e^{-i\omega\alpha\mathbb{1}\otimes a^{\dagger}}\ket{0}~{},$ (66)

and hence Eq. (22) reads

$\dot{\rho}_{\Gamma}=-i\big{[}H_{\rm eff},\rho_{\Gamma}\big{]}+\int d\mu\,{\rm
T}\,\Big{(}\sigma^{+}R\sigma^{-}-\frac{1}{2}\big{\\{}\sigma^{-}\sigma^{+},R\big{\\}}\Big{)},$
(67)

where

$\displaystyle{\rm T}=2g^{2}\sum_{k}\gamma_{k}|v_{k}|^{2}\,,$ (68)
$\displaystyle H_{\rm eff}=-g\sum_{k}\int
d\mu\,\gamma_{k}\Big{[}\imaginary(v_{k}u^{*}_{k})\sigma_{x}+\real(v_{k}u^{*}_{k})\sigma_{y}\Big{]}\,,$

and $v_{k},u_{k}$ are functions on the manifold. Getting their expression
would be the same as solving the exact quantum dynamics of the system and its
enviroment, altogether, which is not an achievable task. However, reminding
Eq. (32), when the environment becomes macroscopic and behaves classicaly we
get

$\dot{\rho}_{\Gamma}\approx-i[\tilde{H}_{\rm
eff},\rho_{\Gamma}]+\widetilde{\rm
T}\Big{(}\sigma^{+}\rho_{\Gamma}\sigma^{-}-\frac{1}{2}\big{\\{}\sigma^{-}\sigma^{+},\rho_{\Gamma}\big{\\}}\Big{)}$
(69)

for some $\widetilde{H}_{\rm eff}$ and $\widetilde{\rm T}$ that depend neither
on $\alpha$ nor on $t$, which makes the above equation (69) the well known
master equation for the Jaynes-Cummings model.

##  5\. Conclusions

In this work we have i) considered the reduced density operator for an OQS,
ii) written its parametric representation with environmental coherent states,
and iii) shown that its time derivative satisfies an equation that can be cast
into a GKSL form, once some assumptions are made.

In particular, we have shown that it is the time derivative of the
distribution $\chi^{2}(\Omega(t))$, representing the probability for the
environment to be in the ECS labelled by the parameter $\Omega(t)$, that
generates a GKSL structure. Moreover, the operators $F_{k},F_{k}^{\dagger}$
that we recognize as the siblings of the usual lindbladians, are defined
locally on the manifold $\mathcal{M}$, that is, every $\Omega\in\mathcal{M}$
has associated to it a set of lindbladians $\\{F_{k}(\Omega)\\}$ entering the
equation for $\dot{\rho}(t)$. Such operators depend on the microscopic details
of the theory and, in principle, can be computed explicitly once the total
hamiltonian of the composite system is given. Clearly, this is a highly non-
trivial task, and can only be done in some specific cases. Let us finally
underline that the use of ECS in our approach is crucial, as continuous
parametric representations are not generally equipped with some inherent
dynamics on the manifold, in the way ECS are. The possibility of associating a
time evolution $\Omega(t)$ to environmental states $\ket{\Omega}$ stems from
the fact that the dynamics of coherent states is known, and it is of classical
type, with a classical-like hamiltonian function that is related to the
original hamiltonian operator. This provides an interpretation that proves
itself appropriate to obtain a GKSL general structure, and motivates further
investigation along these lines, as well as comparison with related recent
works, such as Refs. [33, 34, 35].

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# BLoad: Enhancing Neural Network Training with Efficient Sequential Data
Handling

A S M Iftekhar*, Raphael Ruschel*, Suya You, B. S. Manjunath * represents
equal contribution.A S M Iftekhar, Raphael Ruschel, and B.S. Manjunath are
with the Department of Electrical and Computer Engineering, University of
California Santa Barbara, Santa Barbara, CA, 93106. E-mail: {iftekhar,
raphael251<EMAIL_ADDRESS>You is with Army Research Laboratory,
Intelligent Perception, CISD Los Angeles, CA. Email<EMAIL_ADDRESS>
###### Abstract

The increasing complexity of modern deep neural network models and the
expanding sizes of datasets necessitate the development of optimized and
scalable training methods. In this white paper, we addressed the challenge of
efficiently training neural network models using sequences of varying sizes.
To address this challenge, we propose a novel training scheme that enables
efficient distributed data-parallel training on sequences of different sizes
with minimal overhead. By using this scheme we were able to reduce the padding
amount by more than 100$x$ while not deleting a single frame, resulting in an
overall increased performance on both training time and Recall in our
experiments.

###### Index Terms:

distributed, training, machine learning, multi-GPU

##  I Introduction

The increasing complexity of modern deep neural network models and the
expanding sizes of datasets necessitate the development of optimized and
scalable training methods. Neural networks are commonly trained using multiple
GPUs either within a single machine or distributed across a cluster of nodes.
Traditional distributed training schemes, such as distributed data-parallel
(DDP)[1], have been widely employed. While this scheme is popular, it
struggles with data sequences of varied lengths, like videos of different
durations. To address this challenge, we propose a novel training scheme that
enables efficient DDP training on sequences of different sizes with minimal
overhead.

##  II Problem Statement and Current Limitations

We consider a dataset $\mathcal{D}$ comprising $N$ samples, where each sample
$S_{i\in N}$ represents a video with dimensions $H\times W\times T$. Here, $H$
and $W$ denote the height and width of each frame, respectively, while $T$
represents the duration of the video. Our objective is to train a deep neural
network model efficiently using a DDP scheme while accommodating varying
values of $H$, $W$, and $T$ for each sample $S_{i}$. While our primary focus
is on videos, we expect our method to be applicable to other data types like
audio and text.

Using PyTorch’s Distributed Data-Parallel with datasets of varying lengths can
lead to stalled training without any error message. The root of this issue is
in the gradient synchronization step. Here, each GPU process collects
gradients from other processes to compute an average gradient, which updates
the model. If sequences differ in size, each process gets varying sample
counts, potentially causing a deadlock as processes await others indefinitely,
unable to calculate the gradient.

To illustrate the problem, consider the sample dataset from figure 1. It has 8
sequences with lengths varying from 2 to 6 frames. Initiating a DDP training
with a batch size of 2 using random sampling can produce situations like that
in figure 2. Here, GPU 1 handles two videos, each 2 frames long, while GPU 2
manages two videos, each 6 frames long. After just 2 iterations, GPU 1
completes its batch, leaving it idle, as GPU 2 continues processing. New data
is only retrieved once all GPUs finish their batches. Consequently, when GPU 2
tries to gather gradients from GPU 1, it faces an indefinite wait since GPU 1
has no gradient to return.

Figure 1: Sample dataset with 8 videos of varying length - Each $V_{i}$
represents an individual video, and each yellow square represents a frame.
Figure 2: Deadlock situation when each GPU receives sequences of different
lengths. In this situation, after the third iteration, GPU 1 will not have any
gradients to report, causing GPU 2 to wait without any error message.

A common strategy to resolve this issue involves padding each sample to match
the duration $T_{max}$ of the longest sequence in the dataset (as illustrated
in figure 3). While this method solves the stalling problem, it becomes highly
inefficient when $T_{max}$ is significantly larger than the average sequence
length, resulting in substantial padding and unnecessary computations during
training.

Figure 3: Naive padding solution - Every sequence on the dataset is padded to
match the length of the largest sequence, generally by adding $0$’s or
repeating the last entry of the sequence

Another strategy entails breaking down each data sample into smaller chunks of
size $H\times W\times T_{block}$ and treating each smaller block as an
individual sample, as employed in [2], [3]. While this approach resolves
synchronization issues, it cannot be applied to train neural networks that
incorporate feedback, such DDS[4]. Breaking down the original data sample into
smaller pieces destroys the temporal relationships inherent in the original
sequence, as shown in figure 4.

Figure 4: Sampling solution, where each sequence is trimmed to match a smaller
size, usually the length of the average entry in the dataset. In this
approach, one sequence might be broken into several smaller portions, which
won’t allow the training of models with long temporal support.

##  III Methods

Our proposed method builds upon the padding strategy but significantly reduces
wasteful computations. We create blocks of size $T_{max}$ by concatenating
randomly sampled sequences with length $T_{i}\leq T_{max}$. If we cannot
construct a video of size exactly equal to $T_{max}$, we then build a block as
close to $T_{max}$ and then pad it with $0$’s to fill the block. Figure 5
shows an example of our proposed solution.

Figure 5: Our proposed padding approach - BLoad (as in block load) - aims to
construct sequences of size $T_{max}$ using shorter sequences as building
blocks

Additionally, we create a table containing the starting index of each new
video within each particular block. This table can be useful during training a
recurrent network, such as the DDS, architecture shown in Figure 6 where some
information ($oE_{t-1}$) from iteration $n-1$ is used at iteration $n$. Having
the knowledge of where a new sequence starts enables resetting/discarding the
information from the previous iteration, as it belongs to a different
sequence, correctly maintaining the temporal dependency of the data inside
each block.

Figure 6: DDS architecture from [4] \- In this model, the output of frame
$n-1$ is used as partial input to both encoders during the processing of frame
$n$, resulting in increased performance on video sequences.

For a more technical insight into our method, we’ve provided a pseudocode
outline on 7.

Figure 7: Pseudocode for our proposed padding approach - Note that the
$Random_{*}()$ function returns a random entry of $L_{dict}$ such that
$T_{sampled\\_sequence}\leq remaining\\_frames$

##  IV Experiments & Results

Following the works on [4], we perform experiments on the Action Genome
dataset. This dataset is extensively used in Scene-Graph Detection problems
and contains $7,464$ videos with $166,785$ frames in the training set and
$1,737$ videos with $54,371$ frames in the test set. To evaluate the
differences between each sampling strategy, we retrain DDS using each strategy
mentioned earlier and report the amount of padding added, number of frames
deleted, time per epoch, and performance on the recall@20 metric. The Action
Genome dataset is apt for these experiments, given its wide range of sequence
lengths, from brief 3-frame snippets to as long as 94 frames. Our experiments
were conducted on a machine with 8 NVIDIA A100 with 40GB of memory and are
reported in table I.

TABLE I: Comparison of performance using different training strategies | 0 padding | sampling | mix pad | block_pad
---|---|---|---|---
padding amount | 534831 | 0 | 37712 | 3695
# frames deleted | 0 | 92271 | 40289 | 0
time (per epoch) | 170 min | 18 min | 40 min | 41 min
recall@20 | - | 41.2 | 42.1 | 43.3

From the table, it’s evident that the naive padding solution results in over
$500k$ padding frames—almost $4x$ the original data size. This rendered the
training so inefficient that we chose not to complete it for performance
evaluation. Interestingly, with the sampling strategy, despite discarding
nearly $2/3$ of the data, we achieved results comparable to or even surpassing
several established models such as [5]. We attribute this to the dataset’s
high frame correlation, leading to marginal gains with added frames. We
haven’t delved deeper into this observation as it falls outside this
manuscript’s scope. Our proposed block pad strategy offers clear advantages.
It combines zero frame removal with minimal padding, reducing unnecessary
computations and enhancing performance.

##  V Conclusion

In this white paper, we addressed the challenge of efficiently training neural
network models using sequences of varying sizes. We proposed a novel training
scheme that combines elements of padding and distributed data parallelism to
achieve optimal results. By padding sequences with videos of appropriate
lengths and employing a table of starting indices, our method reduces wasteful
computations while preserving temporal relationships. The proposed approach
opens up new possibilities for training models on diverse data types, such as
videos, audio, and text, with varying sequence lengths. In future research, we
can delve into the method’s applicability to different modalities and test its
efficacy across various deep learning challenges.

## References

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