#  GenderRobustness: Robustness of Gender Detection in Facial Recognition
Systems with variation in Image Properties

Preliminary Draft

Sharadha Srinivasan  Anna University1 Thørväld CircleChennaiIndia and  Madan
Musuvathi  Microsoft ResearchRedmondUSA

(2018)

###### Abstract.

In recent times, there have been increasing accusations on artificial
intelligence systems and algorithms of computer vision of possessing implicit
biases. Even though these conversations are more prevalent now and systems are
improving by performing extensive testing and broadening their horizon, biases
still do exist. One such class of systems where bias is said to exist is
facial recognition systems, where bias has been observed on the basis of
gender, ethnicity, skin tone and other facial attributes. This is even more
disturbing, given the fact that these systems are used in practically every
sector of the industries today. From as critical as criminal identification to
as simple as getting your attendance registered, these systems have gained a
huge market, especially in recent years. That in itself is a good enough
reason for developers of these systems to ensure that the bias is kept to a
bare minimum or ideally non-existent, to avoid major issues like favoring a
particular gender, race or class of people or rather making a class of people
susceptible to false accusations due to inability of these systems to
correctly recognize those people.

In the last 2 years, facial recognition systems have claimed to have made
their gender detection module more accurate and that they have trained the
model on a more diverse dataset. Yet, in this paper, we present our findings
on variations in gender detection of images when subjected to changes in image
properties like brightness, sharpness, contrast as well as on a combination of
these factors as opposed to just a single factor. We aim to test on different
publicly available facial recognition systems for gender bias on a face
dataset, which we have tried to make as diverse as possible in order to
produce accurate results, and generate a comparative study.

††copyright: acmcopyright††journalyear: 2018††doi:
10.1145/1122445.1122456††conference: Woodstock ’18: ACM Symposium on Neural
Gaze Detection; June 03–05, 2018; Woodstock, NY††booktitle: Woodstock ’18: ACM
Symposium on Neural Gaze Detection, June 03–05, 2018, Woodstock, NY††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††copyright: none

##  1\. Introduction

Facial recognition technology has come a long way since its inception about 50
years ago. Even though the first facial recognition system around the 1960s
proved that it could be a viable biometric, it’s efficiency was greatly
hampered by the technology of that era. It was as early as the 1970s that 21
specific subjective markers were introduced to automate the recognition.
Especially in the past decade, it’s applications have become widespread. It’s
uses go beyond just unlocking phones and laptops. Facial recognition systems
now use artificial intelligence, machine learning and deep learning algorithms
to predict a number of qualities from the detected faces. It’s applications
have also been growing exponentially due these features. It is widely used in
law enforcement, retail, marketing and health applications, almost disrupting
the way people live and work today, whether we realize it or not.

Can we concede with facial recognition software companies claiming that these
models are able to accurately identify faces better than humans? On the
contrary, to say that these systems are controversial would be an
understatement. Unfortunately, it has been observed that facial recognition
algorithms vary in their performance across different face types. As mentioned
above, since law enforcement agencies have started using facial recognition to
identify subjects, the accuracy of these algorithms is now a matter of
criminal justice. Not just that, there have been several racial issues,
starting with the 2009 face tracking software and prevalent in systems even
today that can track fairer skin tones but not darker skin tones. This has led
to several companies being called out, leading to outright bans of facial
recognition software in specific places or for specific uses.

Recent studies and experiments showed that most of the popular facial
recognition systems like Amazon Rekognition,Microsoft’s Cognitive Services
Face API, Face++, Google Vision API and IBM’s Watson Visual Recognition API
that identify a list of characteristics like gender, age, ethnicity, emotions
and so on are gender biased. Following these accusations, Google and IBM have
removed their gender recognition module from their systems. The main issue
that was identified was that the training data used to train these models was
not diverse enough.

This paper is motivated by such biases, focussing on gender bias in facial
recognitions systems. Though it has been established that these facial
recognition systems are biased, there is a void in terms of exploration and
analysis of gender bias patterns with variation in image properties. Would
these facial recognition systems be as biased when the images are subjected to
these transformations? If unbiased, then for any image the model should start
with confidently correct, start showing ambiguity uniformly for all images or
identify them correctly till the maximum value of the range.

Firstly, this paper explores the gender bias pattern when image
transformations are applied on properties like sharpness, brightness and
contrast. This paper also defines a robustness value to determine the range
for which these experiments hold valid.

Secondly, Publicly available datasets like Adience, Labelled Faces in the Wild
Home, Flickr-Faces-HQ Dataset (FFHQ) etc have proved to be inefficient while
testing for bias due to their limited diversity. As we conduct experiments on
image transformations, we also aim to create an all-inclusive dataset, to
include every country on the globe. This would help us compartmentalize and
study the bias of face recognitions systems with respect to different
ethnicities.

##  2\. Implementation

This tool, coded in python, is developed primarily for testing of the software
for studying gender bias variations with variations in image The model that we
have chosen to start developing the tool for is the “Amazon Rekognition”
software.

  * •

To test on Amazon Rekognition, a Boto3 client is created that represents
Amazon Rekognition and facilitates requests with the help of the AWS API.

  * •

A diverse dataset must be gathered in order to develop a testing tool that has
appreciable efficacy. We start with a small set of images which included
images from under-represented communities in the other publicly available
datasets like South-East Asians, African Americans, South Asians and included
images of other majority communities.

  * •

To test the Boto3 client, an image can be chosen at random from the dataset.
The path is set and the image is read in “rb” mode which reads the image and
it is then stored as a binary file. This byte file is sent as an argument to
the “DetectFaces API” using the client to generate a response from the facial
recognition software.

  * •

The response is essentially in the form of a dictionary, where the keys
themselves are either a collection of dictionaries or a list of dictionaries.
This response includes information about gender, age , emotion, moustache,
smile, pose and various other details about the face, along with their
confidence values.

This paper will focus on the accuracy and bias pertaining to the gender value
and the corresponding confidence value.

##  3\. Image Transformations

This paper explores the effect of applying image transformations on the gender
value detected by the facial recognition system. The “ImageEnhance” function
of the Pillow library used to perform these transformations. Pillow is a
versatile and popular imaging library. The ImageEnhance module is a
particularly useful module within the Pillow library. The ImageEnahance module
consists of a number of classes that can be used for image enhancement. It
contains in-built functions to change the brightness, sharpness, contrast and
color of the image. While performing image transformations, the input and
output, both would be in the form of images.

The image transformations discussed in this paper would be brightness,
contrast and sharpness, as well as a combination of these factors. The gender
bias becomes even more significant when the results are divided into the
following three categories:

  * •

The gender value is correctly identified throughout all factors

  * •

The gender value is incorrectly identified throughout all factors

  * •

The gender value varies with variation in the factors

###  3.1. Brightness

The ImageEnahance module has a function called Brightness() where the
brightness factor can be passed as a parameter to adjust the brightness of an
image. Where ‘0’ would be a black and white image. When we call this function,
what happens is the value of all pixels is changed by a constant. Adding a
positive constant to all of the image pixel values makes the image brighter.
Similarly, we can subtract a positive constant from all of the pixel values to
make the image darker.

We start with the factor ‘0’ all the way to ‘30’, with log intervals, to
observe the changes in the gender values. The response is stored as a list of
dictionaries. This is then used to visualize the results and find a range for
the brightness factor and also observe any patterns in gender value variation.

###  3.2. Contrast

As mentioned above, brightness refers to the overall lightness or darkness of
an image,whereas, contrast refers to the brightness difference between
different objects or regions of the image. For this, the built in function of
the ImageEnhance module Contrast() is used. To adjust the contrast, we can
modify the slope of the transfer function. A transfer function with a slope
greater than one will increase contrast, while a slope less than one will
decrease it.

We start with the factor ‘0’, which gives a solid grey image, all the way till
250, as some images are still able to be recognized at such high factors. The
result thus obtained for each factor is stored and visualized to look for
patterns and also find a range where the images are recognizable or show
observable variations in gender values.

###  3.3. Sharpness

When the Sharpness() function is called, it increases the contrast between
bright and dark regions to bring out features. During the sharpening process,
a high pass filter is applied to an image. An enhancement factor of 0.0 gives
a blurred image, a factor of 1.0 gives the original image. Any factor above
1.0, increases the sharpness of the image.

Again, as in the other two factors, we start with the factor ‘0’, all the way
till 200 after which the gender value becomes invariant or the image is not
recognized by the facial recognition application.

##  4\. Dataset

##  5\. Results

##  6\. Conclusion

##  7\. References


11institutetext:  Department of Physics, University of Maryland, Baltimore
County, Baltimore, MD 21250, USA
Instituto de Física ‘Gleb Wataghin’, Universidade Estadual de Campinas,
13083-859, Campinas, São Paulo, Brazil

# Nonlinear speed-ups in ultracold quantum gases

Sebastian Deffner E-mail: 1122<EMAIL_ADDRESS>1122

###### Abstract

Quantum mechanics is an inherently linear theory. However, collective effects
in many body quantum systems can give rise to effectively nonlinear dynamics.
In the present work, we analyze whether and to what extent such nonlinear
effects can be exploited to enhance the rate of quantum evolution. To this
end, we compute a suitable version of the quantum speed limit for numerical
and analytical examples. We find that the quantum speed limit grows with the
strength of the nonlinearity, yet it does not trivially scale with the
“degree” of nonlinearity. This is numerically demonstrated for the parametric
harmonic oscillator obeying Gross-Piteavskii and Kolomeisky dynamics, and
analytically for expanding boxes under Gross-Pitaevskii dynamics.

In public perception, the prophesied _quantum advantage_ of novel technologies
has become almost synonymous with anticipated _quantum speed-ups_. This
impression is driven by quantum computing, which indeed can solve certain
problems faster than any classical computer could [1]. At least superficially
this expectation seems to be at variance with the so-called _quantum speed
limits_ (QSL), which are fundamental bounds on the maximal rate with which a
quantum system can evolve [2, 3]. In fact, diverging QSLs can be interpreted
as an herald of classicality [4, 5], as they are deeply rooted in more
rigorous formulations of Heisenberg’s uncertainty relation for energy and time
[6].

The apparent contradiction quickly dissolves, once one realizes that in the
lingo of computer science a “speed-up” simply refers to a smaller number of
required single gate operations, whereas in quantum physics the QSL refers to
the maximal rate with which such a gate operation can be applied [7]. Thus it
also becomes rather obvious why so much research activity has been dedicated
to the study of QSL in virtually all areas of quantum physics, including,
e.g., quantum communication [8, 9, 10, 11, 12, 13], quantum computation [14,
15], quantum control [16, 17, 18], many body physics [19, 20], and quantum
metrology [21, 22]. See somewhat recent reviews on the topic [23, 24].

The original QSLs were formulated for standard quantum mechanics [25], whose
dynamics is described by the Schrödinger equation. However, over the last
decade is has become obvious that there is a variety of “quantum resources”
that can be employed to speed-up quantum dynamics. For instance, is has been
established that judiciously designed open system dynamics permit environment
driven speed-ups [26, 27, 28, 29, 30, 31]. Similarly, the QSL is larger for
non-Hermitian quantum dynamics [32, 33], for which the quantum states can find
literal shortcuts through Hilbert space.

Figure 1:  From left to right: absolute value, real part, and imaginary part
of the solution $\Psi_{\tau}(x)$ of Eq. (1), for the parametric harmonic
oscillator (4) and $\kappa=0$ (blue, dashed line), $\kappa=5$ (purple, dot-
dashed line), and $\kappa=10$ (red, solid line). Other parameters are
$\hbar\omega_{0}=5$, $\hbar\omega_{1}=1$, $m=1$, and $\tau=2$

In the present analysis, we will be focusing on another type of quantum
dynamics, which has not received much attention in the study of the QSL.
Processing information requires some form of interaction between signals. In
photonic systems, such interactions can be enabled by nonlinear optical
processes [34]. Nonlinear optics has become almost ubiquitous in fundamental
science and technological applications [35] and can be realized, e.g., with
single atoms in cavities [36], in atomic ensembles [37], and through atom-atom
interactions [38]. It is expected that nonlinear systems will be able to
outperform linear systems in optical quantum information processing [34]. The
reason is that nonlinear quantum optics can provide both, linear operations
such as storage and manipulation of quantum states, as well as nonlinear
operations such as the generation of quanta and quantum logic between photonic
quantum bits. Recent experimental developments in nonlinear quantum optics
hold the promise to enable the development of universal quantum computers
supporting conditional quantum logic operations, which are now within
technological reach [34]. The natural question arises whether nonlinear
interactions can be exploited as a quantum resource to enhance the QSL. Some
evidence can be found in the literature [39, 40], yet a comprehensive analysis
within the framework of QSL appears to be lacking.

We show that there is indeed a speed-up of the rate of quantum evolution in
nonlinear systems. To this end, we begin with an numerical investigation of a
cold bosonic quantum gas in a harmonic trap, which is described by the Gross-
Pitaevskii equation [41, 42]. We find that the resulting QSL grows with the
strength of the scattering length, meaning that the stronger the nonlinearity
the faster the quantum state can evolve. While such numerical evidence is
interesting, it is not as insightful as analytical treatments. Therefore, we
then solve the Gross-Pitaevskii equation for a 1-dimensional box, whose volume
expands at constant rate. For these dynamics, we obtain an analytical
expression for the QSL, and we find our numerical evidence rigorously
confirmed. Thus, as main result, we show that nonlinear quantum effects can
quantifiably enhance the rate of quantum evolution. We conclude the analysis
by also exploring the QSL for the parametric harmonic oscillator evolving
under Kolomeisky dynamics. Remarkably, we find that the QSL does not trivially
grow with the order of the nonlinearity.

##  1 Numerical case study: parametric harmonic oscillator

In the following, we consider quantum dynamics that are described by the
Gross-Pitaevskii equation,

$i\hbar\,\dot{\Psi}_{t}(x)=\left[-\frac{\hbar^{2}}{2m}\,\partial_{x}^{2}+U\left(x,t\right)+\kappa\,\left|\Psi_{t}(x)\right|^{2}\right]\,\Psi_{t}(x)\,,$
(1)

where, as usual, we denote a derivative with respect to time by a dot.
Further, $U(x,t)$ is a time-dependent, external potential and $\kappa$
measures the “strength” of the nonlinearity.

Equation (1) was first discovered in the description of the propagation of
light in nonlinear optical fibers and planar waveguides [35], and shortly
after in the mean-field description of Bose-Einstein condensates [41, 42]. In
cold quantum gases the nonlinearity arises from an effective treatment of the
interaction of the bosonic particles, whereas in nonlinear optics
$\kappa\,\left|\Psi_{t}(x)\right|^{2}$ describes the polarization dependent
amplitude in the paraxial field equations. Since its first description, the
nonlinear Schrödinger equation (1) has appeared also in many other areas of
physics, such as in hydrodynamics [43], in plasma physics [44], and the
propagation of Davydov’s alpha-helix solitons, which are responsible for
energy transport along molecular chains [45]. Thus, Eq. (1) has become one of
the best studied equations in quantum physics, from a conceptual as well as a
numerical point of view [46, 47, 48, 49].

The Gross-Pitaevskii equation (1) has also found some attention in quantum
control [50, 51, 52, 53], yet a genuinely nonlinear QSL appears to be lacking
in the literature. This may, in parts, be a consequence of the fact that in
the derivation of QSL one typically focuses on geometric measures of
distinguishability [54, 55]. However, it also has been argued [4, 13] that in
a fully dynamical approach the QSL is simply determined by the rate with which
a quantum state evolves, and one can define

$v_{\mathrm{QSL}}\equiv\int dx\left|\dot{\Psi}_{t}(x)\right|^{2}\,.$ (2)

It is easy to see that for linear dynamics, $\kappa=0$, we simply have

$v^{0}_{\mathrm{QSL}}\equiv v_{\mathrm{QSL}}(\kappa=0)=\left\langle
H^{2}(t)\right\rangle/\hbar^{2}\,,$ (3)

where $H(t)$ is the time-dependent Hamiltonian. Therefore, Eq. (2) can be
interpreted as a generalization of the Mandelstam-Tamm bound [2, 13], which
remains applicable for any purity preserving quantum dynamics, such as the
Gross-Pitaeveskii equation (1).

To build intuition, and to gain some insight into possible nonlinear speed-
ups, we now consider the parametric harmonic oscillator, with potential

$U(x,t)=\frac{1}{2}m\omega_{t}^{2}x^{2}\,.$ (4)

To keep things as simple as possible, we further choose $\omega_{t}^{2}$ to be
a linear function of time,

$\omega^{2}_{t}=\omega_{0}^{2}-\left(\omega_{0}^{2}-\omega_{1}^{2}\right)t/\tau\,,$
(5)

for which the linear Schrödinger dynamics is analytically solvable [56]. For
this protocol, we solved Eq. (1) numerically for several values of $\kappa$.
For each of the realizations, we choose the ground state of the linear problem
as initial state, namely

$\Psi_{0}(x)=\left(\frac{m\omega_{0}}{\pi\hbar}\right)^{1/4}\,\exp{\left(-\frac{m\omega_{0}}{2\hbar}x^{2}\right)}\,.$
(6)

The resulting solution is depicted in Fig. 1. We observe that for moderate
values of $\kappa$ the solution of the nonlinear dynamics remains close to the
linear solution, whereas for large values the dynamics is strongly
distinguishable.

In Fig. 2 we plot the corresponding QSL (2). As expected, we observe that the
stronger the nonlinearity, i.e., the larger the value of $\kappa$, the faster
is the evolution of the quantum state.

Figure 2:  Left: QSL (2) for the parametric harmonic oscillator (4) for
$\kappa=0$ (blue, dashed line), $\kappa=5$ (purple, dot-dashed line), and
$\kappa=10$ (red, solid line). Right: QSL (2) for the parametric harmonic
oscillator (4) evaluated at $t=\tau/2$ as a function of $\kappa$. Other
parameters are $\hbar\omega_{0}=5$, $\hbar\omega_{1}=1$, $m=1$, and $\tau=2$.

While this numerical result is interesting and confirms our intuition, more
analytical insight is desirable. In particular, it would be useful to be able
to “design” optimal nonlinearities to meet, e.g., speed requirements.

##  2 Scale-invariant Gross-Pitaevskii dynamics

Therefore, we now continue with a special class of time-dependent problems.
Consider the scale-invariant, nonlinear Schrödinger equation

$i\hbar\,\dot{\Psi}_{t}(x)=\left[-\frac{\hbar^{2}}{2m}\,\partial_{x}^{2}+\frac{1}{\lambda_{t}^{2}}\,U\left(\frac{x}{\lambda_{t}}\right)+\frac{\kappa}{\lambda_{t}}\,\left|\Psi_{t}(x)\right|^{2}\right]\,\Psi_{t}(x),$
(7)

where $\lambda_{t}$ is an external control parameter, such as the volume of an
optical trap confining the BEC. Then, if $\lambda_{t}$ is given for some
arbitrary $a$, $b$, and $c$ by $\lambda_{t}=\sqrt{at^{2}+2bt+c}$, a solution
of Eq. (7) can be written as [57, 51, 52]

$\Psi_{t}(x)=\exp{\left(-\frac{i}{\hbar}\mu_{0}\,\tau(t)\right)}\exp{\left(\frac{im\dot{\lambda}}{2\hbar\lambda}\,x^{2}\right)}\,\Phi(x,\lambda_{t})\,,$
(8)

where $\Phi(x,\lambda_{t})=\Phi(x/\lambda_{t})/\lambda_{t}^{2}$ is a scale-
invariant solution of the instantaneous Gross-Pitaveskii equation (7),
$\mu_{0}$ is the initial chemical potential, and
$\tau(t)=\int_{0}^{t}ds/\lambda_{s}^{2}$. Note that for linear dynamics
$\kappa=0$, the chemical potential reduces to the average energy of the
initial state, $\epsilon_{0}$.

It is then a simple exercise to show that the QSL (2) becomes

$\begin{split}v_{\mathrm{QSL}}&=\frac{\mu_{t}^{2}}{\hbar^{2}}+\frac{m\mu_{t}}{\hbar^{2}}\left(\frac{\dot{\lambda}_{t}^{2}}{\lambda_{t}^{2}}-\frac{\ddot{\lambda}_{t}}{\lambda_{t}}\right)\,\left\langle
x^{2}\right\rangle\\\
&+\frac{m^{2}}{4\hbar^{2}}\left(\frac{\dot{\lambda}_{t}^{4}}{\lambda_{t}^{4}}-\frac{2\dot{\lambda}_{t}^{2}\ddot{\lambda}_{t}}{\lambda_{t}^{3}}+\frac{\ddot{\lambda}_{t}^{2}}{\lambda_{t}^{2}}\right)\left\langle
x^{4}\right\rangle+\frac{\dot{\lambda}_{t}^{2}}{\hbar^{2}}\,\left\langle
p^{2}\right\rangle\,,\end{split}$ (9)

where we introduced the instantaneous chemical potential
$\mu_{t}=\mu_{0}/\lambda_{t}^{2}$, and $\left\langle x^{n}\right\rangle=\int
dx\,x^{n}|\Phi\left(x,\lambda_{t}\right)|^{2}$.

###  2.1 Example: infinite square well

As a case study, we continue with an analytically solvable example. Arguably,
the simplest, non-trivial problem in quantum mechanics is a particle trapped
in an infinite square well. The potential simply is

$U(x)=\begin{cases}0,\qquad\forall x\in[0,\lambda]\\\
\infty,\qquad\text{otherwise}\,.\end{cases}$ (10)

It is easy to see that the one-dimensional box with time-varying length is
scale-invariant. This has been exploited, for instance, in standard
thermodynamic problems such as analyzing the validity of the quantum Jarzynski
equality [58, 59], and to construct explicit expressions for the
counterdiabatic field [50, 60, 52].

As a point of reference, and to build intuition we begin with the linear case
$\kappa=0$. For completeness, recall that the eigensystem is given by

$\psi_{n}(x,\lambda)=\sqrt{\frac{2}{\lambda}}\,\sin{\left(\frac{n\pi}{\lambda}\,x\right)}\quad\text{and}\quad
E_{n}(\lambda)=\frac{\hbar^{2}\pi^{2}}{2m\lambda^{2}}\,n^{2}\,.$ (11)

Moreover, we choose a parameterization in which the box is expanded at
constant rate $v$, and hence $\lambda_{t}=\lambda_{0}+vt$. Accordingly, the
QSL (9) simplifies to read

$v_{\mathrm{QSL}}=\frac{\epsilon_{t}^{2}}{\hbar^{2}}+\frac{m\epsilon_{t}}{\hbar^{2}}\,\frac{v^{2}}{\lambda_{t}^{2}}\,\left\langle
x^{2}\right\rangle+\frac{m^{2}}{4\hbar^{2}}\,\frac{v^{4}}{\lambda_{t}^{4}}\,\left\langle
x^{4}\right\rangle+\frac{v^{2}}{\hbar^{2}}\,\left\langle
p^{2}\right\rangle\,,$ (12)

where $\epsilon_{t}=\epsilon_{0}/\lambda_{t}^{2}$ and $\epsilon_{0}$ is the
initial energy. We immediately observe that $v_{\mathrm{QSL}}$ is a sum of
positive terms, and hence we can continue comparing the nonlinear case to the
linear result term-by-term.

The corresponding nonlinear problem can also be solved analytically [61], and
the stationary states read

$\psi_{n}(x,\lambda)=\sqrt{\frac{\nu
K(\nu)}{\lambda\,(K(\nu)-E(\nu))}}\,\mathrm{sn}\left(\frac{2n\mathrm{K}(\nu)}{\lambda}x\bigg{|}\nu\right)\,.$
(13)

Here $\mathrm{sn}(x|\nu)$ is the Jacobian elliptic function [62], and $K(\nu)$
and $E(\nu)$ are elliptic integrals. The parameter $\nu$ is directly related
to the strength of the nonlinearity $\kappa$. It it is implicitly determined
by

$\mathrm{K}(\nu)\,\left(K(\nu)-E(\nu)\right)=\frac{m\lambda\,\kappa}{4n^{2}\hbar^{2}}\,,$
(14)

which follows from substituting Eq. (13) into Eq. (7) for $U(x)$ as given in
Eq. (10). Accordingly, the generalized eigenvalues can be written as

$\mu_{n}(\lambda)=\frac{\hbar^{2}}{2m\lambda^{2}}\,n^{2}\,(2\mathrm{K}(\nu))^{2}\,(1+\nu)\,.$
(15)

With Eqs. (13)–(15) the QSL (12) can be computed exactly. However, the
resulting expression is rather involved, and hence does not permit to gain
much insight.

Therefore, we now continue to compute corrections to the quantum speed limit
up to linear order in the strength of the nonlinearity $\kappa$. The linear
eigensystem (11) is recovered for $\nu=0$. Thus, we begin by expanding the
left side of Eq. (14) up to linear order in $\nu$ 111See Ref. [62] for
mathematical details of the elliptic integrals, from which we obtain

$\nu\simeq\frac{2m\lambda\,\kappa}{n^{2}\hbar^{2}\pi^{2}}\,.$ (16)

Similarly, the generalized eigenvalues become [61]

$\mu_{n}(\lambda)\simeq\frac{\hbar^{2}\pi^{2}}{2m\lambda^{2}}\,n^{2}\left(1+\frac{3\nu}{2}\right)\,,$
(17)

which can be written in terms of the linear energy eigenvalue $E_{n}(\lambda)$
(11) as

$\mu_{n}(\lambda)\simeq E_{n}(\lambda)+\frac{3}{2}\frac{\kappa}{\lambda}\,.$
(18)

Hence, for the first term in Eq. (12) we have already obtained the leading
order correction, i.e., “the nonlinear speed up”.

The other terms in Eq. (12) require a little more work. We continue with the
second moment of the momentum, which can in fact be computed in closed form.
To further simplify the analysis, we now assume that the system was initially
prepared in its ground state, $n=1$. In this case we have

$\left\langle
p^{2}\right\rangle=\frac{(2K(\nu))^{2}}{3\lambda^{2}(K(\nu)-E(\nu))}\,\left[K(\nu)(\nu-1)+E(\nu)(\nu+1)\right]\,,$
(19)

which in leading order of $\nu$ simply becomes

$\left\langle
p^{2}\right\rangle=\frac{\pi^{2}}{\lambda^{2}}+\frac{\pi^{2}}{32\lambda^{2}}\,\nu^{2}\,.$
(20)

In other words, we obtain

$\left\langle p^{2}\right\rangle=\left\langle
p^{2}\right\rangle_{\mathrm{lin}}+\frac{m^{2}\kappa^{2}}{8\hbar^{4}\pi^{2}}$
(21)

which is again larger than the expression corresponding to linear dynamics,
$\left\langle p^{2}\right\rangle_{\mathrm{lin}}$, and where we used Eq. (16)
for $n=1$.

To compute the second and fourth moment of $x$, we now need to expand the
stationary states (13). For small $\nu$ the Jacobi elliptic function can be
approximated by [62]

$\mathrm{sn}(x|\nu)\simeq\sin{\left(x\right)}-\frac{1}{4}\nu\left(x-\sin{\left(x\right)}\cos{\left(x\right)}\right)\cos{\left(x\right)}\,.$
(22)

and thus we have

$\begin{split}&\psi_{n}(x,\lambda)\simeq
A_{n}\sin{\left(\frac{2n\mathrm{K}(\nu)}{\lambda}x\right)}\\\
&\quad-\frac{1}{4}\nu\left(\frac{2n\mathrm{K}(\nu)}{\lambda}x-\sin{\left(\frac{2n\mathrm{K}(\nu)}{\lambda}x\right)}\cos{\left(\frac{2n\mathrm{K}(\nu)}{\lambda}x\right)}\right)\\\
&\quad\times\cos{\left(\frac{2n\mathrm{K}(\nu)}{\lambda}x\right)}\,.\end{split}$
(23)

Note that $K(0)=\pi/2$ and hence in leading order we recover the
eigenfunctions of the linear case (11). Moreover, $A_{n}$ is the new
normalization coefficient, which can be written in leading order of $\nu$ as

$A_{n}\simeq\sqrt{\frac{2}{\lambda}}-\frac{\nu}{8\sqrt{2\lambda}}\,.$ (24)

Accordingly, the second moment of $x$ becomes,

$\left\langle x^{2}\right\rangle\simeq\left\langle
x^{2}\right\rangle_{\mathrm{lin}}+\frac{3\lambda^{2}}{64\pi^{2}}\,\nu\,,$ (25)

which with Eq. (16) can be written as

$\left\langle x^{2}\right\rangle\simeq\left\langle
x^{2}\right\rangle_{\mathrm{lin}}+\frac{3m\lambda^{3}\,\kappa}{32\hbar^{2}\pi^{4}}\,.$
(26)

We again observe that the leading order in $\kappa$ is additive and positive.
Similarly, we have for the fourth moment

$\left\langle x^{4}\right\rangle\simeq\left\langle
x^{4}\right\rangle_{\mathrm{lin}}+\frac{3(8\pi^{2}-15)\lambda^{2}}{256\pi^{4}}\,\nu\,,$
(27)

and

$\left\langle x^{4}\right\rangle\simeq\left\langle
x^{4}\right\rangle_{\mathrm{lin}}+\frac{3(8\pi^{2}-15)m\lambda^{3}\,\kappa}{128\hbar^{2}\pi^{6}}\,.$
(28)

Hence, we immediately conclude that the quantum speed limit (9) for quantum
boxes that are prepared in their corresponding ground states, and whose volume
is changed at constant rate, is larger for nonlinear dynamics, than for linear
dynamics. These findings are illustrated in Fig. 3.

Figure 3:  Left: QSL (2) for the time-dependent box (10) for $\kappa=0$ (blue,
dashed line), $\kappa=0.25$ (purple, dot-dashed line), and $\kappa=0.5$ (red,
solid line). Right: QSL (2) for the time-dependent box (10) as a function of
$\kappa$. Other parameters are $\hbar=1$, $v=1$, and $m=1$.

In conclusion, the analytically solvable model confirms the numerically found
result, namely that nonlinear quantum dynamics supports faster evolution than
linear dynamics. However, from the analytical result we have also obtained
exact, closed expressions for $v_{\mathrm{QSL}}$ in leading order of $\kappa$

Figure 4:  From left to right: absolute value, real part, and imaginary part
of the solution $\Psi_{\tau}(x)$ for the parametric harmonic oscillator (4)
with linear dynamics (blue, dashed line), Gross-Piteaveskii dynamics (1) (red,
solid line), and Kolomeisky dynamic (29) (green, dot-dashed line). Other
parameters are $\hbar\omega_{0}=5$, $\hbar\omega_{1}=1$, $m=1$, $\kappa=10$,
and $\tau=2$

##  3 Beyond Gross-Pitaevskii dynamics

In our analysis we have so far restricted ourselves to the 1-dimensional
Gross-Pitaevskii equation (1). However, it has been noted [63] that the
description of dilute Bose systems requires a fundamental modification of the
dynamics in low dimensions $d\leq 2$. In fact, such scenarios are better
described by

$i\hbar\,\dot{\Psi}_{t}(x)=\left[-\frac{\hbar^{2}}{2m}\,\partial_{x}^{2}+U\left(x,t\right)+\kappa\,\left|\Psi_{t}(x)\right|^{4}\right]\,\Psi_{t}(x)\,,$
(29)

where the nonlinear terms is of fifth order, rather than third in Eq. (1). For
a in-depth discussion of the significance, properties, and experimental
relevance of the Kolomeisky equation (29) we refer to a recent review article
[64].

For our present purposes the immediate question arises whether making the
dynamics “more nonlinear” allows for a further speed-up of the evolution. To
systematically answer this question, we solved Eq. (29) numerically for the
parametric harmonic oscillator (4) with the linear protocol (5), and for the
same initial state (11). In Fig. 4 we depict the result together with linear
evolution and the Gross-Pitaveskii dynamics.

We observe that the solution of the Kolomeisky dynamics (29) is markedly
different from the linear and Gross-Pitaevskii case (1). However, from the
solution is it not obvious whether the evolution is any faster or slower for
the “more nonlinear dynamics”.

Therefore, we again computed the QSL (9), which is plotted in Fig. 5.

Figure 5:  Left: QSL (2) for the parametric harmonic oscillator (4) with
linear dynamics (blue, dashed line), Gross-Piteaveskii dynamics (1) (red,
solid line), and Kolomeisky dynamic (29) (green, dot-dashed line). Right: QSL
(2) for the parametric harmonic oscillator (4) evaluated at $t=\tau/2$ as a
function of $\kappa$. Other parameters are $\hbar\omega_{0}=5$,
$\hbar\omega_{1}=1$, $m=1$, and $\tau=2$.

We immediately observe that for the present case, the QSL for Kolomeisky
dynamics is almost always smaller than for the Gross-Piteaveskii case. Hence,
we conclude that higher order nonlinearities do generally not enhance the rate
of quantum evolution. However, now the obvious question arises whether there
is an “optimal” nonlinearity for which the QSL becomes maximal. Since
answering this question will require an in-depth, systemic explorations of
physically relevant, nonlinear evolution equations, we leave the answer for
future work.

##  4 Concluding remarks

In the present work we have demonstrated, with numerical and analytical
examples, that nonlinear quantum dynamics can enhance the rate of quantum
evolution. These findings are not only of academic interest, but they may find
application in a variety of practically relevant problems. For instance, we
can foresee that nonlinear quantum speed-ups may be particularly relevant for
quantum communication, quantum computation, and quantum thermodynamics.

Nonlinear effects in classical optics have been well-studied in the context of
communication [65]. However, nonlinear effects in optical fibers are typically
very weak [65], since they depend on the intensity of the transmitted signals
and the interaction length. Thus, exploiting nonlinear effects in quantum
optics appears more realistic than in classical optics [66] . In fact, the
National Institute of Standards and Technology (NIST) has dedicated an entire
research program to the exploration and development of nonlinear quantum
networks [67]. Our findings clearly show that nonlinear effects will enable
faster quantum communication.

Another area, where nonlinear quantum speed-ups may be relevant, is nonlinear
quantum computation. In this framework, unitary gates are replaced with
nonlinear quantum operations [68, 69, 70]. Our results indicate that nonlinear
computing not only has a computational advantage, but also that single gate
operations can be implemented faster than in linear computers. Hence, the
overall processing time can be made shorter, and the quantum computer becomes
less susceptible to environmental noise.

Finally, we have recently shown that Bose-Einstein condensation can boost the
performance of quantum heat engines [71]. However, our previous analysis [71]
was restricted to endoreversible cycles, i.e., finite time operation that is
still slow enough to allow the working medium to reach a state of local
equilibrium. At this point, it is not obvious that the performance boost will
persist for engines that operate far from equilibrium. However, our present
results suggest that, indeed, the power output may be enhanced by
collaborative quantum effects within the working medium, which would
constitute a thermodynamic advantage arising from the effectively nonlinear
quantum dynamics. See also Ref. [72] for similar conclusions.

###### Acknowledgements.

Enlightening discussions with Nathan M. Myers, Akram Touil, Maxwell Aifer, and
Steve Campbell are gratefully acknowledged. S.D. acknowledges support from the
U.S. National Science Foundation under Grant No. DMR-2010127.

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