Article Type: Description (see Introduction for more detail) XX XX 8 Month
Publication Title

# Quantum Computing for All: Online Courses Built Around Interactive Visual
Quantum Circuit Simulator

Juha Reinikainen  Faculty of Information Technology, University of Jyväskylä,
Finland  Vlad Stirbu  Faculty of Information Technology, University of
Jyväskylä, Finland  Teiko Heinosaari  Faculty of Information Technology,
University of Jyväskylä, Finland  Vesa Lappalainen  Faculty of Information
Technology, University of Jyväskylä, Finland  Tommi Mikkonen  Faculty of
Information Technology, University of Jyväskylä, Finland

(2023)

###### Abstract

Quantum computing is a highly abstract scientific discipline, which, however,
is expected to have great practical relevance in future information
technology. This forces educators to seek new methods to teach quantum
computing for students with diverse backgrounds and with no prior knowledge of
quantum physics. We have developed an online course built around an
interactive quantum circuit simulator designed to enable easy creation and
maintenance of course material with ranging difficulty. The immediate feedback
and automatically evaluated tasks lowers the entry barrier to quantum
computing for all students, regardless of their background.

††journal: Publication Name

Quantum computing is a rapidly developing field that has the potential to
transform the way we approach complex computational problems. By harnessing
the principles of quantum mechanics, quantum computers may be able to solve
problems that are currently unsolvable or require an impractical amount of
time to solve using classical computers.

Quantum computing is a highly mathematical and abstract field that requires a
strong foundation in physics, computer science, and mathematics. The concepts
are often difficult to grasp, especially for students who have not had prior
exposure to quantum mechanics or linear algebra. Therefore, it is often seen
as a purely theoretical subject, and it can be challenging to relate the
concepts to real-world applications or to explain how they can be used to
solve practical problems.

Since quantum computing is becoming a practical field, it requires students to
gain hands-on experience with both quantum software and hardware. However, the
learning curve for quantum computing should be smoother and not require deep
programming expertise from the start. Further, even if there are already some
quantum computers available for online access, the computing time may be
limited, especially for students who are not enrolled in specialized programs.
These reasons have motivated the University of Jyväskylä to develop an
interactive quantum circuit simulator as a part of the existing learning
platform.

The rest of paper is structured as follows, aligning with the phases of the
Design Science Research (SDR) methodology [9]. We start by introducing the
problem identification and motivation, followed by presenting the learning
environment in which the solution is integrated. Then, we describe the quantum
simulator implementation, and a demonstration on how it is used to conduct
three specific tasks within the educational process. Subsequently, we discus
how the visual editor meets the teaching goals for the target student
audience. We finish with concluding remarks.

## MOTIVATION

Quantum computing is an interdisciplinary topic that cannot be put under a
single scientific field. For that reason, also students from various study
tracks are eager to learn about quantum computing. However, their different
background and prerequisite knowledge poses an obvious challenge. A physics
student knows about quantum mechanics, but may not be fluent in programming. A
computer science student knows programming, but has not the needed theoretical
background of quantum physics. Further, a third case is e.g. a student in
business school, who understands how technology relates to business, and would
hence keen to know the basics of quantum computing. These student personas,
their educational background and expertise, as well as their expectations are
summarized in Table 1. When developing the new teaching material, the
different personas has served as imaginary student profiles and helped us to
take into account their diverse viewpoints.

Table 1: Student personas, their knowledge, expertise, and expected results
from using the educational system to learn quantum computing (QC).

Student persona |  Background |  Expectations
---|---|---
Physics |  Knows the basics of quantum mechanics |  QC is not only a paper exercise
Software engineer |  Knows programming and development processes |  Gets familiar with the theoretical background of the field
Business |  Knows how to use technology in commercial context |  Understands the challenges of QC
|  |

To lower the entry barrier for both the students and teaching staff, we
decided to extend the learning platform TIM for teaching also the quantum
computing course. As the TIM platform is used at the university for various
courses, especially programming courses, the staff and major part of the
students are already familiar with it. The TIM platform needed an extension to
be able to have interactive quantum circuit exercises and examples. The
interactive quantum circuit simulator is, in fact, the most important part of
the online course. It can also be used as a stand-alone tool when
demonstrating quantum algorithms or other quantum information protocols in
various courses or events.

## LEARNING ENVIRONMENT

The Interactive Material (TIM) [8] is a massive open online course (MOOC)
learning environment. The development of TIM began in 2014 when there was no
ready-made tool suitable for programming courses that would make it easy to
produce and maintain long interactive book-like learning materials. TIM has
been an open-source application under the MIT license from the
beginning111https://github.com/TIM-JYU/TIM. The development of TIM is the
responsibility of the staff at the Information Technology faculty. A large
part of TIM has been developed as theses in computer science education.

The basic idea of TIM is document-based. Everything done in TIM is essentially
documents. Documents consist of blocks. A block can be one or more normal text
paragraphs or an interactive element. An interactive element can be, for
example, a multiple-choice task, a drawing task, a free text field, a
programming code task, an image, a series of images, a video, a mathematically
handwritten task returned as an image, an assisted LaTeX-written answer, or a
new component. The appearance of TIM is adjusted with CSS styles, so users can
be offered ready-made appearance styles or users can create their own style
sheet. Teachers can choose the appearance from ready-made styles or create
their own style for their course.

Figure 1: Information model supporting instructor-student interactions

TIM can be used to create lecture materials according to the original plan,
but in the ten years TIM has developed so that it can handle all the necessary
course bookkeeping from individual tasks done in TIM to the final grade. The
instructor can see all the attempts the student has made on a specific task
and can use it to try to figure out why the student may initially go in the
wrong direction. The instructor can write feedback on the student’s answers.
This way, TIM provides a lot of data for learning analytics. Students can
comment on the material and the instructor can respond to the comments. In
fact, this is currently the most common way to communicate with students using
TIM. The information model that supports this interaction is presented in
Figure 1.

TIM documents can be used to borrow blocks from one document to another, and
thus with a single maintenance, smaller documents suitable for different needs
can be made from one document, or vice versa, smaller documents can be
combined into cohesive entities. Thanks to macros, parts that depend on years
or even tool version numbers can be combined into one place in TIM, thus
speeding up maintenance as things change. The TIM document can be translated
into several different languages using the automatic translation provided by
DeepL222https://www.deepl.com/translator. Naturally, the human must check the
result produced by the translator. If changes are made to the original
document, the translation administrator will be notified and new blocks or a
note of the changed section will be added to the translated document.

When using TIM, the teacher and student can do everything in one system
without the need to switch from one system to another. TIM is suitable and has
been used in all kinds of teaching methods from classroom teaching to MOOC
courses. TIM is at its best in teaching where there are performers in many
different ways. Conditional blocks allow different groups to be offered
slightly different instructions if necessary.

## ARTIFACT DESIGN

We implemented quantum circuit simulator in TIM platform, so that it can be
directly used for developing the quantum technology curriculum.

At the started the project, TIM already had all the essential features for
lecture-oriented course as well as for a MOOC. Our purpose was to design and
build a tool that is functional in both types of courses. The quantum circuit
simulator is implemented as a TIM plugin, and it can be customized to create
different types of exercises. The teacher has options to customize the
different components to be visible or not according to the context it is used
in.

The simulator is usable by different kind of demographics. Using the simulator
doesn’t require neither physics knowledge nor programming knowledge like
Qiskit [10]. On the other hand, the simulator allows the user to export the
circuit to the Qiskit equivalent, or the final quantum state and output
probabilities for further analysis if needed.

### The circuit simulator interface

Figure 2: Quantum circuit simulator interface tour: (1) available gates
toolbar, (2) input qubits and values, (3) maximum circuit steps, (4) circuit
layout grid, (5) qubit output measurements – value and probability, (6)
results probability distribution, and (7) individual execution results table –
inputs and output measurements

The inspiration for our quantum circuit simulator was drawn from the existing
graphical quantum circuit simulators like IBM Quantum Composer [1], Quantum
Länd Circuit Simulator [2], Quantum JavaScript [3], Quirck [4] or Uranium [5].
One of the design goals of our circuit simulator was adaptability so that it
could be modified to fit different types of exercises within the TIM platform.
For this reason, we choose to create a new simulator that incorporates the
relevant visual elements for designing circuits and visualise the results
present in existing simulators.

The circuit simulator contains an editor area that allows the user to
construct the circuit – toolbar with available gates, input qubits names and
values (e.g. $\ket{0}$ or $\ket{1}$), maximum number of circuit steps, circuit
layout grid, and qubit output measurements with probability distribution, and
an results area for visualizing the simulation results – probability
distribution, or individual results table. The annotated quantum circuit
simulator user interface is depicted in Figure 2.

The editor functionality of the simulator allows the user to construct the
circuit from the gates available in the toolbar, by dragging and dropping
gates in the circuit layout grid. The controlled gates can be created by
dragging control nodes to same step as target gate. The controls are connected
by vertical wires to the target gate, while the anti-controls – e.g., controls
that are activated with zero instead of one – are represented with unfilled
circles. The multi-qubit gates are represented as rectangles covering adjacent
lines. The default configuration of the simulator contains a set of common
gates, which can be extended with own custom gates.

The size of the circuit layout grid is determined by input qubit count for
rows and step count for columns, and is immutable. This way user knows exactly
how many qubits and steps the answer consists of, which can help to narrow
down their choices on how to implemented the desired circuit. Even on simple
exercises there are many possible ways to arrange the available gates in the
circuit so this predetermined circuit size might make the exercises more
manageable for the user.

The user can edit the circuit and see the results in real-time, allowing them
to interactively experiment with the behavior of gates to understand how
quantum computation works. The progression of the computation is represented
as discrete steps going from left to right as gates modify the quantum state.
The measurement is done on all qubits at the end in the $\ket{0}$, $\ket{1}$
basis.

The input state of circuit is represented using bits that user can toggle
between 0 and 1. being able to change input bit values makes it possible to
explore the behavior of the circuit especially when working with simple gates
like X and CX. The instructor can decide whether to use braket or bit notation
for inputs and give initial values and names for the qubits. The values of the
qubits can be locked to specific value if changing the value isn’t needed in
the exercise.

(a)

(b)

(c)

Figure 3: Example gate representations in the visual simulator’s toolbar: (a)
a common notation for a CX-gate, (b) a CX-gate using our unified notation, and
(c) a SWAP gate.

The graphical presentation of the circuit is similar to what is used in the
literature [7, 6]. For example, the Control-X gate is typically represented as
$\oplus$. Since our simulator allows to add a control dot to any quantum gate,
we chose to use more unified notation, see Figure 3. First, every gate is
represented by a rectangle with it’s name inside it. A control dot, or several
of them, can be dragged and connected to the gate. The only exception to this
rule is the swap gate which is represented by two X connected by wire.

The result part of the simulator interface provides several widgets that allow
the visualization of the quantum computation. The bar chart widget shows the
probability distribution of the output states. For larger circuits, rows where
output probability is zero can be hidden from the chart to make the results
easier to understand. The measurements widget allows the user to perform shot
measurements and to visualize the sampled output value in a table containing
the measurement number, and the corresponding input and output.

The simulator computes the full state vector allowing for statistical sampling
of the outputs. Samples can be drawn from the theoretical output probability
distribution computed from the final state vector. The output probabilities
shown in the chart can be these exact probabilities or the probabilities can
be based on automatic sampling of certain sample size or from the measurements
made by the user.

The visibility of the elements of circuit simulator interface is configurable,
therefore the user can first be shown a strip down version and then additional
features can be gradually introduced making the simulator more approachable
for new users.

### System architecture

Figure 4: Visual quantum simulator architecture overview (existing TIM
components in gray), and interaction between the components: (1) loading the
exercise, (2) initialize client-side simulator, (3) initialise server-side
simulator if circuit is large, (4) return the results, and (5) render the
results

The quantum circuit simulator implementation consists of a client component
and server component. The client component, written in JavaScript, using
Angular333https://angular.io framework, integrates the circuit visualisation
into the TIM exercise user interface – executed into the web browser main
thread, and the client-side simulator that integrates into the TIM application
engine – executed in a web worker. The server component implements the server-
side circuit simulator, using the Qulacs library [11], and the exercise
assessment. The interaction between the client-side and server-side simulators
is realised via a REST interface. The decomposition of the system is depicted
in Figure 4.

When opening a new task in the exercise, the user interface renders the
circuit editor and initialises the client-side simulator. If the circuit is
too large to be handled locally, the server-side simulator is initialized.
After initialization, the circuit is executed on the simulator and the results
are collected, send to the browser if necessary, then rendered in the user
interface. A new execution process is performed whenever the user modifies the
circuit or changes the input values, allowing the user interface to be always
up to date. To maintain the responsiveness of the user interface, the client-
side simulator in implemented in a web worker. Therefore, even under intense
load the browser main thread is not blocked.

When there are a lot of simulation exercises on one web page the load time
increases making it quite slow especially on slower devices like tablets. This
was mitigates by lazy loading the simulator components. The simulator is fully
rendered and the simulation computation is done when user clicks on the
exercise.

The exercises can be automatically checked for correctness by specifying model
circuit to compare the user’s circuit to. The circuit equality is checked by
iterating over all $2^{nQubits}$ input combinations and checking that both
circuit result in same probability vector. The number of checked inputs can be
reduced by giving regex patterns to match each input bit-strings against and
only running the simulation against these matching bit-string inputs.

### MOOC integration

To support the teaching activities, the visual quantum circuit simulator is
seamlessly integrated into TIM’s MOOC environment. Each task instance is
visualized by an instance of the circuit simulator. When creating a task, the
instructor can control the appearance and behaviour of the simulator using a
YAML-based configuration, depicted in Listing 5. The configuration options can
be grouped into the following categories: the task description (lines 1–2),
the appearance of the simulator’s user interface elements (lines 4–10), the
circuit properties – number of qubits, circuit steps, qubits initial state and
editability, initial and target circuit state, and special gate conditions
(lines 12–35), and the feedback provided to the student (lines 37–40). The
text-based format of the configuration enables the instructor to reuse and
adapt circuits for different learning tasks, lowering the effort to create or
maintain the course material.

The student attempt to solve a task is saved on the server. This allows the
instructor to asses the progress for every student. The instructor can engage
in a conversation with the student using the TIM’s chat function, both sharing
the same attempt to solve a task context. As the visual simulator is
implemented as a TIM plugin, it leverages the interactive capabilities of the
MOOC platform to provide the teaching experience that is not possible with
individual study tools like IBM Quantum Composer.

⬇

header: "Tehtävä"

stem: "Käytä kahta CX-porttia ja etsi niille sellainen muodostelma, että
ensimmäisen bitin arvo q[0] kopioituu kahdelle seuraavalle bitille kun nämä
ovat alkutilassa 0. Kun olet saanut piirin valmiiksi, paina ’Tallenna’ ja
ratkaisusi tarkistetaan."

\parqubitNotation: "bit"

showChart: false

showOutputBits: true

middleAxisLabel: "Askel"

leftAxisLabel: "In"

rightAxisLabel: "Out"

\parnQubits: 2

nMoments: 4

gates: ["X","control"]

samplingMode: matrix

qubits:

- value: 0

editable: true

- value: 0

editable: false

modelCircuit:

- controls:

- 0

editable: true

name: X

target: 1

time: 0

- controls:

- 1

editable: true

name: X

target: 2

time: 1

modelConditions: ["C1X == 2"]

- pointsRule: # tämän alle säännöt miten pisteitä saa

multiplier: 5.0 # Millä luvulla kerrotaan tehtävästä saadut pisteet

\parfeedbackText:

correct: "Oikein"

conditionWrong: "Väärin. Vastauksessa pitää olla kaksi CX-porttia."

feedbackShowTable: false

Figure 5: YAML representation of a circuit in TIM, with localization
properties in Finnish

## DEMONSTRATION

The quantum circuit simulator is a flexible tool to construct various
exercises. In the following we demonstrate three different kind of exercises
that are in used in our online courses.

### Understanding the probabilistic nature of quantum gates

Quantum gates, such as Hadamard gate and square-root-X, do not lead to
determined measurement outcomes. They are the simplest examples of non-
classical building blocks of quantum circuits and for that reason natural
first elements in a quantum computing course. As one of the first
demonstrations of the difference of standard gate based computation and
quantum gate computation, a student is given an exercises where she needs to
find what gates lead to deterministic outcomes. In that kind of exercise, the
simulator is given with the settings as in Figure 6. In that setting the
circuit gives individual measurement outcomes and the non-deterministic nature
of quantum gates is transparent. Pressing ’measure’ several times and getting
different outcomes is very concrete and better for this task than a static
visualization of a probability distribution. Furthermore, both Hadamard and
square-root-X gates offer intriguing demonstration of quantum randomness as
individually used they lead to random outcomes, but two similar gates to
deterministic outcomes.

Figure 6: Task 1 - Understanding probabilistic gates

### Working with quantum circuits

The action of a single quantum gate is simple. To achieve certain task, one
may have to use a circuit composed of several gates. Their collective
performance may be hard to infer. The circuit simulator offers a fantastic
workaround as a student and experiment with gate constellations. The simulator
gives an instant visualization of the output in the form of measurement
outcome distribution.

A real quantum computer has certain native gates and all the other gates must
be build up from those. For that reason, it is important to learn to recognize
circuits with equivalent performance. The simulator enables to give
complicated gate decompositions (see Figure 7), and the task for a student is
to find equivalent circuit with less gates. In the depicted case one can
replace the whole circuit by just one controlled X-gate. The simulator has an
automatic system to check answers, and in this way a student gets an immediate
feedback if her trial is the correct one. A typical exercise does not have a
unique answer, but the automatic evaluator is capable of recognizing all
correct answers from their input-output behaviour. It is, in fact, common that
students find novel solutions that a teacher was not able to foresee.

Figure 7: Task 2 - Working with circuits

### Identifying an unknown quantum gate

Once a student has reached some familiarity of quantum gates has been
achieved, she can proceed to more interesting exercises. With the simulator
the teacher can create arbitrary gates and label them in any desired way. This
makes it possible to form exercises where a student has e.g. identify an
unknown gate, or their combination, by using it in concatenation with other
gates (see Figure 8). This kind of exercises are almost like entertaining
puzzles. As the gate can be any of the previously introduced gates, a student
necessarily has to review previously learned material. Some quantum gates,
like the Z-gate, do nothing in the computational basis unless they are coupled
with some other, appropriately chosen, quantum gates. This quantum feature
makes additional excitement to suitably planned exercises.

Figure 8: Task 3 - Identifying unknown quantum gate

## DISCUSSION

Figure 9: Spectrum of tools that can be used by quantum information and
computing practitioners

The spectrum of experiences that can be used by quantum technology
practitioners (excluding hardware), can be divided into the following areas,
as depicted in Figure 9: paper, paper-like, notebook, and integrated
development environment (IDE). The paper is used mainly by practitioners with
physics and mathematical background and does not necessitate computer systems
to experimenting besides paper or a whiteboard. The paper-like provides a low
threshold for using a computer system that offers an experience close to the
paper, but with instant feedback aided by tools like TIM or IBM Quantum
Composer. The notebook experience is familiar to scientist that are using
Python programming language and quantum development toolkits (e.g., Qiskit,
Cirq444https://quantumai.google/cirq or Pennylane555https://pennylane.ai), and
use tools like Jupyter and IBM Quantum Lab to develop algorithms and perform
experiments. The IDE experience is tailored for software developers that use
advanced tools like Visual Studio Code666https://code.visualstudio.com or
PyCharm777https://www.jetbrains.com/pycharm/. Among our target student groups,
the paper-like experience provided by the TIM’s quantum circuit simulator
offers the optimal experience, as it does not require familiarity neither with
programming languages, nor mathematical concepts, instead leading students to
experiment with circuits by drag-and-dropping gates and instantly see the
results.

Since February 2024, we have been running the first MOOC for early adopters.
These are mainly students from the IT faculty that are keen in learning
quantum computing. Hence, from the student personas described earlier, we have
not covered all, and it is too early to draw definite conclusions. We can,
however, make some preliminary observations.

Given the diverse backgrounds of the students for whom the MOOC is designed,
we anticipated that the difficulty of tasks would vary among students.
However, the additional hints provided after most tasks have effectively
served their purpose, preventing any task from becoming a significant
bottleneck. We estimate the difficulty of tasks by the average number of
attempts students make before arriving at the correct answer. While this
average exceeds five for some tasks, it has not led to a decrease in
motivation. In fact, students appear to appreciate a certain level of
challenge. Since the number of attempts does not impact the course grade,
students are motivated by the opportunity to tackle challenges without the
pressure of exam failure or low grades. In this regard, the MOOC format is
functioning as intended, fostering a supportive environment for learning.

We recall that the simulator’s primary advantage over similar products is its
seamless integration with the existing TIM learning platform. It’s not
confined to a single course but can be utilized across various courses,
allowing for the easy generation of exercises with diverse difficulty levels.
The integration with TIM provides access to all necessary tools for a
university course, including discussion forums, straightforward registration,
and grading. The value of these features has been demonstrated, and the
simulator has fulfilled its intended purpose. The predominant question from
students has been, “When will the next course be available?” This underscores
the clear demand for flexible, self-paced courses at the university level.

Very recently, the MOOC was started in the open university, where it is
offered for free. This will be a real-world test of our primary objective:
making quantum computing accessible to everyone.

## REFERENCES

  * [1] Ibm quantum composer, 2024.  Last accessed 22 February 2024. https://quantum.ibm.com/composer/files/new.
  * [2] Quantum circuit simulator - the quantum länd, 2024.  Last accessed 22 February 2024. https://thequantumlaend.de/quantum-circuit-designer/.
  * [3] Quantum javascript (q.js), 2024.  Last accessed 22 February 2024. https://github.com/stewdio/q.js.
  * [4] Quirk: Quantum circuit simulator, 2024.  Last accessed 22 February 2024. https://algassert.com/quirk.
  * [5] Uranium circuit-editor, 2024.  Last accessed 22 February 2024. https://uranium.transilvania-quantum.org/circuit-editor/.
  * [6] Kean Chen, Wang Fang, Ji Guan, Xin Hong, Mingyu Huang, Junyi Liu, Qisheng Wang, and Mingsheng Ying.  Veriqbench: A benchmark for multiple types of quantum circuits.  arXiv preprint arXiv:2206.10880, 2022.
  * [7] Zhao-Yun Chen, Qi Zhou, Cheng Xue, Xia Yang, Guang-Can Guo, and Guo-Ping Guo.  64-qubit quantum circuit simulation.  Science Bulletin, 63(15):964–971, 2018.
  * [8] Ville Isomöttönen, Antti-Jussi Lakanen, and Vesa Lappalainen.  Less is more! preliminary evaluation of multi-functional document-based online learning environment.  In 2019 IEEE Frontiers in Education Conference (FIE), pages 1–5, 2019.
  * [9] Marcus A. Rothenberger Ken Peffers, Tuure Tuunanen and Samir Chatterjee.  A design science research methodology for information systems research.  Journal of Management Information Systems, 24(3):45–77, 2007.
  * [10] Qiskit contributors.  Qiskit: An open-source framework for quantum computing, 2023.
  * [11] Yasunari Suzuki, Yoshiaki Kawase, Yuya Masumura, Yuria Hiraga, Masahiro Nakadai, Jiabao Chen, Ken M Nakanishi, Kosuke Mitarai, Ryosuke Imai, Shiro Tamiya, et al.  Qulacs: a fast and versatile quantum circuit simulator for research purpose.  Quantum, 5:559, 2021.

Juha Reinikainen is a master’s degree student in mathematical information
technology in the University of Jyväskylä. His research interests include
quantum computing, simulation and software development. Contact him at
<EMAIL_ADDRESS>
Vlad Stirbu is a postdoctoral researcher at the University of Jyväskylä,
Finland. His research interests include: software engineering, software
architecture, quantum software, and software development in regulated
industries. He is a member of IEEE Computer Society. Contact him at
<EMAIL_ADDRESS>
Teiko Heinosaari is a professor in quantum computing at the University of
Jyväskylä. His research encompasses various topics in quantum information
theory and hybrid classical-quantum computing. He is developing novel ways to
teach quantum computing to students who have no prior knowledge of quantum
physics. Contact him at<EMAIL_ADDRESS>
Vesa Lappalainen is a senior lecturer at the University of Jyväskylä with over
40 years of teaching experience. He is the creator of the TIM-platform. His
current research interest is e-learning. Contact him at
<EMAIL_ADDRESS>
Tommi Mikkonen is a professor of software engineering at the University of
Jyväskylä, in Finland. Contact him at<EMAIL_ADDRESS>


# Variations for some Painlevé equations

P.B. Acosta–Humánez111 Instituto Superior de Formación Docente Salomé Ureña -
ISFODOSU, Recinto Emilio Prud’Homme, Santiago de los Caballeros, Dominican
Republic., M. van der Put222University of Groningen, Groningen, the
Netherlands, J. Top†

###### Abstract

This paper first discusses irreducibility of a Painleve equation $P$. We show
that reducibility is equivalent to the existence of special classical
solutions. As in a paper of Morales–Ruiz we associate an autonomous
Hamiltonian $\mathbb{H}$ to a Painlevé equation $P$. Complete integrability of
$\mathbb{H}$ is shown to imply that all solutions to $P$ are special classical
or algebraic, so in particular $P$ is reducible.

Next, we show that the variational equation of $P$ at a given algebraic
solution coincides with the normal variational equation of $\mathbb{H}$ at the
corresponding solution.

Finally, we test the Morales–Ramis theorem in all cases $P_{2}$ to $P_{5}$
where algebraic solutions are present, by showing how our results lead to a
quick computation of the component of the identity of the differential Galois
group for the first two variational equations. As expected there are no cases
where this group is commutative. 333MSC2010: 33E17, 34M55 keywords:
Hamiltonian systems, variational equations, Painlevé equations, differential
Galois groups

## Introduction and Summary

The interesting idea of J.-A. Weil to apply the Morales–Ramis theorem to
Painlevé equations was initiated in [Mo2]. It is also the subject of more
recent papers [A, HS, St2, St3, St4, SC]. The Hamiltonian $H$ of a Painlevé
equation $x^{\prime\prime}=R(x^{\prime},x,t)$ depends on ‘the time’ $t$. In
order to apply the Morales–Ramis theorem, $H$ is changed into a time-
independent Hamiltonian $\mathbb{H}=H+e$. We show (Proposition 2.1) that
complete integrability for $\mathbb{H}$ implies that all solutions of the
equation $x^{\prime\prime}=R(x^{\prime},x,t)$ are special classical solutions
in the sense of H. Umemura [U1, U2, U3, UW] or are algebraic functions. In
fact, one may state that such an equation $x^{\prime\prime}=R(x^{\prime},x,t)$
is not considered as a true Painlevé equation. This is in agreement with [ZF],
see also §2 below.

The normal variational equation(s) of $\mathbb{H}$ along a given explicit
solution are shown (Proposition 3.1) to be equivalent to the variational
equation(s) for $x^{\prime\prime}=R(x^{\prime},x,t)$ along a given solution.
For the equations $P_{2}$ to $P_{5}$, there is a convenient list in [OO2] of
all cases with algebraic solutions, up to Bäcklund transformations, for which
variational equations can be examined. One expects, in accordance with [MR,
MRS], that the variational equations produce differential Galois groups $G$
such that $G^{o}$, the component of the identity, is not abelian. The results
are (see §4):
(1). The first variational equation produces, for almost all cases, $G={\rm
SL}_{2}$.
(2). In some cases the first variational equation produces the differential
Galois group with component of the identity $\mathbb{G}_{m}$. The second
variational equation produces an extension of this group by a unipotent group
$\mathbb{G}_{a}^{m}$. The action by conjugation of $\mathbb{G}_{m}$ on
$\mathbb{G}_{a}^{m}$ is not trivial. Hence $G^{o}$ is not abelian.
(3). The most interesting case is §4.7 which discusses $P_{5}(a,-a,0,\delta)$
and $y=−1$ with first variational equation
$v^{\prime\prime}=−t^{-1}v′+(8at^{-2}+\frac{1}{2}\delta)v$. It follows from
the monodromy theorem [vdP-Si, Prop. 8.12 (2)] that its differential Galois
group is ${\rm SL}_{2}$ unless $a=\frac{(2n+1)^{2}}{32}$ with
$n\in\mathbb{Z}$. For these special values of $a$ the differential Galois
group is $\mathbb{G}_{m}$. Again, for these special cases, the second
variational produces a $G$ with non abelian component of the identity.

These special values for $a$ can be explained as follows. There is a standard
isomonodromy family corresponding to $P_{5}$, see [JM]. Let
$\pm\frac{\theta_{0}}{2}$ and $\pm\frac{\theta_{1}}{2}$ denote the local
exponents of this family for the regular singularities $0$ and $1$. Then
$\theta_{0}-\theta_{1}=\sqrt{8a}$. Thus the special values for $a$ correspond
to a type of resonance between the regular singularities at 0 and at 1.

In [St2] a $P_{5}$ equation with different parameters (but equivalent by
Bäcklund transformations) is studied and the same special values are found.
(4). (This is in part inspired by a discussion of one of us with Juan J.
Morales-Ruiz; we thank him for his question.) Each of the Painlevé equations
is induced by isomonodromy of some family of order $2$ linear differential
equations [vdP-Sa]. The possible singular points of such a family are
$0,1,\infty$ with prescribed singularity. The first variational equation
happens to have the same type of possible singularities; this observation is
made in various examples discussed in Section 4. Our methods considerably
simplify (e.g., compared to earlier similar results by Stoyanova et al. [St2],
[St3], [St4], [SC]) the determination of the Galois group associated to such a
variational equation.

R. Fuchs’ problem, see [OO2, St1], also concerns algebraic solutions of
Painlevé equations. The second order linear differential equations resulting
from this problem seem to be unrelated to the first variational equations.

In Section 1 we observe that if a second order equation $R$ has the Painlevé
property and moreover is reducible, then the induced first order differential
equation $Q$ has the Painlevé property, too. The classification of first order
equations with the Painlevé property has consequences for the special
solutions of $R$.

##  1 Reducibility and special solutions

Consider a Painlevé equation $x^{\prime\prime}=R(x^{\prime},x,t)$ with fixed
parameters. Let $den$ be the denominator of $R(x^{\prime},x,t)$ seen as
element of the field of fractions of $\mathbb{C}(t)[x^{\prime},x]$. Then
$\mathcal{D}:=\mathbb{C}(t)[x^{\prime},x,\frac{1}{den}]$ is a differential
algebra with respect to the differentiation given by $t^{\prime}=1$,
$x^{\prime}=x^{\prime}$, $x^{\prime\prime}=R(x^{\prime},x,t)$.

In the literature (see for example [U1, U2, U3, OKSO]) a Painlevé equation is
called reducible if there is a principal ideal $(F)\neq 0$ which is a
differential ideal (i.e., invariant under differentiation). In fact, one
considers this property not only for $\mathcal{D}$ but in the differential
algebra $K[x^{\prime},x,\frac{1}{den}]$ where $K\supset\mathbb{C}(t)$ is any
extension of differential fields. In the paper [U1] the last property is
called “reducible over the field $K$”.

According to [U1], the irreducibility for a Painlevé equation as above (over
$\mathbb{C}(t)$) follows from:
(1). there are no algebraic solutions, and
(2). $\mathcal{D}$ has no principal differential ideal $\neq(0),(1)$.
We now discuss how to verify these two conditions.

Concerning (1): By the Painlevé Property, an algebraic solution can only be
ramified above $t=\infty$. Hence this solution must be rational. It is easily
seen that no solution of $P_{1}$ in $\mathbb{C}(t)$ exists.

Concerning (2): The algebra $\mathcal{D}$ has unique factorization and one
easily verifies that every prime factor $Q$ of $F$ generates again a
differential ideal. Thus the equation is reducible if and only if
$\mathcal{D}$ has a prime ideal $(Q)$ of height one which is invariant under
differentiation.

Now $Q(x^{\prime},x,t)=0$ is a first order differential equation. It is well
known that $x^{\prime\prime}=R(x^{\prime},x,t)$ has the Painlevé Property. The
solutions of $Q(x^{\prime},x,t)=0$ are also solutions of this Painlevé
equation. Therefore $Q(x^{\prime},x,t)=0$ itself has the Painlevé Property. A
classical result (see [M, Mun-vdP, NNPT] for modern proofs and references to
some of the rich classical literature) implies that $Q(x^{\prime},x,t)$ has
one of the following properties:
(i) Genus 0; it is a Riccati equation; thus $x^{\prime}=a+bx+cx^{2}$ with
$a,b,c\in\mathbb{C}(t)$.
(ii) Genus 1; it is a Weierstrass equation; thus it is equivalent to
$(x^{\prime})^{2}=f\cdot(x^{3}+ax+b)$, where $f\neq 0$ is algebraic over
$\mathbb{C}(t)$, and $a,b\in\mathbb{C}$ are such that the equation
$y^{2}=x^{3}+ax+b$ represents an elliptic curve.
(iii) Genus $>1$; after a finite extension of $\mathbb{C}(t)$, the equation is
equivalent to the equation $x^{\prime}=0$. This is equivalent to the
statement: all solutions of $Q(x^{\prime},x,t)=0$ are contained in a fixed
finite extension of $\mathbb{C}(t)$.

We will call the above three cases “classical special solutions” of the
Painlevé equation. We conclude that the Painlevé equation is irreducible if it
has no classical special solutions. This observation seems to provide a
simplification of the proofs for irreducibility given in [U1, U2, OKSO] et
al..

We observe that most of the cases of special solutions have genus 0. The
corresponding second order linear differential equation has at most
singularities where the Painlevé equation has fixed singularities (for $P_{6}$
the points $0,1,\infty$; for $P_{5},degP_{5},P_{3}$ the points $0,\infty$; for
$P_{4},P_{2,FN},P_{2},P_{1}$ the point $\infty$).
Genus 1 occurs for the cases (a) $\alpha=\gamma=0$ in $P_{3}$, (b)
$\beta=\delta=0$ in $P_{3}$ and (c) $\gamma=\delta=0$ in $P_{5}$ (see §2).
No example of a ‘first integral’ of genus $>1$ is known to us.

##  2 Complete integrability for the Hamiltonian $\mathbb{H}$

$x^{\prime\prime}=R(x^{\prime},x,t)$ is again some Painlevé equation with
fixed parameters. There is a Hamiltonian function $H(y,x,t)$ related to the
given Painlevé equation. There are various possibilities for $H$ but we assume
that it is a rational in the variables $y,x,t$ and moreover is polynomial of
degree 2 in the variable $y$. The usual equations are:

$x^{\prime}(t)=\frac{\partial H}{\partial y}(y(t),x(t),t)\mbox{ and
}y^{\prime}(t)=-\frac{\partial H}{\partial x}(y(t),x(t),t).$

Since $H$ is a polynomial of degree two in $y$, the first equation can be used
to write $y(t)$ as a rational expression in $x^{\prime}(t)$, $x(t)$ and $t$.
Substitution of this expression for $y(t)$ in the second equation will produce
an explicit second order equation for $x(t)$ and this is the given one
$x^{\prime\prime}=R(x^{\prime},x,t)$.

Now $H$ depends on the time $t$. One wants to apply the Morales–Ramis theorem
concerning complete integrability. This leads to a choice of a new Hamiltonian
$\mathbb{H}(y,x,z,e)=H(y,x,z)+e$ which depends on two pairs of variables $y,x$
and $z,e$. The new equations are:

$x^{\prime}(t)=\frac{\partial\mathbb{H}}{\partial y},\
y^{\prime}(t)=-\frac{\partial\mathbb{H}}{\partial x},\
z^{\prime}(t)=\frac{\partial\mathbb{H}}{\partial e}=1,\
e^{\prime}(t)=-\frac{\partial\mathbb{H}}{\partial z}$

###### Proposition 2.1

Suppose that $\mathbb{H}$ is completely integrable. Then all solutions of
$x^{\prime\prime}=R(x^{\prime},x,t)$ are special classical or algebraic. In
particular the equation is reducible.

  * Proof.

$\mathbb{H}$ is a first integral. There is an independent first integral
$E(y,x,z,e)$. We suppose that $E$ is a rational (or algebraic) function of the
4 variables. Now we replace the $e$ in $E$ by $-H(y,x,z)$. The result is a
first integral $F(y,x,z)$ for $H$ and a rational (or algebraic) function
$G=G(x^{\prime},x,t)$ such that $G(x^{\prime}(t),x(t),t)$ is independent of
$t$ for every solution $x(t)$ of the Painlevé equation
$x^{\prime\prime}=R(x^{\prime},x,t)$. Then, taking the derivative with respect
to $t$, one finds that the expression $R(x^{\prime},x,t)\frac{\partial
G}{\partial x^{\prime}}+x^{\prime}\frac{\partial G}{\partial x}+\frac{\partial
G}{\partial t}$ is zero on every solution $(x^{\prime}(t),x(t),t)$ of the
Painlevé equation. It follows that this expression itself is zero.

Consider, as before, the differential algebra
$\mathcal{D}:=\mathbb{C}(t)[x^{\prime},x,\frac{1}{den}]$ with derivation
$F\mapsto F^{\prime}$ given by $t^{\prime}=1,\ (x)^{\prime}=x^{\prime},\
(x^{\prime})^{\prime}=R(x^{\prime},x,t)$. Assume (for convenience) that $G$ is
rational. Thus $G$ lies in the field of fractions $Qt(\mathcal{D})$ of
$\mathcal{D}$ and $G^{\prime}=0$. Let $L\subset Qt(\mathcal{D})$ denote the
field of constants. Then $L(x^{\prime},x,t)$ equals $Qt(\mathcal{D})$ and the
transcendance degree of $L(x^{\prime},x,t)\supset L$ is 2, because
$G\not\in\mathbb{C}$. Therefore there is an irreducible polynomial $Q\in
L(t)[S,T]$ such that $Q(x^{\prime},x)=0$. The coefficients of $Q$ lie in
$\mathcal{D}[\frac{1}{S}]$ for a suitable element $S$.

The solutions of the Painlevé equation correspond to $\mathbb{C}(t)$-linear
differential homomorphism $\phi:\mathcal{D}\rightarrow Mer$, where $Mer$
denotes the differential field of the multivalued meromorphic functions on,
say, $\mathbb{C}\setminus\\{0,1\\}$. Indeed, the homomorphism $\phi$
corresponds to the solution $\phi(x)\in Mer$.

If $\phi(S)\neq 0$, then $\phi(Q)[S,T]$ makes sense. Since the coefficients of
$Q$ are rational functions in $t$ with ‘constant’ coefficients, one has
$\phi(Q)[S,T]\in\mathbb{C}(t)[S,T]$. Moreover
$\phi(Q)(\phi(x)^{\prime},\phi(x))=0$, which means that the solution
$\phi(x)\in Mer$ satisfies a first order differential equation, which has
again the Painlevé Property. We conclude that $\phi(x)$ is a special classical
solution or an algebraic solution.

Consider finally a $\phi$ such that $\phi(S)=0$. Then $\phi$ is also zero on a
prime differential ideal of $\mathcal{D}$ containing $S$. If this is a
principal ideal, then $\phi(x)$ is a special classical solution. If this is a
maximal ideal, then the solution $\phi(x)$ is algebraic. $\Box$

We note that Proposition 2.1 is in agreement with a main result of [ZF]:
The Hamilton system $\mathbb{H}$, associated with any of the equations
$P_{1}$–$P_{6}$, does not admit any first integral which is an algebraic
function of $x,y,z,e$ and independent of $\mathbb{H}$, except in the following
cases:
(a) $\alpha=\gamma=0$ in $P_{3}$, (b) $\beta=\delta=0$ in $P_{3}$ and (c)
$\gamma=\delta=0$ in $P_{5}$.

It is well known that in the cases (a)–(c) all solutions are obtained by
“quadratures”. The ‘first integrals’ (this means here an $F$ such that
$(F)\subset\mathcal{D}$ is a (prime) differential ideal) are actually known.
Namely for $P_{3}$ with $\beta=\delta=0$ they are
$t^{2}(x^{\prime})^{2}+2txx^{\prime}-(C+2\alpha tx+\gamma t^{2}x^{2})x^{2}$
(with arbitrary $C$ and a similar formula for the case $\alpha=\gamma=0$); see
slide 47 of the 2002 lecture [C2] by Clarkson and also the Russian paper
[L67]. For $P_{5}$ with $\gamma=\delta=0$ the ‘first integrals’ are
$t^{2}(x^{\prime})^{2}-(x-1)^{2}(2\alpha x^{2}+Cx-2\beta)$. This is, e.g.,
stated on slide 48 of [C2], see also [L68].

One observes that the above ‘first integrals’ are order one differential
equations having the Painlevé Property. They have genus 1 and are equivalent
to the “Weierstrass differential equation”.

##  3 Several variational equations

Suppose that an algebraic solution $x_{0}=x_{0}(t)$ of
$x^{\prime\prime}=R(x^{\prime},x,t)$ is given. The variational equation VEP
for the Painlevé equation is given by the following formalism. Put
$x=x_{0}(t)+\epsilon v$ with $\epsilon^{2}=0$. Substitution yields the
equation
$x^{\prime\prime}_{0}+\epsilon v^{\prime\prime}=R(x_{0}^{\prime}+\epsilon
v^{\prime},x_{0}+\epsilon v,t)$. The coefficient of $\epsilon$ in this
equation is a second order linear equation for $v$. Explicitly, VEP is the
equation:

$v^{\prime\prime}=\frac{\partial R}{\partial
x^{\prime}}(x^{\prime}_{0},x_{0},t)v^{\prime}+\frac{\partial R}{\partial
x}(x_{0}^{\prime},x_{0},t)v.$

The algebraic solution $x_{0}$ produces an algebraic solution
$y_{0}=y_{0}(t),x_{0}=x_{0}(t)$ for the Hamiltonian equations for $H$. The
variational equation VEH for this Hamiltonian equation is defined by the
following formalism. Put $y=y_{0}+\epsilon w,\ x=x_{0}+\epsilon v$ with
$\epsilon^{2}=0$ in the two Hamilton equations. Thus

$x_{0}^{\prime}+\epsilon v^{\prime}=\frac{\partial H}{\partial
y}(y_{0}+\epsilon w,x_{0}+\epsilon v,t)\mbox{ and }y_{0}^{\prime}+\epsilon
w^{\prime}=-\frac{\partial H}{\partial x}(y_{0}+\epsilon w,x_{0}+\epsilon
v,t).$

The coefficients of $\epsilon$ in these equations yield linear differential
equations for $w$ and $v$ of order one. Moreover, the first equation can be
used to eliminate $w$ as linear expression in $v$ and $v^{\prime}$. Thus we
obtain a second order homogenous differential equation for $v$ which coincides
of course with the earlier VEP.

The algebraic solution $x_{0}=x_{0}(t)$ for the Painlevé equation yields for
$\mathbb{H}$ the solution $y_{0}=y_{0}(t),\ x_{0}=x_{0}(t),\ z_{0}(t)=t,\
e_{0}=e_{0}(t)=-\int\frac{\partial\mathbb{H}}{\partial z}dt$.
The function $e_{0}$ is maybe not always an algebraic function. This is not
important for the sequel.

The variational equation VE$\mathbb{H}$ for $\mathbb{H}$ along this solution
is obtained by the following formalism. Put $y=y_{0}+\epsilon w,\
x=x_{0}+\epsilon v,\ z_{0}=t+\epsilon a,\ e=e_{0}+\epsilon b$ with
$\epsilon^{2}=0$. Substitution these data into the Hamiltonian equations for
$\mathbb{H}$ yields a systems of rank 4 of linear differential equations of
first order. More in detail:

$x_{0}^{\prime}+\epsilon v^{\prime}=\frac{\partial\mathbb{H}}{\partial
y}(y_{0}+\epsilon w,\dots,e_{0}+\epsilon b)\mbox{ and }y_{0}^{\prime}+\epsilon
w^{\prime}=-\frac{\partial\mathbb{H}}{\partial x}(y_{0}+\epsilon
w,\dots,e_{0}+\epsilon b)$ $t^{\prime}+\epsilon
a^{\prime}=\frac{\partial\mathbb{H}}{\partial e}=1\mbox{ and
}e_{0}^{\prime}+\epsilon b^{\prime}=-\frac{\partial\mathbb{H}}{\partial
z}(y_{0}+\epsilon w,\dots,e_{0}+\epsilon b).$

The normal variational equation NVE$\mathbb{H}$ for $\mathbb{H}$ along this
solution is obtained by taking the three dimensional space perpendicular to
the equation for $a$ and dividing out by the tangent line of the curve (i.e.,
the given solution). This means that we are reduced to the case $a=b=0$.

###### Proposition 3.1

The normal variational equation NVE$\mathbb{H}$ of $\mathbb{H}$ coincides with
the variational VEP of the Painlevé equation
$x^{\prime\prime}=R(x^{\prime},x,t)$.

We note that the variational equation VE of a Painlevé equation has at most
the same singularities as this Painlevé equation, i.e., for $P_{6}$ the points
$0,1,\infty$; for $P_{5},degP_{5},P_{3}$ the points $0,\infty$; for
$P_{4},P_{2,FN},P_{2},P_{1}$ the point $\infty$.

##  4 VE for the algebraic solutions of $P_{2},\ldots,P_{5}$

We adopt here the list of special solutions of [OO2, Theorem 2.1]. Further we
will, as in that paper, replace the classical $P_{3}$ by $P^{\prime}_{3}$.
Finally for the degenerate fifth Painlevé equation we will use the $degP_{5}$
of our paper [APT].

###  4.1 $P_{2}(\alpha=0)$ with solution $y=0$

$P_{2}$ reads $y^{\prime\prime}=2y^{3}+ty+\alpha$. The VE for $\alpha=0$ and
$y=0$ reads $v^{\prime\prime}=tv$. This is the Airy equation with differential
Galois group ${\rm SL}_{2}$. This is also present in [A, Mo2, SC].

We skip the Flaschka-Newell $P_{34}$ because of its equivalence to $P_{2}$.

###  4.2 $P_{4}(0,-\frac{2}{9})$ with $y=-\frac{2}{3}t$

VE reads $v^{\prime\prime}=t^{-1}v^{\prime}-\frac{4}{3}t^{2}v$. A basis of
solutions is $\\{e^{\sqrt{-1/3}\ t^{2}},e^{-\sqrt{-1/3}\ t^{2}}\\}$ and the
differential Galois group is $\mathbb{G}_{m}$. The second variational
equations (obtained by putting as solution $-\frac{2}{3}t+\epsilon
v+\epsilon^{2}w$) read

$v^{\prime\prime}-t^{-1}v^{\prime}+\frac{4}{3}t^{2}v=0,\
w^{\prime\prime}-t^{-1}w^{\prime}+\frac{4}{3}t^{2}w=\frac{3}{2}t^{-1}vv^{\prime\prime}-\frac{3}{4}t^{-1}(v^{\prime})^{2}+3tv^{2}.$

For every solution $v_{0}\neq 0$ of the first equation, the second
inhomogeneous equation produces an extension of $\mathbb{G}_{m}$ by an
additive group $\mathbb{G}_{a}$. For instance, the choice $v_{0}=e^{ct^{2}}$
with $c^{2}=-\frac{1}{3}$ leads to the equation
$w^{\prime\prime}-t^{-1}w^{\prime}+\frac{4}{3}t^{2}w=(3ct^{-1}+2t)e^{2ct^{2}}$.
Any special solution $w_{0}$ involves the “error function” ${\rm erf}(t)$. An
element of $s\in\mathbb{G}_{m}\cong\mathbb{C}^{*}$ maps $v_{0}$ to $sv_{0}$
and maps $w_{0}$ to $s^{2}w_{0}+$ a solution of the homogeneous equation.
Hence the component of the identity of the differential Galois group of the
second variational equation is not commutative. Compare also to the next case,
where more details are given.

###  4.3 $P_{4}(0,-2)$ with $y=-2t$

This case is similar to the one discussed in §4.2. VE is
$v^{\prime\prime}-t^{-1}v^{\prime}-4t^{2}v=0$. A basis of solutions is
$\\{e^{t^{2}},e^{-t^{2}}\\}$ and the differential Galois group is
$\mathbb{G}_{m}$. The second VE is

$v^{\prime\prime}-t^{-1}v^{\prime}-4t^{2}v=0,\
w^{\prime\prime}-t^{-1}w^{\prime}-4t^{2}w=\frac{1}{2}t^{-1}vv^{\prime\prime}-\frac{1}{4}t^{-1}(v^{\prime})^{2}-7tv^{2}.$

The Picard–Vessiot field $K$ for this set of equations is an extension of the
Picard–Vessiot field $K_{0}=\mathbb{C}(t,e^{t^{2}})\supset\mathbb{C}(t)$ of
the first equation. The extension $K\supset K_{0}$ is obtained by adding a
particular solution $w_{0}$ of the second inhomogeneous equation for every
solution $v_{0}=ae^{t^{2}}+be^{-t^{2}}\neq 0$ of the first equation. The
differential Galois group of $K/K_{0}$ is an unipotent group and can be seen
to be $\mathbb{G}_{a}^{3}$. The action of $\mathbb{G}_{m}$ on the solutions
$v_{0}$ induces a non trivial action (by conjugation) of $\mathbb{G}_{m}$ on
$\mathbb{G}_{a}^{3}$. Thus the component of the identity of the differential
Galois group of $K/\mathbb{C}(t)$ is not commutative.

The results of [St4] seem to agree with our §4.2 and §4.3.

###  4.4 $P_{3}^{\prime}(D_{6})(a,-a,4,-4)$ and $q=-t^{1/2}$

The variational equation reads
$v^{\prime\prime}+(\frac{1}{4}t^{-2}+\frac{a}{2}t^{-3/2}-4t^{-1})v=0$. At
$t=0$ there is a logarithm present in the local solutions. At $t=\infty$ there
is an exponential present in the local solutions. The differential Galois
group $G$, say over $\mathbb{C}(t^{1/2})$, contains therefore a
$\mathbb{G}_{a}$ and a $\mathbb{G}_{m}$. Thus $G$ contains a Borel subgroup.
The operator corresponding to this equation does not factor over
$\mathbb{C}(t^{1/2})$. One concludes that $G={\rm SL}_{2}$.

###  4.5 $P_{3}^{\prime}(D_{7})(0,-2,2,0)$ and $y=t^{1/3}$

The variational equation reads
$-t^{1/3}v^{\prime\prime}-\frac{1}{3}t^{-2/3}v^{\prime}+(-\frac{1}{9}t^{-5/3}+\frac{3}{2}t^{-1})v=0$.
As in the case 4.4, there is at $t=0$ a logarithm present and at $t=\infty$ an
exponential. The corresponding operator does not factor over the field
$\mathbb{C}(t^{1/3})$. One concludes that the differential Galois group is
${\rm SL}_{2}$.

###  4.6 $P_{3}^{\prime}(D_{8})(8h,-8h,0,0)$ and $y=-t^{1/2}$

The variational equation is
$v^{\prime\prime}+(4ht^{-3/2}+\frac{1}{4}t^{-2})v=0$. Computations similar to
those in the cases 4.4 and 4.5 imply that the differential Galois group is
${\rm SL}_{2}$.

###  4.7 $P_{5}(a,-a,0,\delta)$ and $y=-1$

VE is $v^{\prime\prime}=-t^{-1}v^{\prime}+(8at^{-2}+\frac{1}{2}\delta)v$ and
the operator form is $\delta_{t}^{2}-(8a+\frac{1}{2}\delta t^{2})$ with
$\delta_{t}:=t\frac{d}{dt}$. The differential Galois group $G\subset{\rm
SL}_{2}$ depends on $a$ and $\delta$.

If $\delta=0$, then this group is a subgroup of $\mathbb{G}_{m}$. We skip this
equation because it is a special case of the degenerate fifth Painlevé
equation.

Suppose that $\delta\neq 0$ (then one usually scales $\delta$ to
$-\frac{1}{2}$). We use [vdP-Si] for some facts and terminology and we use the
package DEtools of MAPLE for some computations. The singularity at $t=\infty$
is irregular; its generalized eigenvalues are $\pm(\frac{\delta}{2})^{1/2}t$
and the formal monodromy $\gamma$ is $-id$. On a suitable basis of the formal
solution space at $t=\infty$, the topological monodromy at $t=\infty$ is the
product of $\gamma$ and two Stokes matrices and has the form ${-1\ 0\choose 0\
-1}{1\ 0\choose e_{1}\ 1}{1\ e_{2}\choose 0\ 1}$.

Since there is only one other singular point, namely at $t=0$, and since this
singularity is regular, the group $G$ coincides with the differential Galois
group taking over the field of the convergent Laurent series
$\mathbb{C}(\\{t^{-1}\\})$. The latter is generated by the group
$\mathbb{G}_{m}\cong\\{{c\ \ \ 0\choose 0\ 1/c}|\ c\in\mathbb{C}^{*}\\}$,
$\gamma$ and the two Stokes matrices. The topological monodromy at $t=\infty$
is conjugated to the topological monodromy at $t=0$. Comparing the traces of
these matrices yields $-e_{1}e_{2}-2=e^{2\pi i\sqrt{8a}}+e^{-2\pi
i\sqrt{8a}}=2\cos(2\pi\sqrt{8a})$.

If $e_{1}e_{2}\neq 0$, then $G={\rm SL}_{2}$. Now $e_{1}e_{2}=0$ is equivalent
to $\sqrt{8a}-\frac{1}{2}\in\mathbb{Z}$ or $a=\frac{(2n+1)^{2}}{32}$ for some
integer $n\geq 0$. In these cases $G$ is contained in a Borel subgroup of
${\rm SL}_{2}$ and the operator $\delta_{t}^{2}-(8a+\frac{1}{2}\delta t^{2})$
factors as $(\delta_{t}-F)(\delta_{t}+F)$ over the field $\mathbb{C}(t)$. In
fact $e_{1},e_{2}$ are both zero if $a=\frac{(2n+1)^{2}}{32}$ and $G$ is
generated by $\gamma={-1\ 0\choose 0\ -1}$ and $\mathbb{G}_{m}\cong\\{{c\ \ \
0\choose 0\ 1/c}|\ c\in\mathbb{C}^{*}\\}$.

Example: for $a=\frac{1}{32}$ one finds $F=\frac{1}{2}+\sqrt{\delta/2}\ t$. In
fact, a basis of solutions is $t^{-1/2}e^{\sqrt{\delta/2}\
t},t^{-1/2}e^{-\sqrt{\delta/2}\ t}$. A basis of solutions for
$a=\frac{(2n+1)^{2}}{32}$ and (for convenience) with $\delta=2$ is
$\\{t^{-(2n+1)/2}e^{t}(t^{n}+\cdots),\ t^{-(2n+1)/2}e^{-t}(t^{n}+\cdots)\\}$.
The second polynomial is obtained from the first one by changing the sign of
the terms $t^{k}$ with $k\equiv(n-1)\bmod 2$.

The second variational equation is

$v^{\prime\prime}+t^{-1}v^{\prime}-(8at^{-2}+\delta/2)v=0,$
$w^{\prime\prime}+t^{-1}w^{\prime}-(8at^{-2}+\delta/2)w=\frac{3}{2}vv^{\prime\prime}-(v^{\prime})^{2}+\frac{3}{2}t^{-1}vv^{\prime}-(16at^{-2}+\delta)v^{2}.$

For the case $a=\frac{1}{32}$ a MAPLE computation indicates that the
differential Galois group $G$ of the above two equations has the properties:
$G/G^{o}=C_{2}$, $G^{o}/H=\mathbb{G}_{m}$, $H\cong\mathbb{G}_{a}^{3}$ and the
action of $\mathbb{G}_{m}$ (by conjugation) on $H$ is not trivial. In
particular $G^{o}$ is not commutative. A similar result holds for all cases
$a=\frac{(2n+1)^{2}}{32}$.

We note that the results of [St2] are similar to §4.7 (see also the
“Introduction and Summary” above).

###  4.8 $P_{5}(\frac{s^{2}}{2},-\frac{1}{2},-s,-\frac{1}{2})$ and
$y=-\frac{t}{s}+1$

We note that in [OO2], Theorem 2.1 part 8) there is a typo. According to the
“Clarkson lectures” [C2] slides 52-53, the above choice of parameters
corresponds to a Riccati family of solution $w$ with equation
$w^{\prime}=\frac{s}{t}w^{2}+(1+\frac{1-s}{t})w-\frac{1}{t}$. This Riccati
equation has the rational solution $w=1-\frac{t}{s}$. We refer to [C3, §7,
especially 7.1] for a derivation of this.

Clarkson’s papers [C] and [C3, §5.6] contain a list of rational solutions of
$P_{5}$, e.g., $P_{5}(\frac{1}{2},-\frac{s^{2}}{2},2-s,-\frac{1}{2})$ with
$y=t+s$. We skip these examples since they are equivalent via Bäcklund
transformations to one of the two equations in [OO2, Theorem 2.1].

VE is
$v^{\prime\prime}+\frac{s-2t}{t(t-s)}v^{\prime}+\frac{(s-t)^{3}-s-2t}{t^{2}(t-s)}v=0$.
One observes that:
$\frac{s-2t}{t(t-s)}$ has the form $\frac{f^{\prime}}{f}$ for some
$f\in\mathbb{C}(t)$,
$t=0$ is regular singular with local exponents $1\pm s$,
$t=s$ is regular singular with local exponents $0,0$ and a logarithmic term is
present in the local solutions,
$t=\infty$ is irregular singular with (generalized) local exponents $\pm(t+c)$
for some $c\in\mathbb{C}$.
It follows that the differential Galois group $G$ is contained in ${\rm
SL}_{2}$, contains copies of $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$. If
$G\neq{\rm SL}_{2}$, then $G$ is a Borel subgroup and the operator
corresponding to the differential equation factors over $\mathbb{C}(t)$. One
can verify that this is not the case. Hence $G={\rm SL}_{2}$.

###  4.9 $degP_{5}$ with $\theta_{0}=\frac{1}{2}$ and solution
$q(t)=1-\frac{\theta_{1}}{2t}$

In [OO2] another degenerate fifth Painlevé equation
$P_{5}(\frac{h^{2}}{2},-8,-2,0)$ and solution $y=1+\frac{2t^{1/2}}{h}$ is
considered. We translate this for the degenerate fifth Painlevé equation
$degP_{5}$ of our paper [APT]. VE is the equation

$v^{\prime\prime}+av^{\prime}+bv=0\mbox{ with
}a=\frac{4t-3\theta_{1}}{t(2t-\theta_{1})},\
b=-\frac{32t^{3}-32t^{2}\theta_{1}+8t\theta_{1}^{2}+\theta_{1}}{t^{2}(2t-\theta_{1})}.$

We note that $h=0$ corresponds to $\theta_{1}=0$. The given solution has no
meaning for $h=0$ and, likewise, the solution $q(t)=1$ has no meaning for the
case $\theta_{1}=0$. Therefore we may suppose that $\theta_{1}\neq 0$.

The equation has three singular points $0,\frac{\theta_{1}}{2},\infty$. The
first two singularities are regular singular and $\infty$ is an irregular
singularity with Katz invariant 1.

Since $a$ equals $\frac{f^{\prime}}{f}$ for some $f\in\mathbb{C}(t)$, the
differential Galois group $G$ is a subgroup of ${\rm SL}_{2}$. By
straightforward computations one finds that at $t=0$, the function $\log t$ is
present, that the singularity $t=\frac{\theta_{1}}{2}$ is apparent and that
$e^{4t}$ is present in the formal solutions at $t=\infty$. Thus $G$ contains a
copy of $\mathbb{G}_{a}$ and a copy of $\mathbb{G}_{m}$. If $G\neq{\rm
SL}_{2}$, then $G$ is a Borel subgroup and the operator
$(\frac{d}{dt})^{2}+a(\frac{d}{dt})+b$ factors (or equivalently the Riccati
equation has a rational solution). A straightforward computation shows that
the operator does not factor. One concludes that $G={\rm SL}_{2}$.

## References

  * [A] P. B. Acosta-Humánez, _Nonautonomous Hamiltonian Systems and Morales-Ramis Theory I. The Case $\ddot{x}=f(x,t)$_, SIAM Journal on Applied Dynamical Systems, 8 (2009), 279–297.
  * [APT] P.B. Acosta-Humánez, M. van der Put and J. Top, Isomonodromy for the degenerate fifth Painlevé equation, SIGMA, 13 (2017), 029, 14 pages.
  * [C] P. A. Clarkson, Special polynomials associated with rational solutions of the fifth Painlevé equation, Journal of Computational and Applied Mathematics, 178 (2005), 111–129.
  * [C2] P. A. Clarkson, Painlevé Equations – Nonlinear Special Functions, $87$ slides presented during the IMA Summer Program “Special Functions in the Digital Age, Minneapolis, 22 July - 2 August 2002. http://www.math.rug.nl/~top/Clarkson.pdf
  * [C3] P. A. Clarkson, Painlevé Equations – Nonlinear Special Functions, pages 331–411 in: “Orthogonal Polynomials and Special Functions. Computation and Applications” (F. Marcellán and W. Van Assche, eds.), Lecture Notes in Mathematics, 1883, Springer-Verlag, Berlin, 2006.
  * [HS] E. Horozov and T. Stoyanova, _Non-Integrability of Some Painlevé-VI Equations and Dilogarithms_ , Regular and Chaotic Dynamics, 12 (2007), 622–629.
  * [JM] M. Jimbo and T. Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients II, Physica D, 2 (1981), 407–448.
  * [Ki] A. V. Kitaev, _Quadratic transformations for the third and fifth Painlevé equations_ , Journal of Mathematical Sciences, 136 (2006), 3586–3595.
  * [L67] N.A. Lukashevich, On the theory of Painlevé ′s third equation, Differ. Uravn., 3 (1967), 1913–1923 (Russian).
  * [L68] N.A. Lukashevich, Solutions of the fifth Painlevé equation, Differ. Uravn., 4 (1968), 1413–1420 (Russian).
  * [M] M. Matsuda,  First Order Algebraic Differential Equations. A Differential Algebraic Approach, Lecture Notes in Mathematics, 804, Springer Verlag, Berlin, 1980.
  * [Ma] J. Malmquist, _Sur les équations différentielles du second ordre dont l’intégrale générale a ses points critiques fixes_ , Arkiv. Mat., Astron. Fys., 18 (1922), 1–89.
  * [Mo1] J. Morales-Ruiz, _Differential Galois Theory and Non-Integrability of Hamiltonian Systems_ , Birkhäuser, Basel, 1999.
  * [Mo2] J. Morales-Ruiz, _A Remark About the Painlevé Transcendents_ , Séminaires & Congrès, 14 (2006), 229–235.
  * [MR] J. Morales-Ruiz and J-P. Ramis, _Galoisian Obstructions to Integrability of Hamiltonian Systems_ , Methods and Applications of Analysis, 8 (2001), 33–96.
  * [MRS] J. Morales-Ruiz, J-P. Ramis, and C. Simó, _Integrability of Hamiltonian Systems and Differential Galois Groups of Higher Variational Equations_ , Annales scientifiques de l’Ecole normale supérieure, 40 (2007), 845–884.
  * [Mun-vdP] G. Muntingh, M. van der Put, _Order one equations with the Painlevé property_ ,  Indag. Math. (N.S.), 18 (2007), 83–95.
  * [NNPT] L. X. Ngo Chau, K. A. Nguyen, M. van der Put, and J. Top, Equivalence of differential equations of order one, J. Symbolic Comput., 71 (2015), 47–59.
  * [OKSO] Yousuke Ohyama, Hiroyuki Kawamuko, Hidetaka Sakai and Kazuo Okamoto, Studies on the Painlevé Equations, V, Third Painlevé Equations of Special Type PIII(D7) and PIII(D8) J. Math. Sci. Univ. Tokyo 13 (2006), 145–204.
  * [OO1] Yousuke Ohyama and Shoji Okumura, A coalescent diagram of the Painlevé equations from the viewpoint of isomonodromic deformations, J. Phys. A: Math. Gen., 39 (2006), 12129–12151.
  * [OO2] Yousuke Ohyama and Shoji Okumura, R. Fuchs’ problem of the Painlevé equations from the first to the fifth, Contemporary Mathematics, 593 (2013), 163–178.
  * [O] K. Okamoto, Studies on the Painlevé equations I, sixth Painlevé equation, Ann. Mat. Pura Appl., 146 (1987), 337–381.
  * [vdP-Sa] M. van der Put and M.-H. Saito, Moduli spaces for linear differential equations and the Painlevé equations, Ann. Inst. Fourier (Grenoble), 59 (2009), 2611–2667. ́
  * [vdP-Si] M. van der Put, M.F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der mathematischen Wissenschaften, 328, Berlin: Springer Verlag, 2003.
  * [St1] T. Stoyanova, _A Note on the R. Fuchs’s Problem for the Painlevé Equations_ , Preprint, 2012, arXiv:1204.0157.
  * [St2] T. Stoyanova, _Non-Integrability of Painlevé V Equations in the Liouville Sense and Stokes Phenomenon_ , Advances in Pure Mathematics, 1 (2011), 170–183.
  * [St3] T. Stoyanova, _Non-integrability of Painlevé VI equations in the Liouville sense_ , Nonlinearity, 22 (2009), 2201–2230.
  * [St4] T. Stoyanova, _Non-Integrability of the Fourth Painlevé Equation in the Liouville–Arnold Sense_ , Nonlinearity, 27 (2014), 1029–1044.
  * [SC] T. Stoyanova and O. Christov, _Non-Integrability of the Second Painlevé Equation as a Hamiltonian System_ , C. R. Acad. Bulgare Sci., 60 (2007), 13–18.
  * [U1] H. Umemura, Second proof of the irreducibility of the first differential equation of Painlevé, Nagoya Math. J., 117 (1990), 125–171.
  * [U2] H. Umemura, Birational automorphism groups and differential equations, Nagoya Math. J., 119 (1990), 1–80.
  * [U3] H.Umemura, Irreducibility of the First Differential Equation of Painlevé, Nagoya Math. J., 117 (1990), 231–252.
  * [UW] H. Umemura and H. Watanabe, Solutions of the second and the fourth Painlevé equations, I, Nagoya Math. J., 148 (1997), 151–198.
  * [ZF] H. Zoladek and G. Filipuk, _Painlevé equations, elliptic integrals and elementary functions_ , Journal of differential equations, 258 (2015), 1303–1355.


First, we prove the following vector norm error bound of quantum simulation:

######  Lemma 9 (Vector norm error bound).

Let $H_{1}$, $H_{2}$ be two Hermitian operators and $H=H_{1}+H_{2}$. Then, for
any $t>0$ and an arbitrary vector $\ket{\varphi}$, we have

$\displaystyle\left\|e^{-iH_{1}t}e^{-iH_{2}t}\ket{\varphi}-e^{-iHt}\ket{\varphi}\right\|\leq\frac{t^{2}}{2}\sup_{\tau_{1},\tau_{2}\in[0,t]}\left\|[H_{1},H_{2}]e^{-iH_{2}\tau_{2}}e^{-iH_{1}\tau_{1}}\ket{\varphi}\right\|.$
(139)

###### Proof.

By [26, Proposition 15], we have the variation-of-parameter formula

$\displaystyle
e^{-iH_{1}t}e^{-iH_{2}t}=e^{-iHt}+\int^{t}_{0}\mathrm{d}\tau_{1}\int^{\tau_{1}}_{0}\mathrm{d}\tau_{2}\leavevmode\nobreak\
e^{-iH(t-\tau_{1})}e^{-iH_{2}\tau_{1}}e^{-iH_{2}\tau_{2}}[H_{1},H_{2}]e^{-iH_{2}\tau_{2}}e^{-iH_{1}\tau_{1}}.$
(140)

Thus, for an arbitrary vector $\ket{\varphi}$, we have

$\displaystyle\left(e^{-iH_{1}t}e^{-iH_{2}t}-e^{-iHt}\right)\ket{\varphi}$
$\displaystyle\qquad=\int^{t}_{0}\mathrm{d}\tau_{1}\int^{\tau_{1}}_{0}\mathrm{d}\tau_{2}\leavevmode\nobreak\
e^{-iH(t-\tau_{1})}e^{-iH_{2}\tau_{1}}e^{-iH_{2}\tau_{2}}[H_{1},H_{2}]e^{-iH_{2}\tau_{2}}e^{-iH_{1}\tau_{1}}\ket{\varphi}.$
(141)

Since the spectral norm of the vector in the integrand is upper bounded by

$\displaystyle\sup_{\tau_{1},\tau_{2}\in[0,t]}\left\|[H_{1},H_{2}]e^{-iH_{2}\tau_{2}}e^{-iH_{1}\tau_{1}}\ket{\varphi}\right\|,$
(142)

and
$\int^{t}_{0}\mathrm{d}\tau_{1}\int^{\tau_{1}}_{0}\mathrm{d}\tau_{2}=\frac{t^{2}}{2}$,
we obtain the desired vector norm error bound (139). ∎

Second, we observe the following fact:

######  Theorem 6 ([12, Theorem 2, informal]).

For Schrödinger equations of the form

$\displaystyle i\frac{\partial}{\partial t}u+\Delta u+V(x,t)u=0,$ (143)

defined over an arbitrary finite-dimensional space with periodic boundary
condition, let $u(x,t)$ be the solution at time $t$. If $V(x,t)$ is smooth in
space and periodic in time, and the initial condition $u(x,0)$ is smooth, then
we have

$\displaystyle\|\nabla u(t)\|_{2}\leq C(\log t)^{\alpha}\|\nabla u(0)\|_{2},$
(144)

where $C$ and $\alpha$ are absolute constants.

######  Remark 5.

The original Theorem 2 in [12] actually proved the logarithmic growth in
Sobolev norm $\|u(t)\|_{H^{s}}$ for all $s>0$, while we only cite the special
case $s=1$. The $\|\nabla u(0)\|_{2}$ term was absorbed in the constant factor
in the original statement, while we feel necessary to expand it out because it
may introduce dependence on $n$ and $r_{0}$. It is worth noting that the
theorem was proven for two-dimensional Schrödinger equations with quasi-
periodic potential field $V(x,t)$, while it has been made clear in the context
that this result holds for arbitrary-dimensional cases if $V$ is periodic.
Bourgain also explicitly discussed the periodic-$V$ case in [13].

######  Corollary 1.

For a quadratic function of the form
$f_{q}=\frac{1}{2}(\mathbf{x}-\tilde{\mathbf{x}})^{T}\mathcal{H}(\mathbf{x}-\tilde{\mathbf{x}})+F$
where $\mathcal{H}$ is a Hermitian matrix and $F$ is a constant, consider the
Schrödinger equation of the form

$\displaystyle i\frac{\partial}{\partial
t}\Phi=\left[-\frac{r^{2}_{0}}{2}\Delta+\frac{1}{r^{2}_{0}}f_{q}\right]\Phi,$
(145)

with periodic boundary conditions and initial condition $\Phi_{0}(x)$ defined
in (7) (i.e., the initial state of the quantum simulation Algorithm 1), then
we have

$\displaystyle\|\nabla\Phi(t)\|_{2}\leq C\sqrt{\frac{n}{r_{0}}}(\log
t)^{\alpha},$ (146)

where $C$ and $\alpha$ are absolute constants.

###### Proof.

Note that the constant $F$ just adds a global phase to the solution which does
not influence either $\|\Phi(t)\|_{2}$ or $\|\nabla\Phi(t)\|$, and the
Schrödinger equation is translation-invariant under
$\mathbf{x}\to\mathbf{x}-\tilde{\mathbf{x}}$, we may assume without loss of
generality that $f_{q}=\frac{1}{2}\mathbf{x}^{T}\mathcal{H}\mathbf{x}$.

Define a new function
$u(\mathbf{x},t)=\Phi\left(\frac{r_{0}\mathbf{x}}{\sqrt{2}},t\right)$, and it
is straightforward to verify that

$\displaystyle i\frac{\partial}{\partial t}u+\Delta
u-\frac{1}{r^{2}_{0}}f_{q}\left(\frac{r_{0}\mathbf{x}}{\sqrt{2}}\right)u=0.$
(147)

Note that the function $f_{q}(\mathbf{x})$ is quadratic, so
$\frac{1}{r^{2}_{0}}f_{q}\left(\frac{r_{0}\mathbf{x}}{\sqrt{2}}\right)=\frac{1}{2}f_{q}(\mathbf{x})$,
which is a constant multiple of $f_{q}$. Thus, we may directly invoke Theorem
6 to yield

$\displaystyle\|\nabla u(t)\|_{2}\leq C\left(\log t\right)^{\alpha}\|\nabla
u(0)\|_{2},$ (148)

where the $\|\nabla u(0)\|_{2}$ can be directly calculated as follows:

$\displaystyle\|\nabla u(0)\|_{2}$
$\displaystyle\leq\left(\sum^{n}_{j=1}\int_{\mathbb{R}^{n}}|u_{x_{j}}(\mathbf{x},0)|^{2}\leavevmode\nobreak\
\mathrm{d}\mathbf{x}\right)^{1/2}=\frac{r_{0}}{\sqrt{2}}\left(\sum^{n}_{j=1}\int_{\mathbb{R}^{n}}|(\Phi_{0})_{x_{j}}(r_{0}\mathbf{x}/\sqrt{2},0)|^{2}\leavevmode\nobreak\
\mathrm{d}\mathbf{x}\right)^{1/2}$ (149)
$\displaystyle=\frac{\sqrt{r_{0}}}{2^{1/4}}\left(\sum^{n}_{j=1}\int_{\mathbb{R}^{n}}|(\Phi_{0})_{x_{j}}(\mathbf{x},0)|^{2}\leavevmode\nobreak\
\mathrm{d}\mathbf{x}\right)^{1/2}$ (150)
$\displaystyle=\frac{\sqrt{r_{0}}}{2^{1/4}}\frac{1}{2r^{2}_{0}}\left(\sum^{n}_{j=1}\int_{\mathbb{R}}\frac{1}{\sqrt{2\pi}r_{0}}e^{-(x_{j}-\tilde{x}_{j})^{2}/2r^{2}_{0}}(x_{j}-\tilde{x}_{j})^{2}\leavevmode\nobreak\
\mathrm{d}x_{j}\right)^{1/2}=\frac{1}{2^{5/4}}\sqrt{\frac{n}{r_{0}}}.$ (151)

Absorbing the $2^{-5/4}$ factor into the absolute constant $C$, we complete
the proof. ∎

Now, we are ready to prove Lemma 3, our result of bounding the deviation from
perfect Gaussian in quantum evolution.

See 3

###### Proof.

Define the following (Hermitian) operators:

$\displaystyle A=-\frac{r^{2}_{0}}{2}\Delta,\leavevmode\nobreak\
\leavevmode\nobreak\ B=\frac{1}{r^{2}_{0}}f,\leavevmode\nobreak\
\leavevmode\nobreak\ B^{\prime}=\frac{1}{r^{2}_{0}}f_{q},$ (152)
$H=A+B,\leavevmode\nobreak\ H^{\prime}=A+B^{\prime},\leavevmode\nobreak\
E=H-H^{\prime}=\frac{1}{r^{2}_{0}}(f-f_{q}).$ (153)

Let $\ket{\Phi(t)}=e^{-iHt}\ket{\Phi_{0}}$ be the wave function generated by
the quantum simulation with potential field $f$ and evolution time $t$, and
similarly, $\ket{\Phi^{\prime}(t)}:=e^{-iH^{\prime}t}\ket{\Phi_{0}}$ as the
wave function generated by the quantum simulation with potential field $f_{q}$
and the evolution time $t$.

By Lemma 9, and notice that $E$ is a scalar-valued function, we have

$\displaystyle\left\|e^{-iEt_{e}}\ket{\Phi^{\prime}(t_{e})}-\ket{\Phi(t_{e})}\right\|_{2}$
$\displaystyle\leq\frac{t^{2}_{e}}{2}\sup_{\tau_{1},\tau_{2}\in[0,t]}\left\|[H^{\prime},E]e^{-iE\tau_{2}}e^{-iH^{\prime}\tau_{1}}\ket{\Phi_{0}}\right\|_{2}$
(154)
$\displaystyle=\frac{t^{2}_{e}}{2}\sup_{\tau_{1}\in[0,t]}\left\|[H^{\prime},E]e^{-iH^{\prime}\tau_{1}}\ket{\Phi_{0}}\right\|_{2}.$
(155)

Denote $\ket{\Psi(\tau_{1})}:=e^{-iH^{\prime}\tau_{1}}\ket{\Phi_{0}}$. Note
that $[H^{\prime},E]=[A+B^{\prime},E]$ and $B^{\prime}$ commutes with $E$, we
have

$\displaystyle\sup_{\tau_{1}\in[0,t]}\left\|[H^{\prime},E]\Psi(\tau_{1})\right\|_{2}$
$\displaystyle=\frac{1}{2}\sup_{\tau_{1}\in[0,t]}\left\|[-\Delta,f-f_{q}]\Psi(\tau_{1})\right\|_{2}$
(156)
$\displaystyle=\frac{1}{2}\sup_{\tau_{1}\in[0,t]}\left\|-\Delta(f-f_{q})\Psi(\tau_{1})-2\nabla(f-f_{q})\cdot\nabla\Psi(\tau_{1})\right\|_{2}$
(157)
$\displaystyle\leq\frac{1}{2}\|\Delta(f-f_{q})\|_{\infty}+\|\nabla(f-f_{q})\|_{\infty}\|\nabla\Psi(\tau_{1})\|_{2}.$
(158)

The second equality follows from the fact that $[-\Delta,g]\varphi=-(\Delta
g)\varphi-2\nabla g\cdot\nabla\varphi$ for smooth functions $g$ and $\varphi$.
The last step follows from the triangle inequality (and the fact that
$\|\Psi(\tau_{1})\|=1$). By the $\rho$-Hessian Lipschitz condition of $f$, we
have

$\displaystyle|\Delta(f(\mathbf{x})-f_{q}(\mathbf{x}))|$
$\displaystyle=\left|\tr(\nabla^{2}f(\mathbf{x})-\nabla^{2}f_{q}(\mathbf{x}))\right|=\left|\tr(\nabla^{2}f(\mathbf{x})-\nabla^{2}f(\tilde{\mathbf{x}}))\right|$
(159) $\displaystyle\leq n\|\nabla^{2}f-\nabla^{2}f(\tilde{\mathbf{x}})\|\leq
n^{3/2}\rho M,$ (160)

where the second equality holds because $f_{q}$ is a quadratic form and
$\nabla^{2}f_{q}(\mathbf{x})=\mathcal{H}=\nabla^{2}f(\tilde{\mathbf{x}})$.
Note that the diameter of the hypercube domain is $n^{1/2}M$, and the last
step follows from the $\rho$-Hessian Lipschitz condition. It turns out that

$\displaystyle\|\Delta(f-f_{q})\|_{\infty}\leq n^{3/2}\rho M.$ (161)

Next, we bound the $L^{\infty}$-norm of the gradient of $f-f_{q}$:

$\displaystyle\|\nabla f-\nabla f_{q}\|_{\infty}$
$\displaystyle\leq\sup_{\mathbf{x}}\|\nabla f(\mathbf{x})-\nabla
f_{q}(\mathbf{x})\|=\sup_{\mathbf{x}}\|\nabla
f(\mathbf{x})-\mathcal{H}(\mathbf{x}-\tilde{\mathbf{x}})\|$ (162)
$\displaystyle\leq\sup_{\mathbf{x}}\|\nabla
f(\mathbf{x})\|+\sup_{\mathbf{x}}\|\mathcal{H}(\mathbf{x}-\tilde{\mathbf{x}})\|,$
(163)

where the last step uses the triangle inequality. Note that
$\tilde{\mathbf{x}}$ is a stationary point of $f$, so $\nabla
f(\tilde{\mathbf{x}})=0$. By the $\ell$-smoothness condition of $f$, we obtain

$\displaystyle\sup_{\mathbf{x}}\|\nabla
f(\mathbf{x})\|=\sup_{\mathbf{x}}\|\nabla f(\mathbf{x})-\nabla
f(\tilde{\mathbf{x}})\|\leq\ell\sup_{\mathbf{x}}\|\mathbf{x}-\tilde{\mathbf{x}}\|\leq\ell
n^{1/2}M.$ (164)

Meanwhile, the $\ell$-smoothness of $f$ implies that
$\|\nabla^{2}f(\mathbf{x})\|\leq\ell$ for all $\mathbf{x}\in\mathbb{R}^{n}$,
therefore $\|\mathcal{H}\|\leq\ell$ and

$\displaystyle\sup_{\mathbf{x}}\|\mathcal{H}(\mathbf{x}-\tilde{\mathbf{x}})\|\leq\ell
n^{1/2}M.$ (165)

Plugging (164) and (165) to (163), we end up with

$\displaystyle\|\nabla(f-f_{q})\|_{\infty}\leq 2\ell n^{1/2}M.$ (166)

The upper bound for $\sup_{\tau_{1}}\|\nabla\Psi(\tau_{1})\|_{2}$ is given by
Corollary 1. Combining all three bounds, we end up with

$\displaystyle\left\|e^{-iEt_{e}}\ket{\Phi^{\prime}(t_{e})}-\ket{\Phi(t_{e})}\right\|_{2}\leq\left(\frac{\sqrt{n}\rho}{2}+\frac{2C\ell}{\sqrt{r_{0}}}(\log
t_{e})^{\alpha}\right)\frac{nMt^{2}_{e}}{2}.$ (167)

In what follows, we will simply write $\Psi^{\prime}$ for
$\Psi^{\prime}(t_{e})$. We also denote
$\ket{\Psi^{\prime\prime}}:=e^{-iEt_{e}}\ket{\Psi^{\prime}}$. Note that
$e^{-iEt_{e}}$ is actually a scalar function with modulus $1$, hence the two
wave functions $\ket{\Psi^{\prime}}$ and $\ket{\Psi^{\prime\prime}}$ yield the
same probability density, i.e.,
$|\Psi^{\prime}|^{2}=|\Psi^{\prime\prime}|^{2}$. By the definition of total
variation distance,

$\displaystyle TV(\mathbb{P}_{\xi},\mathbb{P}_{\xi^{\prime}})$
$\displaystyle=TV(|\Psi|^{2},|\Psi^{\prime\prime}|^{2})$ (168)
$\displaystyle=\frac{1}{2}\int_{x\in\mathbb{R}^{n}}\left|\Psi\overline{\Psi}-\Psi^{\prime\prime}\overline{\Psi^{\prime\prime}}\right|\mathrm{d}x$
(169)
$\displaystyle\leq\frac{1}{2}\int_{x\in\mathbb{R}^{n}}\left|(\Psi-\Psi^{\prime\prime})\overline{\Psi}\right|\mathrm{d}x+\frac{1}{2}\int_{x\in\mathbb{R}^{n}}\left|\Psi^{\prime\prime}(\overline{\Psi}-\overline{\Psi^{\prime\prime}})\right|\mathrm{d}x$
(170)
$\displaystyle\leq\left(\int_{x\in\mathbb{R}^{n}}|\Psi-\Psi^{\prime\prime}|^{2}\mathrm{d}x\right)^{1/2}\leq\left(\frac{\sqrt{n}\rho}{2}+\frac{2C\ell}{\sqrt{r_{0}}}(\log
t_{e})^{\alpha}\right)\frac{nMt^{2}_{e}}{2}.$ (171)

∎

###  A.3 Variance of Gaussian Wave Packets

Although the variance of the Gaussian wave packet $\sigma(\lambda;t)$ is
explicitly given in (5), it is a bit heavy to use in analysis. In this
subsection, we prove several lemmas that can be utilized to estimate the
variance $\sigma(\lambda;t)$. Based on these lemmas, it is then possible to
quantify the performance of Algorithm 2.

######  Lemma 10.

When $\lambda>0$,

$\min\Big{\\{}1,\frac{1}{2\alpha}\Big{\\}}\leq\sigma(t;\lambda)\leq\max\Big{\\{}1,\frac{1}{2\alpha}\Big{\\}}.$
(172)

When $\lambda<0$, let $\alpha=\sqrt{-\lambda}$,

$\frac{1}{\sqrt{2}}\varphi(t;\alpha)\leq\sigma(t;\lambda)\leq\varphi(t;\alpha),$
(173)

with $\varphi(t;\alpha)=\frac{1}{2\alpha}\sinh(\alpha t)+\cosh(\alpha t)$.

###### Proof.

The first estimate follows from $\cos 2\alpha t\in[0,1]$, while the second
estimate follows from the inequality

$\displaystyle\frac{a+b}{2}\leq\sqrt{\frac{a^{2}+b^{2}}{2}}\leq\frac{a+b}{\sqrt{2}}.$
(174)

∎

######  Lemma 11.

When $\lambda<0$,

$\displaystyle\sigma^{2}(t;\lambda)\geq 1+\frac{t^{2}}{4}.$ (175)

###### Proof.

Recall (5) that $\sigma(t;\lambda)$ equals to:

$\displaystyle\sigma^{2}(t;\lambda)=\frac{(1-e^{2\alpha
t})^{2}+4\alpha^{2}(1+e^{2\alpha t})^{2}}{16\alpha^{2}e^{2\alpha t}},$ (176)

in which $\alpha=\sqrt{-\lambda}$. The equation above can be converted to:

$\displaystyle\sigma^{2}(t;\lambda)$
$\displaystyle=\frac{(1+4\alpha^{2})e^{4\alpha
t}+(1+4\alpha^{2})-2(1-4\alpha^{2})e^{2\alpha t}}{16\alpha^{2}e^{2\alpha t}}$
(177) $\displaystyle=\frac{(1+4\alpha^{2})e^{2\alpha
t}+(1+4\alpha^{2})e^{-2\alpha t}-2(1-4\alpha^{2})}{16\alpha^{2}}.$ (178)

We denote $\mu:=2\alpha t$. Note that $\mu>0$. By the Taylor expansion of
$e^{\mu}$ with Lagrange form of remainder, there exists real numbers
$\zeta,\xi\in(0,\mu)$ such that

$\displaystyle e^{\mu}$
$\displaystyle=1+\mu+\frac{\mu^{2}}{2}+\frac{\mu^{3}}{6}+\frac{e^{\zeta}}{24}\mu^{4};$
(179) $\displaystyle e^{-\mu}$
$\displaystyle=1-\mu+\frac{\mu^{2}}{2}-\frac{\mu^{3}}{6}+\frac{e^{-\xi}}{24}\mu^{4}.$
(180)

Adding these two equations, we have

$\displaystyle e^{\mu}+e^{-\mu}\geq
2+\mu^{2}+\frac{\mu^{4}}{24}(e^{\zeta}+e^{-\xi})\geq
2+\mu^{2}+\frac{\mu^{4}}{24}(1+e^{-\mu})\geq 2+\mu^{2}.$ (181)

In other words,

$\displaystyle e^{2\alpha t}+e^{-2\alpha t}\geq 2+(2\alpha t)^{2},$ (182)

which results in

$\displaystyle\frac{(1+4\alpha^{2})e^{2\alpha t}+(1+4\alpha^{2})e^{-2\alpha
t}-2(1-4\alpha^{2})}{16\alpha^{2}}$
$\displaystyle\geq\frac{(1+4\alpha^{2})(2+4\alpha^{2}t^{2})-2(1-4\alpha^{2})}{16\alpha^{2}}$
(183) $\displaystyle\geq\frac{16\alpha^{2}+4\alpha^{2}t^{2}}{16\alpha^{2}}$
(184) $\displaystyle=1+\frac{t^{2}}{4};$ (185)

or equivalently,

$\displaystyle\sigma^{2}(t;\lambda)\geq 1+\frac{t^{2}}{4}.$ (186)

∎

In the proof of Proposition 1, we will also use the following fact about
multivariate Gaussian distributions:

######  Lemma 12 ([44, Proposition 1]).

Let $A\in\mathbb{R}^{m\times n}$ be a matrix, and let $\Sigma:=A^{T}A$. Let
$\mathbf{x}=(x_{1},\cdots,x_{n})$ be an isotropic multivariate Gaussian random
vector with mean zero. For all $t>0$:

$\displaystyle\mathbb{P}\Big{(}\|A\mathbf{x}\|^{2}>\tr(\Sigma)+2\sqrt{\tr(\Sigma^{2})t}+2\|\Sigma\|t\Big{)}\leq
e^{-t}.$ (187)

###  A.4 Existing Lemmas

In this subsection, we list existing lemmas from [48, 47] that we use in our
proof.

First, we use the following lemma for the large gradient scenario of gradient
descent method:

######  Lemma 13 ([47, Lemma 19]).

If $f(\cdot)$ is $\ell$-smooth and $\rho$-Hessian Lipschitz, $\eta=1/\ell$,
then the gradient descent sequence $\\{\mathbf{x}_{t}\\}$ satisfies:

$\displaystyle f(\mathbf{x}_{t+1})-f(\mathbf{x}_{t})\leq\eta\|\nabla
f(\mathbf{x})\|^{2},$ (188)

for any step $t$ in which quantum simulation is not called.

The next lemmas are frequently used in the large gradient scenario of the
accelerated gradient descent method:

######  Lemma 14 ([48, Lemma 7]).

Consider the setting of Theorem 4. If we have $\|\nabla
f(\mathbf{x}_{\tau})\|\geq\epsilon$ for all $\tau\in[0,\mathscr{T}]$, then
there exists a large enough positive constant $c_{A0}$, such that if we choose
$c_{A}\geq c_{A0}$, by running Algorithm 3 we have
$E_{\mathscr{T}}-E_{0}\leq-\mathscr{E}$, in which
$\mathscr{E}=\sqrt{\frac{\epsilon^{3}}{\rho}}\cdot c_{A}^{-7}$, and $E_{\tau}$
is defined as:

$\displaystyle
E_{\tau}:=f(\mathbf{x}_{\tau})+\frac{1}{2\eta^{\prime}}\|\mathbf{v}_{\tau}\|^{2}$
(189)

where $\eta^{\prime}=\frac{1}{4\ell}$ as in Theorem 4.

Note that this lemma is not exactly the same as Lemma 7 of [48]: to be more
specific, they have an extra $\iota^{-5}$ term appearing in the $\mathscr{E}$.
However, this term actually only appears when we need to escape from a saddle
point using the original AGD algorithm. In large gradient scenarios where the
gradient is greater than $\epsilon$, it does not make a difference if we
ignore this $\iota^{-5}$ term.

######  Lemma 15 ([48, Lemma 4 and Lemma 5]).

Assume that the function $f$ is $\ell$-smooth. Consider the setting of Theorem
4, for every iteration $\tau$ where quantum simulation was not called, we have

$\displaystyle E_{\tau+1}\leq E_{\tau},$ (190)

where $E_{\tau}$ is defined in (189) in Lemma 14.

The correctness of these two lemmas above is guaranteed by two mechanisms. If
the function does not have a large negative curvature between $\mathbf{x}_{t}$
and $\mathbf{y}_{t}$ in the current iteration, the AGD will simply make the
Hamiltonian decrease efficiently. Otherwise, the Negative-Curvature-
Exploitation procedure in Line 3 of Algorithm 3 will be triggered (same as in
[48]) and decrease the Hamiltonian by either finding the minimum function
value in the nearby region of $\mathbf{x}_{t}$ if $\mathbf{v}_{t}$ is small,
or directly resetting $\mathbf{v}_{t}=0$ if it is large.