Table Of ContentLecture Notes in Physics
Alto Osada
Rekishu Yamazaki
Atsushi Noguchi
Introduction
to Quantum
Technologies
Lecture Notes in Physics
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Alto Osada · Rekishu Yamazaki ·
Atsushi Noguchi
Introduction to Quantum
Technologies
AltoOsada RekishuYamazaki
KomabaInstituteforScience CollegeofLiberalArts
TheUniversityofTokyo InternationalChristianUniversity
Meguro-ku,Tokyo,Japan Mitaka-shi,Tokyo,Japan
AtsushiNoguchi
KomabaInstituteforScience
TheUniversityofTokyo
Meguro-ku,Tokyo,Japan
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
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ForYukiandRay
Preface
What is quantum technology? Is it a fancy bandwagon fresh from the academics?
Definitely,thisisnotthecase.Letustracethehistoryinreverseorderfrom2022.
Now the quantum technology is known to everybody, thanks to the many-qubit
quantum “computers” developed by several companies and laboratories. Some of
themareavailableonthecloudplatformandareusedfortheeducationalandbusi-
ness use, while quantum communication and quantum sensing/metrology are also
underintenseinvestigation.Whydoesquantumcomputergathersuchanattention?
An answer is that such a novel computer based on the unconventional working
principle—quantum mechanics—is expected to outperform conventional comput-
ers, or classical computers as a counterpart of the quantum computer, when a
certaincomputationaltaskisgiven.Acelebratedquantumalgorithmofprimefac-
torization was invented by Peter Shor in 1994 [1]. In 1995, he invented another
landmark of the first proposal of a quantum error correction code [2]. We would
like to stress that this very work ignited the experimental exploration of quantum
computer which had been distinguished from a mere analog computer intolerable
toerrorsoccurringinananalogmanneraswell.Todate,anumberofbreathtaking
progress have been made in various quantum systems, especially in ion-trap and
superconducting-circuit systems.
Readers might have heard of another fancy word of quantum teleportation in
which a quantum state is transferred between remote parties by a bit tricky use
of the “spooky interaction” of quantum particles, or the quantum entanglement, a
conceptcoinedbackin1935byEinstein,Podolsky,andRosen[3].Quantumtele-
portationwasproposedin1993byBennetetal.[4],whichwasthenfollowedbyits
realizationsin1997–1998[5,6]usingopticalphotons.Thesestudieslaythebasis
of quantum communication. Optical photons are indispensable for the quantum
technologiesthatdemandthetransferofquantumstatesfromoneplacetoadistant
place,whichisindeedmadepossiblebytheinventionofopticalfiberswithexcel-
lent property of extremely low-loss propagation at room temperature. This nature
can also be utilized for another quantum technology of quantum cryptography,
pioneered by Bennett and Brassard in 1984 [7].
Quantum technologies, or even quantum mechanics, have established their rel-
evance in the development of Atomic, Molecular, and Optical (AMO) physics
during the twentieth century. Furthermore, the spectroscopic and control tech-
niques frequently used in AMO physics originate in the Nuclear Magnetic
vii
viii Preface
Resonance(NMR)experimentssincethelate1930swithcelebratedpioneerssuch
as Rabi, Ramsey, Bloch, and Purcell. NMR technique is widely applied in our
daily life, for instance to Magnetic Resonance Induction (MRI) tomography and
toidentificationofmaterialsusingchemicalshiftsofNMRspectra,bothofwhich
can be regarded as early forms of quantum sensing.
Whatwewanttosayisthatquantumtechnologyisnot(orshouldnotbe)justa
short-lived trend, a scam business, nor a mathematical game in the Hilbert space,
but a technology that has been pursued substantially with physics and devices in
the history of understanding and controlling matters in real life, at least at the
current technological levels. Therefore, the name of the game is to know what
physical objects we want to address in quantum technologies and how to do that.
Abstracted, perfect (error-corrected) logical qubits are not at our hand—even for
a single logical qubit. Realization of them is a “holy grail,” at the time in 2022.
Various physical systems are used to aim it, and numerous experimental tech-
niques are developed for them. Readers might be overwhelmed when they think
about which quantum system to learn, since different quantum systems can have
different theoretical backgrounds and experimental techniques. However, we, the
researcherinthisfield,knowthattherearemanytheoriesandconceptsincommon
amongthevariousquantumsystems,suchasquantumoptics,quantumelectronics,
and so on.
The purpose of this book is to summarize such common basics of quantum
technologies as a foundation for cultivating a comprehensive view of the state-of-
the-art quantum systems. The book is roughly divided into three parts.
(cid:129)
In Part I, we describe how the quantum states are represented based on
linear algebra, which is frequently used in quantum technology. We also
cover the fundamentals of quantum mechanics necessary for understanding the
basic quantum manipulations. It is an excerpt of what is to be collected and
summarized in a concise form.
(cid:129)
Part II starts with a description of two-level systems and electromagnetic
waves, and how they interact. We describe the interaction between the fun-
damental quantum systems, two-level systems, electromagnetic waves, and
harmonic oscillators. We also describe the theory relating itto the actual quan-
tum experiments and how they are realized. In particular, we cover details of
the “resonator”, which is an almost indispensable tool in the current quantum
technology.
(cid:129)
In Part III, we describe an overview of the quantum operations, quantum mea-
surements,quantumerrorcorrections,andanintroductiontothestate-of-the-art
quantum technologies.
Readers who are familiar with analytical mechanics, statistical mechanics, quan-
tum mechanics, and electromagnetism should be able to read this book by using
thosetextbooksasreferences.Inparticular,PartIisareviewofquantummechan-
ics, so readers who are quite familiar with quantum mechanics may start reading
Preface ix
from Part II and continue reading while returning to Part I as necessary. How-
ever, Part I also includes some details of the methods and concepts that are often
seen in quantum technology, so it is recommended that you read them at some
point.Inthissense,thesubjectofthisbookissuitedforthethird-andfourth-year
undergraduate students who have been studying physics. We also hope that some
ambitiousfirst-andsecond-yearstudentstryreading.Itmaybeareviewformany
graduate students and researchers, but comprehensive coverage of the quantum
technologiescompiledheremayprovideanewinsight.Inaddition,thisbookmay
beusedbytheresearcherswhowishtostudyquantumtechnologies,butcurrently
working outside the field. We also warn readers to be aware that there may be
some parts that lack theoretical rigor because we prioritize the description of the
quantum processes as a whole, instead of focusing on the details of the quantum
theories, which you may find in other literature. Lastly, problems are provided in
the last part of each chapter, except for Chaps. 1 and 11. The authors strongly
recommend the readers to tackle with them and check the solutions which are
electronically distributed on the Springer website.
Oneimportanttopicthatweareomittinginthisbookistheadvancementofthe
controlandfabricationsystems,formingafundamentalbasisofthestate-of-the-art
experimental quantum technologies. Digital controlling systems including Direct-
Digital-Synthesis (DDS) and Field-Programmable Gate Array (FPGA), low-noise
amplifiertechnologies,andsingle-photondetectorsareonlyafewoftheexamples
that have been made in the lab by the researchers, but these high-end products
are now readily available commercially. Ultra-high vacuum systems and dilution
refrigerators to provide an environment with extremely low background noise are
also available. Furthermore, the advancement in the field of material sciences and
micro/nano fabrication technologies have allowed researchers to develop a “De-
signer’sQuantum”,suchasQuantumdots,NV-center,andsuperconductingqubits,
with various novel materials and fabrication techniques. These fundamental bases
for the experiment might differ depending on the quantum system of the choice,
and we encourage the reader to refer to these details in the future.
Lastly, we would like to show our greatest appreciation to people who have
helped us preparing these materials. In writing this book, Masato Shigefuji, Yuki
Nakajima, and Genya Watanabe, members of Noguchi Laboratory in The Uni-
versity of Tokyo, Yoshimi Rokugawa, Sorato Nakano, Yutaro Nakai, and Tsubasa
Karino, members of Shu Lab in International Christian University, cooperated in
calculation check and proofreading of the text. Yuta Masuyama from National
Institute of Quantum Science and Technology gave us very useful advice on the
content. We would love to mention that Akiyuki Tokuno, Sridevi Purushothaman,
and Satish Ambikanithi from Springer worked really hard for the publication of
this book and Akiyuki in particular has made invaluable advice and comments.
We would like to express our gratitude here.
Meguro-ku, Japan Alto Osada
Mitaka-shi, Japan Rekishu Yamazaki
Meguro-ku, Japan Atsushi Noguchi
Contents
PartI QuantumStatesandQuantumMechanics
1 Introduction ........................................................ 3
1.1 Common Language in Quantum Information .................. 3
1.2 Various Quantum Systems .................................... 5
1.3 Electromagnetic Waves for Quantum Operations ............... 5
1.3.1 Development of Electromagnetic Wave Source ........ 6
1.4 Concept of Temperature ....................................... 8
2 LinearAlgebra ..................................................... 13
2.1 Vector Space .................................................. 13
2.1.1 Vectors in Three-Dimensional Vector Space ........... 13
2.1.2 Inner Product ......................................... 14
2.1.3 Orthonormal Basis .................................... 14
2.1.4 Vector Components ................................... 15
2.1.5 Norm of a Vector ..................................... 15
2.1.6 Outer Product ........................................ 16
2.1.7 Expansion for Multidimensional System .............. 16
2.2 Matrix and Operator .......................................... 17
2.2.1 Matrix Element ....................................... 17
2.2.2 Transpose Matrix ..................................... 17
2.2.3 Matrix Multiplication ................................. 18
2.2.4 Square Matrix ........................................ 18
2.3 Eigenvectors and Eigenvalues ................................. 20
2.4 Summary of Vector Characteristics in Index Notation .......... 21
3 WavefunctionandNotationsinQuantumMechanics ............... 25
3.1 Equation of Motion in Classical and Quantum Mechanics ...... 25
3.2 Wavefunction ................................................. 27
3.2.1 Inner Product of Wavefunction ........................ 27
3.2.2 Continuous and Discretized Wavefunction ............. 28
3.3 Operator ...................................................... 30
3.4 Dirac Notation ................................................ 30
3.5 Matrix Representation ......................................... 32
xi
