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Further Study

This course has introduced you to all of the important concepts in quantum computing, and no doubt you now are champing at the bit to learn more.

There are many very good popular science books on quantum computing in many languages, but if you have grasped most of what we have discussed in this course, you will find they cover much of the same material, and some will not be as up-to-date as this course. Still, for pure pleasure or to get a different perspective, you may find them useful. Our own personal favorite is Ultimate Zero and One, by Williams and Clearwater.

Go Try It Yourself: Keio and IBM

Our biggest piece of advice, if you are interested in the field, is to go try it yourself. This requires a certain amount of effort on your part to learn more about constructing circuits and interpreting the data that the machine gives you, but it will be an enlightening experience.

Sound impossibly hard? Lucky for you, IBM has connected several quantum computers to the web, allowing you to login to a website and create your own simple circuits and try them out.

Some of IBM’s quantum computers are available for free. Their most advanced machines, however, are only available to IBM Q Network members. Keio University has joined IBM’s Q Network as a hub. Keio is currently the only such hub in Asia, and our students (both undergrads and graduate students) and visiting researchers are using the machines regularly.

Just as we were completing the preparation for this course, a large team from Los Alamos National Laboratory posted a preprint of a paper online, with the title, “Quantum algorithm implementations for beginners,” which includes implementations of twenty quantum computing algorithms on IBM’s computers. Despite the title, it is targeted at those with a high comfort level with the mathematics of quantum computing. “Beginners” here primarily refers to professional physicists, mathematicians or computer scientists who are unfamiliar with quantum algorithms. However, after you work through some of IBM’s introductory exercises, you may wish to check into this paper.

If you have enjoyed learning online, and are looking for short materials on a variety of topics, we recommend:

  • Smarter Every Day
  • The Physics Girl
  • Veritasium
  • Ph.D. Comics

All are available online. The latter three have produced short videos on the key concepts in quantum computing, superposition and entanglement, that you may find enjoyable and helpful.

Online Advanced Courses

Besides our own, you can now study quantum mechanics, quantum chemistry and quantum computing in more mathematical depth in other online courses:

  1. Umesh Vazirani’s course goes the next step beyond this one in depth, covering both quantum mechanics and quantum computing.
  2. Alain Aspuru-Guzik’s course focuses on quantum mechanics as it applies to chemistry.
  3. Isaac Chuang and Peter Shor now have a course on quantum information science, as well.

Any of these would make a good follow-on to this course.

Math

We have been as rigorous and careful as we can in this course, without wading into deep mathematical waters. The next step in your study does involve learning how to swim in the deep end. Of course, physics, especially quantum mechanics and wave mechanics, are important, as is fundamental computer science and engineering, but they all come back to math.

Most important, in the short run, are the following, if you have not already studied them:

  1. Linear Algebra (vectors and matrices, including eigenvalues, eigenvectors, and tensor products)
  2. Probability (initially, discrete probability; later, continuous)

If you are in high school, you may be introduced to these topics. In college, you will have the opportunity to study them more in depth. Fortunately, some of the best books on quantum computing work from the assumption that readers have various backgrounds, and include introductory material such as this either in the main text or an appendix.

We saw the Fourier transform when we discussed Shor’s algorithm; it is essential not only to quantum computing, but to many fields. Its behavior is also far more complex than the examples we presented here; by all means, we encourage you to study it in more detail.

As you progress, still more math is helpful:

  1. Group theory (necessary to really understand Shor’s algorithm)
  2. Basic calculus (for continuous probability, as well as the physics)
  3. Differential equations (to complete quantum mechanics, including understanding the device and state variable physics)

Quantum Computing

By far the most influential book in the field is:

  1. Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.

Popularly referred to as “Mike & Ike”, every quantum computing laboratory has at least one battered copy, and most researchers have their own personal copy as well. The field has grown dramatically in breadth since the book’s initial publication, but the fundamental material in the book is hard to beat. Highly recommended for those who are serious about learning more.

The other most common starting point, written by another of the field’s pioneers, is John Preskill’s online notes, which continue to evolve.

Another important book is:

  1. A. Yu. Kitaev and A. H. Shen and M. N. Vyalyi, Classical and Quantum Computation, Graduate Studies in Mathematics Series, Amer Mathematical Society, 2002.

Kitaev et al. focus on the algorithms and the math, without bothering with the physics. Their rather different take on the algorithms is helpful for deepening your understanding, and the tables of mathematical notation are especially valuable if you are not a working mathematician and find the notation unfamiliar. It is, however, not an easy book.

For a somewhat gentler introduction, targeted at computer scientists and engineers interested in the algorithms but with less background in physics, we have recently been using

  1. Eleanor G. Rieffel and Wolfgang H. Polak, A Gentle Introduction to Quantum Computing, The MIT Press, 2014.

Two remarkable, unique, and funny books are:

  1. Scott Aaronson, Quantum Computing Since Democritus, Cambridge University Press, 2013.
  2. Jonathan P. Dowling, Schroedinger’s Killer App: Race to Build the World’s First Quantum Computer, CRC Press, 2013.

And we would be improperly modest if we failed to mention Van Meter’s own book on quantum repeater networks:

  1. Rodney Van Meter, Quantum Networking, Wiley-ISTE, 2014.

This book is appropriate for networking engineers and the like who know nothing about quantum mechanics or quantum computing.

Physics

Wave mechanics and electricity and magnetism are fundamental to understanding quantum mechanics. To be serious about quantum computing, you should take courses in these areas.

One book suitable for the youngest of students in this course, but a pleasure to read at any age:

  1. J.P. McEvoy, Oscar Zarate, Introducing Quantum Theory, A Graphic Guide

If you are interested in the topic of entanglement, another wonderful and unique book is the partially fictionalized narrative,

  1. Louisa Gilder, The Age of Entanglement: When Quantum Physics Was Reborn, Vintage, 2009.

More advanced topics in optics:

  1. Eugene Hecht, Optics, 5th edition, Pearson, 2016.
  2. Bahaa E. A. Saleh and Malvin Carl Teich, Fundamentals of Photonics, 2nd edition, Wiley-Interscience, 2007.
  3. Christopher Gerry and Peter Knight, Introductory Quantum Optics, Cambridge University Press, 2004.

Additional References

We have also built on some materials from other sources, including research papers.

  1. The graphical “dial” notation for states was inspired by Richard Feynman’s popular lectures on quantum electrodynamics.
  2. The factoring of 21 using Shor’s algorithm is worked out by Lavor et al.
  3. Bacon and van Dam produced a good description of other, more recent algorithms, at about the same level of description as this course, for Communications of the ACM.
  4. The U.S. National Science Foundation sponsored a workshop in 2016, titled “Quantum Information and Computation for Chemistry,” chaired by Alan Aspuru-Guzik (Harvard University) and Michael Wasielewski (Northwestern University). The report on this workshop provided much of the information on quantum algorithms.
  5. Thaddeus Ladd’s encyclopedia article, “Optical Quantum Dot Qubits,” in Juelich, provided valuable background on both quantum dots and quantum optics.
  6. Bennett’s notes on the history of reversible computation.
  7. Wikipedia has an excellent list of Bell inequality violation experiments.
  8. Van Meter’s Ph.D. thesis covers the performance of Shor’s algorithm in detail.
  9. Emma Strubell’s lecture notes on quantum algorithms include a detailed example of Grover’s algorithm.
  10. Michael Biercuk’s excellent article on the state of the industry in mid-2017.
  11. Wikipedia has a rough list of more than 75 companies involved in quantum computing.
  12. Schuld, Sinayskiy and Petruccione with an excellent summary of quantum machine learning.
  13. Andrew Childs gave an early view of the HHL algorithm, Nature Physics 2009, available here or here.
  14. Scott Aaronson on quantum machine learning, especially HHL, in Nature Physics, 2015.
  15. DiVincenzo’s criteria are best presented in a paper in Fortschritte der Physik, a version of which is available here.
  16. Seth Lloyd’s original molecular quantum computer design appeared in Science, in 1993.
  17. A good place to start learning more about the variational quantum eigensolver (VQE) is Talia Gershon’s blog posting.
  18. The quantum approximate optimization algorithm (QAOA) was created by Farhi, Goldstone and Gutmann.
  19. For an ever-evolving catalog of quantum algorithms, see the Quantum Zoo.
  20. John Preskill put online a set of notes on noisy, intermediate-scale quantum technology (NISQ) that is changing how we talk about the near-term prospects for quantum computers. Highly recommended!

Credits

  • Executive producer: Keiko Okawa
  • Producer: Motoki Yasui
  • Video director: Takahiro Niibe
  • Cameraman: Akihiko Matsuzawa
  • Lead educator: Rodney Van Meter
  • Educator: Takahiko Satoh
  • Technical contributors and web application developers: Shota Nagayama, Hideo Daikoku, Takaaki Matsuo, Kotone Itaya, Takafumi Oka, Keiko “Kiki” Shigeta, Takahiko Satoh
  • Web app software testing and debugging: Keiko Okawa, Takaaki Matsuo, Rodney Van Meter
  • Animations: Akihiko Matsuzawa, Keiko “Kiki” Shigeta
  • 3-D models produced by Shinnosuke Ozawa, Takahiko Satoh and Rodney Van Meter in conjuction with the research group of Prof. Hiroya Tanaka, Keio Shonan Fujisawa Campus
  • Bass guitar: Shinnosuke Ozawa
  • Inverted qubit: Shin Nishio
  • Motorcycle tour: Scott Alexander
  • Technical adviser on quantum dots: Thaddeus Ladd
  • Technical adviser on quantum chemistry: James Whitfield
  • Technical adviser on Fourier transform: Jin Mitsugi
  • Technical adviser on the industry: Simon Devitt
  • Technical adviser on ion traps: Tracy Northup
  • Technical adviser on machine learning algorithms: David Meyer

Get extra benefits, upgrade your course

You can now get extra benefits by upgrading this course, including:

Unlimited access to the course: Go at your own pace with unlimited access to the course for as long as it exists on FutureLearn.

A Certificate of Achievement: To help you demonstrate your learning we’ll send you a Certificate of Achievement when you become eligible.

Find out more




さらなる学習

このコースでは量子コンピューティングの重要な概念を一通り紹介しました。あなたは今、より多くのことを学ぶためにちょっとした努力をしています。

量子コンピューティングに関する科学の本は様々な言語で出ていますが、このコースで議論したことの大部分を把握していれば、大体は似たような内容で、ものによっては最新の内容ではないことに気がつくでしょう。それでも、新しい視点を得るためには、それらが役に立つでしょう。私たちの個人的なお気に入りは、WilliamsとClearwater著の Ultimate Zero and One です。

あなた自身で試してみよう:慶應とIBM

私たちのできる最良のアドバイスは、この分野に興味があるならば、自分で量子計算してみることです。回路を構成し、出力されたデータを理解するためには、ある程度の努力を必要としますが、良い経験になります。

不可能に思えますか?運が良いことに、IBMはいくつかの量子コンピュータをWebに接続しており、Webサイトから量子回路を作成して試すことができるシステムがあります。

IBMの量子コンピュータの一部は無料で利用可能です。ただし、最新のマシンは、IBM Qネットワーク・メンバーのみが使用できます。慶應義塾大学は、IBMのQネットワークにハブとして参加しました。慶應義塾は現在、アジア唯一のハブであり、学生(学部生と大学院生)と訪問研究者は毎日マシンを使用しています。

また、Los Alamos National Laboratoryの大規模なチームがオンラインで論文を発表しました。タイトルは「初心者のための量子アルゴリズムの実装」で、IBMコンピュータ上の20個の量子コンピューティングアルゴリズムの実装が含まれています。このタイトルにもかかわらず、すでに量子コンピューティングの数学で高度な知識を有する人を対象としています。ここでの「初心者」とは、主に、量子アルゴリズムに精通していない専門の物理学者、数学者、またはコンピュータ科学者を指します。IBMの量子コンピュータでいくつか回路を試した後、この論文をチェックしてみてください。

オンラインで学べる科学

オンラインで学ぶことを楽しんでいて、さまざまなトピックに関する短めの資料を探している場合は、以下をお勧めします。

  • Smarter Every Day
  • The Physics Girl
  • Veritasium
  • Ph.D. Comics

すべてがオンラインで利用可能です。このうち後ろ3つは、量子コンピューティング、重ね合わせ、量子もつれに関する短いビデオを制作しています。

オンライン上級コース

上級オンラインコースを学びたい場合はこのコースのほかに、量子力学、量子化学、量子コンピューティングをより数学的に勉強することができます。

  1. Umesh Vazirani’s course 量子力学と量子コンピューティングの両方をカバーします。
  2. Alain Aspuru-Guzik’s course 化学に応用される量子力学に焦点を当てています。
  3. Isaac Chuang and Peter Shor 現在、量子情報科学のコースを持っています。

これらのコースはこのコースの続きとして良いでしょう。

数学

このコースでは、数学的に深い領域に立ち入ることはありませんでした。あなたの次のステップは、より深い数学的な領域を学ぶことです。もちろん、物理学、特に量子力学と波力学は、コンピュータ科学と工学の基礎として重要ですが、それらはすべて数学に帰着します。

あなたがまだ勉強していないなら、短期的に最も重要なのは次の分野です。

  1. 線形代数学 (ベクトルと行列, 固有値, 固有ベクトル, 及びテンソル積)
  2. 確率(最初に, 離散型確率; そして, 連続確率)

あなたが高校生なら、これらの領域に関する基礎的なことを習うかもしれませんが、大学では、もっと深く勉強する機会があります。幸いにも、量子コンピューティングに関する良著のいくつかは、読者が様々な背景を持っていることを前提としており、本文や付録にこの記事のような紹介資料が含まれています。

Shorのアルゴリズムについて議論したとき、フーリエ変換に触れました。そしてそれは量子コンピューティングだけでなく、多くの分野に不可欠です。その振る舞いは、ここで示した例よりはるかに複雑です。是非、詳細を勉強してください。

あなたが成長するにつれて、さらに数学が役に立つでしょう:

  1. 群論 (Shor’s algorithmの本質的な理解に必須)
  2. 微分積分学の基礎 (連続確率, 物理学)
  3. 微分方程式 (量子力学の完全な理解)

量子コンピューティング

現時点で最も影響力のある本は次のとおりです。

  1. Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.

一般に「Mike&Ike」と呼ばれます。世界中の量子コンピューティングの研究所には、少なくとも1冊Mike&Ikeがあり、ほとんどの研究者は自分用のものをもっています。この本の出版以来、この分野は幅広く劇的に成長してきましたが、本の基本的な内容は普遍的なものです。もっと学習することを真剣に考えている方にお勧めです。

他の分野の先駆者によって書かれた最も一般的なもう一つの出発点は、John Preskill’s online notesであり、現在も進化し続けています。

もう一冊の重要な本は次は次の通りです。

  1. A. Yu. Kitaev and A. H. Shen and M. N. Vyalyi, Classical and Quantum Computation, Graduate Studies in Mathematics Series, Amer Mathematical Society, 2002.

Kitaev et al. は物理学を避けつつアルゴリズムと数学に焦点を当てました。アルゴリズムの学習はあなたの理解を深めさせるのに役立ちます。数学的表記のテーブルは、あなたが数学者として働いていない人には概念が良く分からないかもしれません。これは、簡単な本ではないかもしれません。

アルゴリズムに興味があるが物理学の知識が少ないコンピュータ科学者やエンジニアを対象としたやや優しい紹介については、

  1. Eleanor G. Rieffel and Wolfgang H. Polak, A Gentle Introduction to Quantum Computing, The MIT Press, 2014.

著名で独特で面白い本は次の2つです。

  1. Scott Aaronson, Quantum Computing Since Democritus, Cambridge University Press, 2013.
  2. Jonathan P. Dowling, Schroedinger’s Killer App: Race to Build the World’s First Quantum Computer, CRC Press, 2013.

Van Meter自身の量子リピータネットワークに関する本も紹介します。

  1. Rodney Van Meter, Quantum Networking, Wiley-ISTE, 2014.

この本は、量子力学や量子コンピューティングについて何も知らないネットワーキングエンジニアなどに適しています。

物理学

波と電磁気は量子力学を理解する上で基本的で不可欠なものです。量子コンピューティングについて真剣に考えるには、これらの分野のコースを受講する必要があります。

若い学生に適した本ですが、どの年齢層でも読むことができます。

  1. J.P. McEvoy, Oscar Zarate, Introducing Quantum Theory, A Graphic Guide

あなたが量子もつれに興味を持っているならば、部分的に架空の物語である、すばらしくユニークな本があります。

  1. Louisa Gilder, The Age of Entanglement: When Quantum Physics Was Reborn, Vintage, 2009.

光学に関する高度な本:

  1. Eugene Hecht, Optics, 5th edition, Pearson, 2016.
  2. Bahaa E. A. Saleh and Malvin Carl Teich,* Fundamentals of Photonics*, 2nd edition, Wiley-Interscience, 2007.
  3. Christopher Gerry and Peter Knight, Introductory Quantum Optics, Cambridge University Press, 2004.

参考文献

これらの他にも論文などの文献も参考にしています。

  1. The graphical “dial” notation for states was inspired by Richard Feynman’s popular lectures on quantum electrodynamics.
  2. The factoring of 21 using Shor’s algorithm is worked out by Lavor et al.
  3. Bacon and van Dam produced a good description of other, more recent algorithms, at about the same level of description as this course, for Communications of the ACM.
  4. The U.S. National Science Foundation sponsored a workshop in 2016, titled “Quantum Information and Computation for Chemistry,” chaired by Alan Aspuru-Guzik (Harvard University) and Michael Wasielewski (Northwestern University). The report on this workshop provided much of the information on quantum algorithms.
  5. Thaddeus Ladd’s encyclopedia article, “Optical Quantum Dot Qubits,” in Juelich, provided valuable background on both quantum dots and quantum optics.
  6. Bennett’s notes on the history of reversible computation.
  7. Wikipedia has an excellent list of Bell inequality violation experiments.
  8. Van Meter’s Ph.D. thesis covers the performance of Shor’s algorithm in detail.
  9. Emma Strubell’s lecture notes on quantum algorithms include a detailed example of Grover’s algorithm.
  10. Michael Biercuk’s excellent article on the state of the industry in mid-2017.
  11. Wikipedia has a rough list of more than 75 companies involved in quantum computing.
  12. Schuld, Sinayskiy and Petruccione with an excellent summary of quantum machine learning.
  13. Andrew Childs gave an early view of the HHL algorithm, Nature Physics 2009, available here or here.
  14. Scott Aaronson on quantum machine learning, especially HHL, in Nature Physics, 2015.
  15. DiVincenzo’s criteria are best presented in a paper in Fortschritte der Physik, a version of which is available here.
  16. Seth Lloyd’s original molecular quantum computer design appeared in Science, in 1993.
  17. A good place to start learning more about the variational quantum eigensolver (VQE) is Talia Gershon’s blog posting.
  18. The quantum approximate optimization algorithm (QAOA) was created by Farhi, Goldstone and Gutmann.
  19. For an ever-evolving catalog of quantum algorithms, see the Quantum Zoo.
  20. John Preskill put online a set of notes on noisy, intermediate-scale quantum technology (NISQ) that is changing how we talk about the near-term prospects for quantum computers. Highly recommended!

クレジット

  • Executive producer: Keiko Okawa
  • Producer: Motoki Yasui
  • Video director: Takahiro Niibe
  • Cameraman: Akihiko Matsuzawa
  • Lead educator: Rodney Van Meter
  • Educator: Takahiko Satoh
  • Technical contributors and web application developers: Shota Nagayama, Hideo Daikoku, Takaaki Matsuo, Kotone Itaya, Takafumi Oka, Keiko “Kiki” Shigeta, Takahiko Satoh
  • Web app software testing and debugging: Keiko Okawa, Takaaki Matsuo, Rodney Van Meter
  • Animations: Akihiko Matsuzawa, Keiko “Kiki” Shigeta
  • 3-D models produced by Shinnosuke Ozawa, Takahiko Satoh and Rodney Van Meter in conjuction with the research group of Prof. Hiroya Tanaka, Keio Shonan Fujisawa Campus
  • Bass guitar: Shinnosuke Ozawa
  • Inverted qubit: Shin Nishio
  • Motorcycle tour: Scott Alexander
  • Technical adviser on quantum dots: Thaddeus Ladd
  • Technical adviser on quantum chemistry: James Whitfield
  • Technical adviser on Fourier transform: Jin Mitsugi
  • Technical adviser on the industry: Simon Devitt
  • Technical adviser on ion traps: Tracy Northup
  • Technical adviser on machine learning algorithms: David Meyer

Get extra benefits, upgrade your course

You can now get extra benefits by upgrading this course, including:

Unlimited access to the course: Go at your own pace with unlimited access to the course for as long as it exists on FutureLearn.

A Certificate of Achievement: To help you demonstrate your learning we’ll send you a Certificate of Achievement when you become eligible.

Find out more

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Understanding Quantum Computers

Keio University