Skip main navigation
We use cookies to give you a better experience, if that’s ok you can close this message and carry on browsing. For more info read our cookies policy.
We use cookies to give you a better experience. Carry on browsing if you're happy with this, or read our cookies policy for more information.

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.


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.


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, Introducting 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!


  • 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

Complete the post-course survey

We would also be very grateful if you can take the time to complete our post-course survey, which asks you some questions about your experience on the course in order to help us keep improving our courses.

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

Share this article:

This article is from the free online course:

Understanding Quantum Computers

Keio University

Contact FutureLearn for Support