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Don't get discouraged, it's possible to get to a reasonable understanding with some sustained effort. However, you need to get the following into your head as quickly as possible:

Popular level explanations of anything quantum are a waste of your time.

Go back and read that again. You will never get close to understanding the field if you rely on someone else managing to "find the right metaphors" for you. Quantum computing is a mathematical field, and if you want to understand a mathematical field, you need to do mathematics. This sounds super scary, but it's actually no problem! Math is not what you think it is, and is actually a lot of fun to learn. You just need to put some work in. This just means maybe doing an hour or so of learning every day before you go to work, or afterwards.

Let's look at a little bit of a roadmap that you can follow to get to a reasonable understanding of quantum computing / quantum information. This is pretty much the path I followed, and now I am just about to submit my PhD thesis on quantum computational complexity. So I guess it worked out OK.

1. You can get really far in quantum computing with some basic understanding of linear algebra. Go to Khan Academy and watch their fantastic introduction.
   
   If Sal asks you to do an exercise, do the exercise.
   
   
2. Now you know what a vector is, can kind of grasp what a vector space is, and have some good intuition on how matrix-vector and matrix-matrix multiplication works, then you can probably make a reasonable start on this great intro book: https://www.amazon.co.uk/Quantum-Computing-Computer-Scientists-Yanofsky/dp/0521879965
   
   Start from the start, take it slowly, and do all of the exercises. Not some of the exercises, do all of the exercises. If you don't know a term, then look it up on wikipedia. If you can't do an exercise, look up similar ideas on Google and see if you can muddle your way through. You need to get good at not being scared of mathematics, and just pushing through and getting to an answer. If there is an explanation that you don't understand, look up that concept and see if you can find somebody else's explanation that does it better. Do the first few intro chapters, then dip in to some of the other chapters to see how far you get. You want to get a pretty good coverage of the topics in the book, so you know that the topics exist and can increase your exposure to the math involved.
   
   
3. If you manage to get through a reasonable chunk of the book from point 2), then you can make a start on the bible: Quantum information and computation by Nielsen and Chuang (https://www.amazon.co.uk/Quantum-Computation-Information-10th-Anniversary/dp/1107002176/ref=pd_lpo_sbs_14_img_1?_encoding=UTF8&psc=1&refRID=S2F1RQKXKN2268JJF3M2). Start from the start, take it slowly, and do all of the exercises.
   
   Nielsen and Chuang is not easy, but it's doable if you utilise some of the techniques I mention in point 2): Google for alternative explanations of concepts that the book explains in a way that confuses you, do all of the exercises, and try to get good coverage throughout the whole book. Make sure you spend time on the early linear algebra and basic quantum chapters, because if you get good at that stuff then the world is your oyster.
   
   Edit:
   
   Just remembered two more excellent resources that really helped me along the way
   
   A) Quantum mechanics and quantum computation, a video lecture course by Umesh Vazirani (YouTube playlist here) is fantastic. Prof. Vazirani is one of the fathers of the field of quantum computing, with a bunch of great results. His lecture course is very clear, and definitely worth devoting serious attention to. Also, he has a wonderful speaking voice that is very pleasant to listen to...
   
   B) Another lecture course called "Quantum Computing for the determined", this time given by Michael Nielsen (YouTube playlist here). In my opinion Nielsen is one of the best scientific communicators alive today (see also his unrelated discourse on neural networks and machine learning, really great stuff), and this series of videos is really great. Communicating this sort of stuff well to non-practitioners is pretty much Nielsen's whole jam (he quit academia to go on and write about science communication ), so it's definitely worth looking at.

Hi, I work on programming quantum computers. I studied in Canada (PhD from the Institute for Quantum Computing at the University of Waterloo) and I now live and work in Sydney, Australia. Your TL;DR is actually a bit different from the rest of your post, so I'll answer the questions in the TL;DR first.

I started getting interested in quantum computing when I was an undergraduate in Physics. I began with an interest in quantum entanglement and did a couple of summer research projects in the subject. I did my Master's degree with my supervisor for the last of those projects, and even wrote my first paper based on that work.

Quantum entanglement is of course very important in quantum computing but the study of the subject is more under the heading of quantum information theory. I switched over to quantum computing when I was deciding where to go for my PhD, and decided that I wanted to do the PhD to answer one simple question to myself: how far away are we, really, from a quantum computer? While I was finishing my PhD, the opportunity in Sydney came up and I decided that I liked the work happening here. I was (and am) interested in simulating quantum fields on a quantum computer, and have gotten interested in simulating physics in general (doesn't have to be quantum) as well as solving problems on a quantum computer in general (doesn't have to be physics).

We're talking about close to half my life at this point, so it's hard to summarise that story in any reasonable way. But if I had to try, I'd say that I followed my nose. I was interested in stuff, so I found ways to learn as much as I could about that stuff from the best people I could find who would give me the time of day or, better yet, a pay check. One of the nice things about doing science as a student is that there are plenty of people willing to pay you to study science if you know how to ask nicely.

Training a scientist is a long and arduous process, from the perspective of the student, the teacher, and the society as a whole. Take your time to learn properly. Don't let the bumps in the road stop you!

With the motivational stuff out of the way, my best advice is to learn everything. I mean everything. Physics, maths, computer science, engineering, chemistry, philosophy, sociology, history, everything. I know you can't possibly become an expert in all of that, but get at least a passing knowledge in whatever strikes your interest. When you hit on the thing that you simply can't stop thinking about, the thing that you literally lose sleep over, then you've found the topic for your PhD thesis. Find a supervisor and work on that as hard as you can for as long as you can until they tell you to get out and get a real job.

If that's not the advice you're looking for, then I'll try another piece. Go study functional analysis. You can't possibly understand quantum physics without knowing some functional analysis. If you're serious about quantum physics, this is now your bible. And when you give up on that book (and you will give up on that book), read this. When you're done, read this.

I feel like I gained traction coming from statistics by ping-ponging between these three books. Nielsen and Chuang is a great place to start, especially the first two chapters. There’s a lot that will go over your head but you will pick up enough. Then Aaronson like you have been doing for a different perspective. Then McMahon holds your hand a bit on the computations, which will help if you aren’t familiar with quantum mechanics, as I was not. When you get stuck, switch books. I feel like once I bought all three books and started going back and forth and reading previous chapters again that is when things started to click and I gained some maturity. I have a long way to go but this has been the greatest self-learning journey I’ve been on in the past year. I hope you get as much as I have. Good luck.

https://www.amazon.com/Quantum-Computation-Information-10th-Anniversary/dp/1107002176/ref=nodl_

https://www.amazon.com/Quantum-Computing-Explained-David-Mcmahon/dp/8126564377/ref=mp_s_a_1_fkmrnull_1?crid=382OF32JOGTRH&keywords=quantum+computing+explained+mcmahon&qid=1551223235&s=gateway&sprefix=quantum+computing+explained&sr=8-1-fkmrnull

If you have a good grasp on classical computer science, pick up a copy of this. There are numerous PDFs around the internet, pretty sure there is one on libgen.io. It has been sort of the "bible" of quantum computing for years now, and will teach you everything you need.

This will give you enough of an understanding of what kinds of problems QCs are good at, and how they work on the logical level. You'll be easily able to start writing code that can run on any of the current quantum computers, and understand languages like Q# more easily. You'll also be able to, if you're into this sort of thing, design your own algorithms.

The only thing I'll mention in terms of prerequisites for the book is: If you have linear algebra down, awesome, that will help. If you have dealt with complex numbers/complex exponentials before, even better. If you have some understanding of probability and statistics, even BETTER. You don't NEED understanding of those topics to learn from the book though - it covers them to the necessary degree. You'll be better off if you study those things along the way though.

I began by just plowing through wikipedia articles and trying to get the basic concepts in broad strokes. I would suggest that you do that.

I found this book, that is totally awesome: Quantum computing for computer scientists

Quantum computing for the determined, already mentioned here, is also awesome.

There is also IBMs quantum experience web site, that contains a front-end for programming a real 5-qubit quantum computer. It is free of charge, you just need to register with an email address in order to access it. Soon (as in a few days) there will also be a 16-qubit computer to play with at that site. Playing with the real thing is very good for study morale, at least for me..

Full disclosure: I'm a software engineer at Microsoft

Here's a few resources I found useful. I just started learning quantum computing recently too:

• Book: Q is for Quantum by Terry Rudolph. A great introduction for those without a maths background.
  
  
• Textbook: Quantum Computation and Quantum Information. It’s already been suggested for good reason, and +1 for the comment to ask if you get stuck with anything.
  
  
• Youtube: Professor Vazirani’s online lecture videos. I think he’s great.
  
  
• Online course: MIT Quantum Information course taught by Shor and Chuang
  
  
• Meetups: Find a meetup in your area if you can, I've been going to the London Quantum Computing meetup for any Londoners reading this
  
  
  Shameless plug time (sorry), I’ve written a blog that aims to explain the background material for a quantum “hello world” and show you how to write that in Q#. I'm learning too so would love your thoughts/feedback/hate mail if you do give it a read.
  

Those are two very different areas. More succinctly, do you have a passion for how to use QC or how to build them? QC are really only good for very specific tasks, so research in how to use them is desperately needed. From a sheer number of job opportunities (in the long run), I think ML will eventually provide more jobs.

I found this book to be a great intro to ML on QC: https://www.amazon.com/Quantum-Machine-Learning-Computing-Mining/dp/0128100400

Look at research done by https://www.researchgate.net/profile/Maria_Schuld and https://www.researchgate.net/profile/Peter_Wittek

This page has some decent resources https://codeforces.com/blog/entry/65063

Also there is a free QC MIT course https://ocw.mit.edu/courses/mathematics/18-435j-quantum-computation-fall-2003/

Last but not least I am trying to put together a QC learning resource https://stevefroehlich.github.io/ I have a graduate degree in CS so I'm trying to make it a resource for people like us that come from a CS background. I picked up the standard text book https://www.amazon.com/Quantum-Computation-Information-10th-Anniversary/dp/1107002176?SubscriptionId=AKIAILSHYYTFIVPWUY6Q&tag=duckduckgo-ffab-20&linkCode=xm2&camp=2025&creative=165953&creativeASIN=1107002176 and realized I am missing some of the core Linear Algebra concepts (Basis, Vector Space, Hamiltonian matrix, ect) so that is where my site starts. Its a work in progress and should get better/more helpful as I add more to it.

I would also like to mention "Quantum Mechanics: The Theoretical Minimum" by Leonard Susskind.

Lectures are available online on YouTube.

https://www.youtube.com/playlist?list=PL701CD168D02FF56F

Lectures go nicely with the book of same name.
Book: https://www.amazon.com/Quantum-Mechanics-Theoretical-Leonard-Susskind/dp/0465062903

I found this book and lecture series a nice and gentle entry into the field. Sort of like preparation for Mike and Ike.

The best introductory book on quantum computing is, imho, is Quantum Computation and Quantum Information by Nielsen and Chuang.

Despite the name, you don't actually need much quantum mechanics to get started and we teach it to comp sci students without any background in QM. I wouldn't bother with books on QM directed towards physics students.

As what you do need; a good understanding of calculus and linear algebra. Also, comp sci concepts like algorithms, gates, circuits, complexity theory.

As a beginner myself what is helping me most is the book Quantum Computing Explained by David MacMahon. It has almost no Physics but is pretty heavy on the Math.

I have also started to look at these online resources and find them promising:

• Quantum Machine Learning
  
• Quantum Computing for the determined
  
• Lectures on Quantum Computation
  
• Quantum Computing for the very curious
  
  I am also taking MIT's online quantum computing series that has a 4 courses with lectures from Prof. Shor and Prof. Chuang among others. Unfortunately I am not getting much out of it. The most heavily recommended book on QC is Neilsen & Chuang but again it is not working well for me. It is too abstract with not enough examples to really explain the concepts. Of course YMMV :-)

Quantum games. It is a simple topic and might be quite interesting for young students. As a reference you can use paper by my collaborator:
An invitation to Quantum Game Theory

There are also two recent papers about the subject (both paywalled)

• Using Quantum Games To Teach Quantum Mechanics, Part 1
  
• Using Quantum Games To Teach Quantum Mechanics, Part 2
  
  A very good book that presents a simple introduction to quantum computing is one by David McMahon.
  
  You can use simulators as a teaching aid - a list of them you can find on quantiki.org
  
  Hope this helps.

There was a similar post a while back and I listed out some resources. Here they are, edited for your case:

Intro book (good to start with or use as a companion in your case): https://www.amazon.com/Quantum-Computation-Information-10th-Anniversary/dp/1107002176 (you can probably find this online)

Youtube playlist by Nielson: https://www.youtube.com/playlist?list=PL1826E60FD05B44E4

Quantum cryptography: https://www.edx.org/course/quantum-cryptography-0

Understanding the basics of how computers works: https://www.coursera.org/learn/digital-systems

I added a basic course for digital systems and how computers work. I guess you don’t need any intro courses on Quantum physics. The cryptography course really starts off with basic stuff and it’s easy to pick up.

I would recommend just getting started right away with these and looking for resources on classical computing as you make your way through the resources. You can hit me up if you want to get started with something more advanced like quantum error correction and fault tolerant quantum computing.

I think that this is fantastic. Maybe not popular, but really great.
https://www.amazon.com/Foundations-Quantum-Mechanics-Photons-Computers/dp/0763776289

Useful resources for getting into quantum computing:

• Intended-for-normal-people-but-not-too-deep youtube videos. Usually stuff talking to researchers is leaps and bounds better than anything else. For example:
  
  • PHD comics animating John Preskill and Spiros Michalakis talking about quantum computers
    
  • Veritaseum interviewing Andrea Morello as part of a video
    
• The youtube video series Quantum Computing for the Determined. Twenty two videos, each 10-15 minutes long, done in a Khan academy style.
  
• A simple quantum circuit simulator to fiddle around in.
  
• Online pdf of the textbook Quantum Computation and Quantum Information, also available more legitimately from Amazon.

...also I recommend to you the textbook by Kaye, Laflamme, and Mosca. Alternatively, if you're feeling daring you can pick up the QC bible. And if you're feeling too cheap to buy books, you can find lecture notes from a lot of QC courses posted online. Check out John Preskill's website, or maybe MIT open courseware.

Nielsen and Chuang is the standard reference, as mentioned by others. I think that the book of Kitaev, Shen, and Vyalyi is a better introduction from a computer science point of view.
https://www.amazon.com/Classical-Quantum-Computation-Graduate-Mathematics/dp/0821832298

If you want some lectures to watch, I recommend the EdX course on quantum cryptography by Thomas Vidick and Stephanie Wehner
https://www.edx.org/course/quantum-cryptography-caltechx-delftx-qucryptox-0.

Like if you treat it as a black box? Hmmm I think this book (https://www.amazon.ca/Introduction-Quantum-Computing-Phillip-Kaye/dp/019857049X) might get into it, though it has been a while since I read it.

Here's a basic example of two-photon quantum computing:

You have an ion (quantum dot). Shoot it with a photon that either goes down the channel to it, or goes somewhere else. Now, shoot the same quantum dot with another photon. If the dot is excited, then stimulated emission will occur, sending two photons down the line. If the dot is NOT excited, then the second photon will simply be absorbed.

This is the most basic implementation of a CNOT gate; if a |0> is in the top wire, the qubit in the bottom will be flipped. If a |1> is in the top wire, it will be flipped.

If you're really interested in these sort of implementations and the algorithms involved in QC, check out this book.

https://www.amazon.com/Quantum-Mechanics-The-Theoretical-Minimum/dp/0465062903/ref=pd_sim_14_4?ie=UTF8&refRID=1TT0F2A70RBV36TT3D0S Is supposed to be good, as is https://www.amazon.com/How-Teach-Quantum-Physics-Your/dp/1416572295 . Then you can work up to https://www.amazon.com/Introduction-Quantum-Mechanics-David-Griffiths/dp/0131118927 and then the gold standard for an introduction to quantum computing is https://www.amazon.com/Quantum-Computation-Information-10th-Anniversary/dp/1107002176

I haven't entirely read this book, but I remember that it explains Grover's algorithm well. It might be what you are looking for because it states that it's target audience is undergraduate computer science students.

Mike and Ike is still a great intro. Restricting to finite-dimensional systems really does let you work with concrete representations that cover most of the unintuitive bits of quantum mechanics with mere linear algebra.
https://www.amazon.com/Quantum-Computation-Information-10th-Anniversary/dp/1107002176/

Quantum Computing for Everyone - Chris Bernhardt/MIT PRESS

Read Aaronson's book first

Specifically written for high schoolers by their physicist father:

http://qisforquantum.org/

https://www.amazon.com/dp/B074DYJTKN/

About the Book

COMPUTING. ENTANGLEMENT. REALITY. Books containing these three words are typically fluff or incomprehensible; this one is not. “Q is for Quantum” teaches a theory at the forefront of modern physics to an audience presumed to already know only basic arithmetic.

Is it an o’reilly one ? Programming Quantum Computers: Essential Algorithms and Code Samples https://www.amazon.co.uk/dp/1492039683/ref=cm_sw_r_cp_api_i_Ie8NDbXAPRKRP