Lesson 01: Single Systems | Understanding Quantum Information & Computation 

Top tier education like this being free is not a given, I want to thank you from the bottom of my heart as this is amazing!

Finally the rigorous approach we needed and with beautiful animations.Thank you Qiskit.

I think this series can easily become a cult in the quantum information education space: concise and simple to digest, even when the topic is elusive for the classically formed brain. Thank you IBM for making quantum this accesible.

What a terrific lesson! His explanations are clear and complete. He does not leave me wondering how conclusions are drawn because he lays it all out so clearly. I might have thought that this would make the lesson tedious, but the opposite is true. Because the lesson is so clear, the material just flows, and an hour and 10 minutes is over in no time at all. Finally, he does make a few comments about deeper things that he does not prove. But those will be covered in future lessons, he assures us, or with a little extra effort, I can discover them on my own. In this way, he gives little victories to his audience. Quite remarkable, thanks.

When someone explains you the meaning of the math used in physics or any science, the subject becomes doubly interesting. I am sure this lecture series will fit to that category. Thanks to the creators of the course and looking forward to the journey of this lecture series.

He has such a soothing voice and elucidates concepts in such a lucid manner!

Before adding a comment, decided to read a few others below. As it happens, they ALL say exactly what I wanted to say. Every time I come back, especially when I pause the video and actually "do the math(s)", I "get" something new that was only vaguely (or more likely, not to any degree at all) understood. Such a privilege to have this freely available wealth of the real kind of deep learning to digest, each at our own pace and foundation background (or even lack thereof). Thank you for being such a clear and cogent guide to us grasshoppers. Long live the Copenhagen Interpretation!

The mathematics is usually put aside or ignored when this topic is popularly discussed - so correcting this omission with this series will prove to be extremely interesting and important. Can't wait for more.

Really wonderful explanation of the basis of quantum managing information and operations. John Watrous is very clear in his words and in examples shown. Thank you to all the Qiskit team for releasing this educational jewel for free.

I'm a quantum entusiast for some time now, I've seen a couple attempts to explain quantum in simple but very precise terms. This is by far the best explanation I've seen. It's step by step, no skipping because something will be covered later or is too difficult, which made me have multiple "aha!" moments. Thank you very much for this and kudos to Mr Watrous!

This is awesome! How he begins with Classical Information and smoothly guides into Quantum System Information, explaining key Quantum information concepts along the way, is very beautifully done. It is very easy to understand and digest. Thank you so much for teaching this in a very clear and concise manner. Thank you, John!

Such a great teacher. Perfect pace and explains everything completely without overdoing it. I really appreciate the scaffolding for future topics.

This is masterfully taught. Absolutely awesome course.

I appreciate this in depth response to what I've questioned for a while and sought a thorough and concise explanation of. You being the only educator who I've found that has offered straightforward, complete, concise and clear information on this topic. your ability to conceptualize and clarify the essential necessary knowledge base without attempting to over simplify it is refreshing and helpful. Grateful and thankful for sharing your knowledge on the matter to all interested. Grateful 🙏👌

You sort of explained here in 2 minutes what I couldn't understand after hours and hours of the MIT course. Thank you!

One of the best explanations so far. Kudos to Prof. John Watrous for breaking the course down to dummies 👍

I've been longing for a in depth explanation of Quantum Computing and you provided it flawlessly. I'm excited for the next installment!

When talking about the Euclidean norm I think it's helpful to mention that when we multiply the coefficient with it's complex conjugate. For example in the (1+2i)/3 |0> - 2/3 |1> so "the absolute value squared" mentioned means that we would take (1+2i)/3 * (1-2i)/3 = ( 1^2 - (2i)^2 )/9 = 5/9 I'm just putting this comment here incase it's helpful to anyone trying to figure out how to get 5/9 😃 Thank you for putting this content out! It's very well done!!💥👌

You are a genius!!! I've always been frustrated at the notation part, but you made me to proceed to the next step! Thank you so much

Amazing lecture. So clear and engaging. Thank you!

Thank you,honestly speaking such an excellent basic approach is not followed by other courses ,this one hour video will save people days of not having understood more complex theoretical concepts in the future.

One of best explanation by sir !! It's people like prof John Watrous who explains these complex topic in a easy and precise way thank you qiskit

Thanks a lot IBM, qiskit and professor Watrous for this amazing course!

I have so much respect for this man here. He was able to teach for an hour straight with crystal clear information. Not sure if he had a teleprompter though

Fantastic! I always wanted to know why we had to use unitary matrix and you finally explained it !

Perfect mathematical intro to QC at the very basic level! Thanks a lot 👍

Nicely presented. Looking forward to the series. Thank you!

Thank you for all the videos always very interesting and well explained!

Excellent explanations, clear and simple. Thank you.

Amazing lecture and a crystal clear knowledge. I am very happy I discovered Qiskit. And now I need more. :)

Excellent Lecture, I am mind blown. Looking forward to build up!

A very good lecture. Hope to watch lecture 2 ASAP. Thanks to Prof. John Watrous giving sixh an interesting lectures.

Discovered just today this very interesting introductory lesson: I think I will follow the entire course together with the reading of the M&M's book. Great! Many thanks to Qiskit.

🎯 Key Takeaways for quick navigation: 00:00 🎓 The video series aims to provide a comprehensive understanding of quantum information and computation, focusing on the technical details of quantum information and its applications. 01:02 🔬 Lesson one focuses on quantum information for single systems, laying the foundation for understanding quantum information for multiple systems and quantum algorithms. 02:01 🔄 Classical information serves as a starting point to understand quantum information, with quantum information being an extension of classical information. 03:41 🧪 There are two descriptions of quantum information: simplified and general. The simplified description focuses on vectors and unitary matrices, while the general description is more powerful, including density matrices and noise modeling. 08:02 📊 Classical states are configurations that can be unambiguously described, and they are represented by a finite set called sigma. 11:41 🧪 The Dirac notation is introduced to describe vectors, using "ket" notation for classical states and standard basis vectors. 15:51 🔍 Measuring a system in a probabilistic state results in knowledge of the classical state with probabilities transitioning to certainty (probability 1) for the observed state. 19:30 🔄 Deterministic operations on classical systems are described by functions and corresponding matrices, where the output depends entirely on the input classical state. Matrix-vector multiplication can represent the effect of deterministic operations on probabilistic states. 24:35 🧮 Quantum information can be represented using column vectors called quantum state vectors with complex number entries, and the Euclidean norm of these vectors must equal one. 27:12 🔄 The inner product, or bracket, of a bra vector and a ket vector is an important concept in quantum information, and it represents the multiplication of a row vector and a column vector. 31:18 🎲 Probabilistic operations in quantum information can introduce randomness or uncertainty, and they are represented by stochastic matrices, which are matrices with non-negative real entries that sum to one in each column. 35:41 🔄 Composing probabilistic operations in quantum information is done by multiplying the corresponding stochastic matrices in the reverse order, and the order of operations matters. 40:03 🌌 Quantum states are represented by quantum state vectors, which are column vectors with complex number entries, and their Euclidean norm must be equal to one, making them unit vectors. 46:35 🃏 Quantum state vectors can represent quantum states of various systems, not just qubits, and they satisfy the condition that the sum of the absolute values squared of their entries equals one. 48:16 🧬 Dirac notation can be used for arbitrary vectors in quantum physics. Kets represent column vectors, and bras represent row vectors. Any name can be used inside a bra or ket to refer to a vector. 49:18 🧩 When using Dirac notation for arbitrary vectors, the bra vector is the conjugate transpose of the corresponding ket vector. This involves transposing the vector and taking the complex conjugate of each entry. 51:22 📊 Measurements in quantum systems provide a way to extract classical information from quantum states. Standard basis measurements are the simplest and most basic type of measurement. 52:28 📈 The outcomes of a measurement in quantum systems are classical states, and each outcome has a probability associated with it. The probabilities are the absolute value squared of the entries in the quantum state vector. 55:08 🌌 When a quantum system is measured, its state may change, and the new state will be the one corresponding to the classical outcome of the measurement. 56:13 🔀 Unitary operations in quantum physics are represented by unitary matrices. These matrices describe how quantum states of systems can be changed. Unitary operations preserve the Euclidean norm of quantum state vectors. 59:28 ⚙️ Compositions of unitary operations are represented by matrix multiplication, and the order of multiplication is from right to left. Unitary matrices are closed under multiplication, resulting in another unitary matrix. 01:09:36 🔄 The combination of Hadamard, S, and Hadamard operations gives rise to a square root of NOT operation, which is an example of how quantum operations behave differently from classical operations.

Thank you for this! And very helpful that you are explaining complex details with clear and great examples.

There is a funny convention about firsts and the number one: usually we associate them with each other. "1" reprents the first of a series, and the first of a series is identified with the digit "1." I suppose this is some sort of daring innovation by IBM here, using the numbering "01" to identify the second in a series and telling us to go find the first video by ourselves. Congratulations, IBM! It's wonderful to see this pioneering spirit here in RU-vid.

This is an amazing introduction for me and it came at the right time since I'm taking a quantum computing course this winter

Plain and simplified, Thank you Qiskat 😁

Awesome and extremely professional!

thank you for the beautiful explanation of a complicated topic. I'm new to quantum computing and this material helps a lot!

I am finding it super helpful to watch this lecture in parts, and then reading up the associated Qiskit textbook sections.

smart speaker with a lucid crisp and simple ecture. thank you. God obless

thanks a lot to creators for giving such relevant content

EXACTLY what I needed, thank you so much

Thank you for this very clear and easy to folllow exposition.

Thank you very much. I am looking forward to the next lessons

Thank you so much Qiskit, Mr. John Watrous

Nice. Looking forward to the next lessons.

Dr. Watrous, just wanted to say I really appreciate the note you added (starting at 47:43) about the symbols being put inside ket being ambiguous (not always depicting a classical state). This is something not mentioned at all generally and something that always confused me whenever I read any QC material.

Very good explanation, for sure I will continue. Thank youu

Very clear presentation.

Thanks, I finished reviewing it 3 times😊

I highly recommend Needham's book on _Visual Differential Geometry and Forms._ There we find that Dirac's bras are _1-forms,_ which come to us from the mathematician Hodge, whose work was inspired by Maxwell's equations. (Chapter 32 3.5) I have often argued that we need to have a closer look at electromagnetism (EM) in order to get on with machine vision. Hodge shows us the way.

Thank you Qiskit for this ! ❤️

Thank you !! Great teacher

Muchas Gracias señor John Watrous.

Highly appreciated.

Great stuff man!

Eagerly looking forward to Lesson 2

great stuff! thank you very much

Great lecture!. But beginners may have to check the link above to explore more at their own pace.

sir, you are making history.

Thank you for this very cear explanations :)

Excellent - thanks!

I really need this ❤

This is just amazing

The definition of classical information for an event "e", is -log(P[e]). Please explain in other next video the information of quantum state |e> or desnity matrix Σ{|e>

It's like watching a new baby being born. It's a real exciting time to be alive, to witness and to be apart of the birth of this new quantum technology. SOSSTSE SCIENTIFIC TECHNOLOGY SOLUTIONS. ❤❤❤🎉🎉🎉

Excellent!

Very informative 1

Thank you!

Great lesson! When future lessons will be available, there is a scheduled program? There Will be also recitations on specific subjects ? Thx a lot!!!

Thank you DR.Watrous ❤

Wow! All very cool, man if only Qiskit were to release all videos in a unit one month apart, with a short break between units. Now that would be something else! 😅

Thank you.

thank you 😊

32:51 I realized it's easier to understand the operations by looking at the matrix as a transformer, with the input from the column vector with first column as input 0 and second column as input 1 and output as the row vectors, with first row as output 0 and second row as output 1.

For those who needs to manipulate (Like me) and before using Qiskit (Which I intend) you may use Python sympy, create matrices ket, H, S gate, T gate, yes with complex number, and so on and play with them. Sure it's good to do it on paper but with symbolic tool it is fun too. And you may verify if you were correct on paper too ... There is also a book Qiskit / IBM which I ordered. Since I'd like to play a little wit a real quantum computer.

The characteristic of an n-dimensional manifold is that each of the elements composing it (in our examples, single points, conditions of a gas, colors, tones) may be specified by the giving of _n_ quantities, the "coordinates," which are continuous functions within the manifold. ~Weyl We express the fact that _n_ parameters are necessary and sufficient for a unique characterization of the configuration of the system by saying that it has "_n_ degrees of freedom." ~Lanczos

good intro. when are other lessons coming?

It is hard to start but once we get it, it is pretty easy

On "deterministic operations", how did you arrive at the Matrices M1 to M4? Time: 24m.27s

Thank you IBM

thanks

Was the matrix described @24:02 correct on the screen? Didn't seem to tie up with what was being said.

Of course, with matrices describing operations, you can map every input to every output. But the downside is, that they usually get very big. Especially when Tensors-/Kroneckerproduct come into play. Without tools like numpy, you're completely lost.

I understood the topics discussed in the video. But when I moved to the Qiskit examples in the reading, I didn't understand anything. Is it because I am lacking in the topics discussed in the video or is it because of my unfamiliarity with the Numpy? Do you think I should rewatch the entire video again?

23:22 I just cant get it? where can find more information to understand how this matrices came out?

I have a question about the quantum information part, when you have a vector ket something, how do you know if it is a standard basis vector or a column vector in general, do you have to understand that based on how it's used?

how often are you going to release lecture?

Could you let us know when the second lesson release will be? What's the release schedule?

Wow the last line of lecture sounded like a magical door is opened of bizarre realities.

Like ket 0...ket 1 shouldn't be read like that ket 1 is the vector that has zero in the entry corresponding to the classical state one, which is the first entry and a 1 for all other entries?

is there any link to the lecture notebook?

Hi sir my self krishna i am a system administrator i am interested in learning quantum computing but any book or videos to remember the basics.

Professor can you please explain ket 0 and ket 1 again. I didn't understand it.

Wonderful .. and, clear.. When is the next lesson foreseen?

I think at 47:40 the state can't be a qstate, simply because the Euclidean norm equals half not one.

Does anybody know what latex font was used in the slides?? It looks so clean, I need to hunt it down at all costs.

You say that /you/ can't give a reason behind this definition of [the vector representation of "pure", finite dimensional] quantum states, other than that physicists have found that it just works, but there is a good reason. The trouble is that after finding that it just works the vast majority of physicists famously "shut up and calculated" and didn't particularly care about understanding why it works and what it really means. Among the few who did care were some mathematical physicists, following von Neumann, who discovered that quantum theory is just a natural algebraic reformulation and generalisation of [the Kolmogorovian model of] probability theory.