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.
The secret is to understand is sometimes to ignore some things....Deal on what we have now and improve on it. Binary in quantum computer does not going to go away for sometime. It only describe the state of ONE ATOM versus a series of atoms describe to produce an effect of a "GATE."
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!!💥👌
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.
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!
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.
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 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 🙏👌
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.
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.
When i was a collage student i was into maxwell's studies and at the same time studying dirac's theorems and i wondered if they are related in some kinda way, so that was the FACT. Thank you so much Brian!! Also, if Maxwell himself would live 10-25 years longer, he could also define quantum physics with his hands way years before in my opinion..
If you have questions of a technical nature, one alternative to leaving them in the comments is to ask them on Quantum Computing Stack Exchange (quantumcomputing.stackexchange.com/). Many knowledgeable people hang out there and can answer questions - and also markdown makes communicating on a mathematical level much easier.
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!
🎯 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 so much for this lesson! It was very clear, simple and easy to understand, but not too easy. Requires some basic mathematics knowledge, but I loved it. Seriously, kudos to Mr. Watrous (& Qiskit of course) for making this video:))
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
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!
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?
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?
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.
At 24:00, the matrix maps the 'ket a' to another 'ket b' = 'ket, f(a)', which is either equal to or not equal to 'ket a'. So, the table is comparing the result of the operation on two different possible 'ket a' states, either 'ket 0' or 'ket 1'. I was quite confused by this at first, thinking that the columns in these charts were elements of the vector corresponding to 'ket a', which is not the case.
@@yukihirasoma3935 They are correct as they appear. If you could explain why *you* think the transposes would be right I would be interested to hear that, because I'd like to understand where the confusion lies. My explanation for why the transpose doesn't work for M_1 is to consider the action of the transpose on the vectors |0> and |1>. We have M_1^T |0> = |0> + |1> (which isn't a probabilistic state because the "probabilities" sum to 2) and M_1^T |1> = 0 (not |0> but 0 - the zero vector - which is also not a probabilistic state). Similar for M_4 - and of course M_2 and M_3 are symmetric so we agree on them. To be clear, I'm talking about multiplying a column vector by a matrix from the left. Could it be that you're thinking about multiplying a row vector by a matrix from the right?
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. ❤❤❤🎉🎉🎉
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.
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.
Confusion I see in the experts. Binary language represents Alphabet and Number system. Quantum Computer is get data of 0 or 1 on an atom versus several atoms producing a electrical current describe as 0 or 1 depending on the timing.
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.
John thank you very much for taking the time to explain such a complex topic in such an understandable way. I really appreciate your hard work and dedication.
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.
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.
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
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?
at minute 42:18 it is stated that they are the sum of the absolute values..... but they are not absolute values, the ak are complex numbers.... so they should be amplitudes. Moreover it would not make sense to square an absolute value.... Is that so?
The absolute value of a complex number is a well-defined quantity and this terminology is standard. (Magnitude and modulus are also standard terms that mean the same thing in this context.) You can definitely square the absolute value of a complex number - but in practice it's generally simplest to compute the absolute value squared by multiplying the complex number to its conjugate (and then the absolute value without the square is the square root of the number you get, which will always be a nonnegative real number). See en.wikipedia.org/wiki/Complex_number#Complex_conjugate,_absolute_value_and_argument for more information.
I don't understand why you need two classical states to be useful for 'storing' information. If there were just one state, wouldn't that still be something?
I'm assuming that the absence of a state (or system) isn't an option, in case that's what you're thinking about. For example, if you put a sticker that can't be removed on your dishwasher that reads "dirty," it's useless (unless you never run the dishwasher); it only has one state so it can't store any information about the dishes. If it was a magnet, though, it would be a different story, because then I guess you could put it on the counter when the dishes are clean - but I'd say that's a second state of the magnet.
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>
This is a very beautiful and informative series for a general reader. I have a small suggestion for Prof. John: if he can upload the detailed PDF of every lecture as if he has added the detailed text explanation, then it will be more helpful to everybody. Because everyone can take the printout and read it. @Qiskit
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! 😅
Thanks for this video. Unfortunately, however, you lost me at 20:00 where you introduce the terms "M |a>" and "M(b,a) = ....". I also note that you introduced a large unclosed curly bracket but without explaining what this notation means. As the title of this video starts with the word "Understanding", I would therefore respectfully suggest that all math notation should be explained when introduced so as to help non mathematicians *understand* the content.
I have to start somewhere, and if I started at the very beginning in terms of mathematics and notation it would make for an extremely long video (which of course would cause others to complain). So it's a balance that unfortunately isn't ideal for everyone, and I don't see a practical way to make it so within the video itself. The path forward I suggest is to tackle the things that are giving you trouble one by one. 1. M |a> refers to the vector we obtain by multiplying the matrix M to the vector |a>. 2. M(b,a) refers to the (b,a) entry of the matrix M, meaning the entry whose row corresponds to the classical state b and whose column corresponds to the classical state a. 3. The large unclosed curly brace is a standard mathematical notation for expressing different *cases*. What it means here is that M(b,a) is equal to 1 if b = f(a), and it's equal to 0 if b != f(a) (!= meaning not equal). The examples of the four functions from one bit to one bit that follow starting at 21:49 might help to clarify these things.
... a little por question: (SHS)^2 = X also (HTH)^4 = X ...this means the matrix Ptheta are about something like "rotation"? S=Pπ/2 T=Pπ/4 and if we use the π denominator's value in combination with H I'm back to X (not) Pauli matrix ...
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.
