University of California - Introduction lyrics

UMESH VAZIRANI: Hello. I'm Umesh Vazirani at UC Berkeley. And I'm delighted to welcome you to this course on Quantum Mechanics and Quantum Computation. I'm sure many of you know that quantum computation starts with this remarkable discovery that quantum systems are exponentially powerful. So a major goal of quantum computation is to harness this exponential power to solve interesting computation problems. So in this overview I want to tell you about what you can expect to learn from this course, and how this course is organized. OK. So let's start with trying to understand more precisely what it means when we say that quantum systems are exponentially powerful. Imagine that we have a small of quantum system over a few hundred particles, like a few hundred electrons, or photons, or something. So let's say, for definiteness we had a system of 500 particles. Now if we could harness all the computational power inherent in the system that quantum mechanics promises us, then in each cycle of the resulting quantum computer we would be able to carry out exponential in 500s, say 2 to 500 steps. Now how big is 2 to the 500? It sounds like a large number. But the interesting thing is 2 to the 500 is and possibly large number. So 2 to the 500 is much larger than estimates for the total number of particles in the universe. It's also much larger than estimates for the age of the universe in femtoseconds. In fact, it's much larger than the product of these two quantities. So what this means is that if you could harness this computational power, then there's no way that in the cla**ical universe we couldn't match it, even if you were able to use the entire resources of the universe in that computation. But now, of course, the difficulty lies in harnessing this power. And there are several challenges. And these are the challenges we'll speak about in this course. So first we have to pick the right computational problems. So not every computational problem can be sped up by quantum computation. Probably the most famous example of a computational problem that can be sped up is what's called the factoring problem, where you're given a number n and you want to write it, factorize it into its prime power factors. ------------------------------------------------------------------------------- ------------------------------------------------------------------------------- OK. Now even if you have the right computation problem, designing a quantum algorithm is a tricky task. And actually, for those of you who are already familiar with the design of cla**ical algorithms, quantum algorithms look very, very different. And they have very different design principles. So we'll talk about things like the quantum Fourier transform, and a completely new style of designing algorithms. We'll also speak about, of course, what are the limits of quantum computers? What are those problems that cannot be solved, or we believe cannot be solved quickly, even if we had quantum computers. And then there's, of course, the challenge of building a quantum computer. So this is something that hundreds of scientists are working on around the world today. It's a very difficult challenge. And in this course I'll just touched upon, briefly, the kinds of systems and the principles that go into designing such quantum computers. So that's what we'll cover in quantum computing. But of course, in order to study this, you'll have to learn the basics of quantum mechanics. And this brings me to the other part of the course, which is an introduction to quantum mechanics. Now the way we'll study quantum mechanics in this course is in terms of a very simple building block, which comes from quantum computation, which is that of a qubit. So just as a bit, it's the simplest representation of information in the cla**ical world. A qubit is the simplest quantum system that we can think of. And describing quantum mechanics, the basic principles of quantum mechanics, in terms of qubits, greatly simplifies the presentation. So what this would mean is that within three to four weeks we'll be in the position where we can start studying quantum computation, the basic notions of quantum computation, as well as quantum algorithms, designing quantum algorithms. ------------------------------------------------------------------------------- ------------------------------------------------------------------------------- Now there's another advantage to studying quantum mechanics this way, which is that we'll be able to jump right into some of the most counterintuitive aspects of quantum mechanics. So in particular, we'll right away, probably the second week of the course, start talking about entanglement, which is one of the most mysterious aspects of quantum systems. And once we are into entanglement, we'll actually talk about various manifestations of it. We'll talk about bell inequalities, and we talk about things like quantum teleportation. And so very quickly, very soon in this course, you'll start grappling with the counterintuitive aspects of quantum mechanics. This is quite important because, of course, quantum computation exploits the most counterintuitive aspects of quantum mechanics. Now, there's a sense in which this way of learning quantum mechanics might actually be a good thing, independent of whether you are interested in quantum computation or not. So here's a quote from Niels Bohr, who is the famous physicist who discovered the structure of that. And he talks about how quantum mechanics is a very counterintuitive theory. And anyone who's not shocked by quantum mechanics has not understood it. Another way of understanding this quote is that, if you really want to deeply understand quantum mechanics, then you have to grapple with the quantum counterintuitive aspects of the theory. And so for those of you who haven't really studied quantum mechanics before, this way of approaching it, this emphasis on the one simple systems, which illustrate the most counterintuitive aspects of the theory, this might be the right way to start studying the subject. And then later if you're interested you can go on to take a standard course in quantum mechanics to learn the physics of the hydrogen atom, et cetera. And for those of you who've already studied quantum mechanics, this treatment might deepen your appreciation of quantum mechanics. -------------------------------------------------------------------------- -------------------------------------------------------------------------- So this finally brings me to the required background for this course, and the teaching philosophy of the course. So in terms of required background, we've really tried to design this course to make it as broadly accessible as possible. So to people from computer science, physics, math backgrounds. And so the prerequisites have been pared down to the minimal possible prerequisites. So basically there are two prerequisites. The main one is that you must have a solid background in basic linear algebra. And the second requirement is pretty simple one. You should be able to an*lyze the running time, you must have seen how to an*lyze the running time of a simple algorithm. You know any simple algorithm, like sorting, or multiplying integers. How do you count the number of steps of the algorithm as a function n, the size of the input? So something very basic. OK let me also say a few words about the philosophy of the course, or how it's going to be taught. So there's one interesting aspect of it, which is what I call a Kanban approach to mathematical notation of math concepts. So this is an approach to manufacturing that the Japanese had. A Kanban means just in time. It's a just in time approach. So in the '80s they came up with this approach where, instead of creating large inventories of raw materials and parts, what they would do is make the inventories as small as possible and have these parts supplied just in time. This made things much more efficient. OK. So what I after here is that you're going to be seeing a lot of interesting new concepts, and many of these concepts are going to be paradoxical. And understanding them, wrapping your mind around these concepts and being able to intuitively understand them is going to be quite challenging. And so what I want to do is, I don't want to overload you with mathematical notation at the same time that you're grappling with these concepts. What the course will do is it will adopt Kanban approach to mathematical concepts and notation. And so, to the extent possible, when a new concept is introduced, I'll introduce it as naively as possible. So that you're confronted with it, and you can build an intuition for it. And then, of course, things will be made precise. I'll push them off as late as possible. But in a way that it's still understandable, and everything can be made precise. Finally, let me just say that probably the most important thing that you can bring to this course, make sure you bring your imagination and your ability to think, grapple with these concepts, some of which are going to be quite mind bending. So it should be an exciting eight weeks. And I hope you enjoy it.