 So this lecture is part of an online course on the theory of numbers. In fact, it's Going to be a little bit off topic because instead of talking about the theory of numbers. I'm going to talk about a Problem that came up in the last lecture, which is how good are quantum computers and are they better than classical computers? So the background for this is that last lecture we we were discussing RSA cryptography, which is quite often used for sending messages over the internet and We've pointed out that you could crack RSA cryptography if you could factor large numbers where large means Maybe several hundred digits On a classical digital computer, this is still out of reach for numbers with maybe a thousand digits however Sure came up with an algorithm for quantum computers Which can be used for factorizing really large numbers if you've got a sufficiently big quantum computer So this brings us to the problem of how good our current quantum computers So obviously this lecture is probably going to be out of date in a few years time because quantum computers are improving all the time So this is a talking about what quantum computers are like at the beginning of the year 2021 so if you look at recent statements in the news about what Quantum computers can do you can see statements claiming They're you know thousands of times faster than classical computers. They've achieved quantum supremacy they can be used for directing traffic in large cities and so on and All of these statements are not exactly false, but I want to show they're all really rather misleading Maybe in a few years They will be correct. I hope so there are a lot of really smart people working on quantum computers and There seems to be fairly steady progress and there's a lot of money in it So in five or ten years time these statements might be Less misleading Now it's actually quite difficult to get hold of a quantum computer They tend to cost several million dollars in there reasonably large, but I've actually managed to borrow a Device that can do quantum computation and in fact it will beat all Current digital supercomputers at quantum computation. I've just got it here in my room and here it is as you can see it is a small teapot and You may be a little bit skeptical of my claim that this is an advanced computational device. So let me explain In order to test computational devices, you need to choose a problem and the problem I'm really interested in is known as the teapot problem and the teapot problem is the following suppose you take a teapot and You drop it on the floor The teapot problem asks calculate how many pieces does the teapot break into and if you think about it, this is an incredibly difficult problem for a digital computer to solve because You need to simulate a teapot and a teapot has Several million million million million atoms and you need to solve a Schrodinger equation for that and so on and it's It's just way beyond the ability of any current digital computer to solve this problem. However the teapot Can solve in one second? Well, yeah, well since there's a borrowed teapot. I don't think I wish to actually do that but You can imagine it falling down um So the teapot can beat any digital computer at this problem however Does undoubtedly occur to you that claiming a teapot is an advanced computational devices really really stupid So what is wrong with my argument? well The problem with the argument I gave about teapots being advanced computational devices is the first of all the problem I chose is completely and utterly pointless. I mean who cares how many Pieces of teapot breaks into but the major problem is it is highly biased towards teapots It was specially selected to be a problem that teapots are very good at solving but digital computers can't solve So by selecting a problem you can make anything look better than almost anything else You know, suppose I want to prove that an anteater is smarter than Einstein. Well, that's easy I just administer an intelligence test to the anteater and to Einstein and see who does better and the intelligence test I choose is how many ants can you catch in one minute? Okay, yeah, so if you choose the test you can make anything look good and Now let's get back to quantum computers Well, if you look at all the claims about quantum computers being better than classical ones They all suffer from this problem That they're being run on a problem that is been very specially selected to make quantum computers look good and that's just pointless So we have the teapot test If someone claims that a quantum computer is better than a classical computer Does that argument also show that teapots are better at computation than classical computers? And if so, you don't need to take much notice of this claim So let's apply this test to some of the claims in About quantum computers first of all we have this phrase quantum supremacy and You know quantum computers have achieved quantum supremacy. It sounds really impressive That means does that mean they're better than classical computers? Well, no, it doesn't because if you look at the definition of quantum supremacy It turns out to mean that there is some problem at which quantum computers are better than classical ones. Well, as we've just seen this is Sort of useless. You can always find some problem which things are better than classical computers my teapot for example has Attained teapot supremacy over classical computers because there's a problem that can solve better than classical computers So quantum supremacy is a kind of really misleading term It doesn't actually mean that quantum computers are better at anything useful the point is Quantum computers or for that matter teapots may be really hard to simulate on a classical computer that doesn't mean they're useful at computation So This doesn't actually mean that the tests are of quantum computers are useless there are various tests of What quantum computers can do for example one that's been in the news a bit recently is Boson sampling and these are actually really good useful tests for what they were designed for What they were designed for was to check whether quantum computers are really doing something quantum in their calculations and not just that they're not just classical computers and Tests like boson sampling absolutely fine for that. You know that they're testing how If your quantum computer is really doing what it's supposed to be doing and they're also good for comparing one quantum computer to another Quantum computer, but they're absolutely useless for comparing a quantum computer to a classical computer because they're so biased towards quantum computers Let's have another example of applying the teapot test So as I mentioned it's been in the news recently that quantum computers were being Used for traffic flow. It wasn't quite clear whether they were actually being used for directing traffic or whether this was some sort of Researchers being a little over optimistic, but let's assume that quantum computers really are being directed used to direct traffic and Let's see whether this part is the teapot test so first of all Current quantum computers are actually quite small They have somewhere between 50 or 2000 qubits and the number of qubits they have depends on how gullible you are about hype from quantum computer Manufacturers, but let's give them the benefit of the doubt and say we've got a few thousand qubits Well, this is pretty obviously nowhere near enough to simulate a large city And there's just not enough room for all the data So what is presumably going on is there is a digital computer which is doing most of the random routine bookkeeping, you know for measuring traffic and This computer probably consults a quantum computer Regularly to in order to solve some little optimization problem that there's nothing wrong with this This is a this is a perfect reasonable setup for quantum computer You have it as a sort of oracle for a digital computer and the digital computer does all the routine stuff in the oracle Computers occasionally will solve the really difficult problems So so first of all we don't have a quantum computer. We have a quantum computer sitting on top of a digital computer Second observation is come on. Let's be serious Traffic directing traffic is not brain surgery Optimizing traffic is a really Straight-forward routine thing to do that a digital computer can do just fine Even if it doesn't have a quantum computer sitting on top of it In fact, if you think about it Using a quantum computer for traffic flow is a really weird thing to do Here you've got the world's most advanced computational device and All you can think of doing with it is directing Russia traffic. I mean come on, you know You could you give these people a time machine and all they would use it for is going back to watch episodes of some soap opera that they missed so This is not part of the teapot test because although a digital computer plus a quantum computer can direct traffic a Digital computer can also direct traffic just fine without a quantum computer helping it So, you know, I could take this digital computer that directs traffic put a teapot on top of it and Claim that my traffic directing device involves a teapot or I could put a quantum computer on top of it and claim it involves a quantum computer either case it's Not saying anything useful about quantum computers So what is a good test to show that a quantum computer really is doing better than a classical computer? Well, it's one I mentioned earlier. We can take the problem of I'm factorizing large numbers. So If you take a number with say a hundred or a thousand digits if a quantum computer Managed to factorize that without too much help from a digital computer Then you can genuinely claim that quantum computers are now better than digital computers So does prior integer factorization pass the teapot test? In order to answer this we have the following question Does a teapot help you to factorize large numbers? Well, it's completely stupid, of course It doesn't so so this passes the teapot test if a quantum computer can factorize large numbers Then it really is doing better than a classical computer Okay, that's all about quantum computers Next lecture we will be going back to serious number theory and probably talking about quadratic reciprocity