#teapot #test #quantum #computers

So this lecture is part of an online course on the theory of numbers in fact it’s um 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 and so the background for this is that last lecture 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 um however shaw 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brings us to the problem of how good are current quantum computers so obviously this lecture is probably going to be out of date in a few years time because quantum computers are improving all the time so this is a talking about what quantum computers are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 um 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 um less misleading um now it’s actually quite difficult you get hold of a quantum computer they tend to cost several million dollars and they’re 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 super computers 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 so as 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 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 so the teapot can beat any digital computer at this problem however it does undoubtedly occur to you that claiming a teapot is an advanced computational device is 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 a 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 that’s easy i just administer an intelligence test 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 has been very specially selected to make quantum computers look good and that’s just pointless so um 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 in about quantum computers um 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 it 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 um 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

Um so um this doesn’t actually mean that the tests are of quantum computers are useless um 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 um what they were designed for was to check whether quantum computers are really doing something quantum in their calculations and not just that 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 um let’s have another example of applying the teapot test

Um 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 um research as 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 passes the teapot test so first of all current quantum computers are actually quite small they are somewhere between 50 or 2000 cubits 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 um this computer probably consults a quantum computer regularly in order to solve some little optimization problem there’s nothing wrong with this this is a this is a perfectly reasonable setup

For quantum computer you have it as a sort of oracle for digital compute and the digital computer does all the routine stuff and the oracle computers occasionally will solve the really difficult problems so um so first of all we don’t have a quantum computer we have a quantum

Computer sitting on top of a digital computer um the second observation is come let’s be serious um traffic directing traffic is not brain surgery optimizing traffic is a really straightforward 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 rush hour 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 um so um this is not passing 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 um we can take the problem

Of um factorizing large numbers so um if you take a number with say a hundred or a thousand digits if a quantum computer manages 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

Prime integer factorization past the teapot test in order to answer this we have the following question um 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’ll be going back to serious number theory and probably talking about quadratic reciprocity

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