Quantum Computing
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00:00
Quantum field theoryMathematicsStochasticHiggs mechanismRSA (algorithm)AlgorithmNumberCryptographyDivision (mathematics)Serial portFactorizationRange (statistics)CoprocessorFamilyBitQubitCAN busWaveDifferent (Kate Ryan album)LengthMatter waveInformation securityNumberInternetworkingNP-hardNegative numberLine (geometry)Musical ensembleUniverse (mathematics)RootPoint (geometry)Random accessQubitSingle-precision floating-point formatWaveMultiplication signPhysicistVolume (thermodynamics)Operator (mathematics)Series (mathematics)Water vaporComplex numberQuantum mechanicsIntegerReduction of orderType theoryModulo (jargon)Functional (mathematics)Vapor barrierDivisorDrop (liquid)Right angleLoop (music)BitCuboid1 (number)Process (computing)Semiconductor memoryPublic-key cryptographyCoprocessorPrime numberDoppelspaltDivision (mathematics)Logical constantComputerCryptographyQuantum field theoryParallel computingPrime factorPhysicalismQuantum computerSlide ruleComputer-assisted translationTerm (mathematics)Set (mathematics)Parallel portModul <Datentyp>ResultantMultilaterationNatural numberQuantum field theoryComputer animation
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WavePattern languagePhysical systemAuto mechanicQuantum field theoryComplex (psychology)Nichtlineares GleichungssystemScalable Coherent InterfaceSuperposition principleRight angleDesign by contractUniverse (mathematics)WordRotationExecution unitCorrelation and dependenceParallel portWeb pageBitTheoryDifferent (Kate Ryan album)Interpreter (computing)Superposition principleNichtlineares GleichungssystemPoint (geometry)State of matterAlpha (investment)Term (mathematics)Universe (mathematics)Student's t-testDistribution (mathematics)QubitThread (computing)Presentation of a groupComputerQuicksortWordQuantum field theorySuite (music)Similarity (geometry)Form (programming)Point cloudQuantum entanglementAtomic nucleusDiscrete groupAtomic numberPhysicalismWave functionQuantum mechanicsAngleMultiplicationPower (physics)Single-precision floating-point formatMultiplication signWaveFreewareRevision controlRight angleEvoluteQuantum stateComplex (psychology)Structural loadConjugacy classSpacetimeWebsiteMixture modeloutputParallel computingAlgorithmCoefficientQuantum computerCASE <Informatik>Game theoryPhysical systemMeasurementMechanism designLogical constantDisk read-and-write headResultantTheory of relativityInfinityScaling (geometry)MereologyAnalytic continuationSquare numberVirtual machinePosition operatorException handlingComputer animation
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AlgorithmEmailUniverse (mathematics)Prime idealPrimality testFactorizationCompilation albumComputerAbsolute valuePhysicsQuantum computerQuantum field theorySequenceFrequencyWitt algebraFourier seriesSuperposition principleCategory of beingElement (mathematics)Number theorySequenceSeries (mathematics)FrequencyTransformation (genetics)Prime factorGoodness of fitNumberTheoryBitQuantum field theoryUniverse (mathematics)State of matterPhysical systemComputerIntegerModulo (jargon)Operator (mathematics)DivisorQuantum computerMultiplicationInterpreter (computing)Fourier seriesMultiplication signConstructor (object-oriented programming)Category of beingAlgorithmResultantFactorizationSuperposition principlePhysicalismImplementationQuantum mechanicsElement (mathematics)2 (number)Power (physics)Game controllerQuantum stateBuildingDegree (graph theory)Order (biology)Division (mathematics)Different (Kate Ryan album)Core dumpComputer programmingCubeModul <Datentyp>CausalityPrime numberResource allocationCase moddingPoint (geometry)Mainframe computerQubitEndliche ModelltheorieComputer animation
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Menu (computing)BitCycle (graph theory)FrequencyPersonal identification numberDirection (geometry)Fourier seriesQuicksortUniverse (mathematics)Quantum field theoryMultiplicationSet (mathematics)Multiplication signRandom walkDifferent (Kate Ryan album)Arrow of timeCircleFourier transformComputer animation
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MultiplicationUniverse (mathematics)CASE <Informatik>InternetworkingNumberGoodness of fitResultantComputerFrequencyReverse engineeringComputer animation
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Universe (mathematics)TwitterMultiplication signGodComputer animationLecture/Conference
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AlgorithmMaxima and minimaQubitNumberPhysical systemSelf-organizationMultiplication signFault-tolerant systemQuantum computerQuantum field theoryComputer animationLecture/Conference
Transcript: English(auto-generated)
00:06
Thank you, good afternoon. This was a talk I didn't really expect to give because I thought it would be rejected since it doesn't really have any Python in it, or very much Python. But I put it in to encourage me to learn more about quantum computing. And as I learned a bit more,
00:21
I decided to change the subtitle to a gentle-ish glance into a likely future. And then as I actually came to preparing the slides, I decided all I could really possibly aim to do is to explain why everything you've ever read about quantum computing is probably wrong unless you've read the real stuff. So I'm gonna try and give you one or two really important corrections
00:42
to a lot of what I think is misinformation. It's great to see so many people here. I'm assuming some of you are here for, in Edinburgh for EuroPython. Maybe quite a lot of you. That's not actually why I'm here, I live here. I came to Edinburgh in 1986 because I was invited by Peter Higgs, humble brag,
01:00
the guy they built the LHC for to discover the Large Hadron Collider and had the privilege of being taught quantum field theory by him. And I say that not to imply that everything I get wrong is his fault at all because he's a genius, but just to say I have spent a lot of time thinking about this. And what you're gonna find during this talk is at various points, I'm gonna say things that are completely ridiculous
01:21
and obviously wrong. And you're gonna think these things are completely impossible. And you're gonna be reminded of Alice in Wonderland or rather Through the Looking Glass where she said there's no use trying. One can't believe impossible things. I am gonna ask you to believe things that will appear to be impossible. It's entirely up to you whether you do, but it's something that everyone who tries
01:40
to come to terms with quantum physics has to do. And there's no way around it, I'm afraid. If you have read or learned anything about quantum computing, probably the most likely thing you've heard is that if it ever really works, cryptography is in trouble and SSL, the security of the internet is in trouble. And there's quite a lot of truth to that.
02:01
Not complete truth, not all cryptography is in trouble, but certainly cryptography based on the difficulty of factorizing big semi-prime numbers is in trouble which is what SSL is based on and RSA public key cryptography. So yeah, the basic idea with public key cryptography is that it's easy to multiply two numbers together
02:24
but it's very hard to figure out what the two prime factors are if you have a semi-prime number, i.e. one that has exactly two prime factors. But we need to be rather precise. When I say it's hard, that's actually not true at all. It's incredibly easy. In fact, it's slow. We can write just five lines of Python
02:40
that will factorize any semi-prime number. This is only really using two things. It's using the fact that we know that if we have two facts and number one of them can't be bigger than the square root of the number because obviously if we multiply two things that are bigger than the square root of the number, we're gonna get something bigger than the number. The other thing about this
03:00
incredibly simplistic approach to it is it uses one difficult or one slow operation which is this percent operator here which is integer division, right, integer modular division. And division is slow. So if we didn't care about how long each individual step takes, there might actually be a more interesting way to do it
03:20
which would be dramatically less efficient but would parallelize well. So we might just write a function that given our semi-prime number n, our big number n, and two possible factors just returns true if it's true if the two factors multiply together to give it and false otherwise. And then we could just do a double loop, one over the small numbers and one over the big numbers.
03:40
The one's bigger than the square root and one of those if it is indeed semi-prime would come back with our result. And if we had a parallel computer, if we had a really big parallel computer like one with at least n squared processes which is probably really quite big if n's big, then things would be quite nice because we could just on one processor try,
04:01
well, does 15, I know 15's not a very big number but does 15 equals two times three, does 15 equals two times four and so on and one of these would find our answer. So this would be a nice way if we had a sufficiently large parallel computer, I'm sorry, I'm moving away from the mic, if we had a sufficiently large parallel computer to solve the problem. But there is a detail missing
04:21
and it's quite an important detail which is that it's no good I just finding the answer on processor seven especially if we've actually got say 100 quadrillion processors because we're probably not connected to all of them. So what we have to do at the end of the operation is we have to collect our answer. We have to arrange for the answer somehow to filter back to the actual processor we're talking to, the host processor if you like.
04:41
And that's not difficult at all. Even if we have 100 quadrillion, we can do a reduced type operation and reduce them in pairs and get the answer back but it is something we have to remember and that will become important in about 15 minutes. So hold this thought, we're going to do some physics in a minute.
05:01
You probably all know what bits are. I'm sure you all know what bits are. Bits are boxes that contain zeros or ones, right? And if it's a bit in random access memory, then you can set that value to either one or zero and you can read that value back and find out whether it's one and zero and if nothing goes wrong, the same thing that you set it to is what you'll get back
05:21
when you read it back any time later. And so the crucial thing of course is that the same bit in random access memory, in memory we can change, can have different values at different times. Hopefully that's not news to anyone. So what about quantum computing? Well quantum computing is based on quantum bits and quantum bits are boxes
05:41
you can put zeros and ones in as well. It's just a slightly funny shaped boxes that Paul Dirac, great British physicist, designed and called kets. These actually have a very specific meaning but for our purposes, just think of it as a box that can contain a zero or a one or possibly both.
06:00
Not like half but like maybe it's partly a zero and it's partly a one at the same time. And we would write that in fact as one over root two of the ket zero plus one over root two of number one. Why one over root two not half? Well it turns out we square these numbers. In fact we take the, if they're complex numbers which they tend to be, we take the complex conjugate and multiply that
06:22
by the thing to get our probability. But it doesn't matter, it's just the way we write it. But I want to go back to this important point that a single qubit, a single quantum bit can be simultaneously partly one and partly zero. And that's so important I'm gonna shout it as well. A single qubit can be both one and zero
06:43
at the same time in some sense. And you're gonna say, well that makes no sense. This is impossible. And I'm gonna say, I completely agree that makes no sense, it's impossible. It just turns out that's how the universe works. And what I'm gonna try and do is tell you why physicists, even physicists who've thought about this
07:01
an awful long time, whom it still makes absolutely no sense, believe this. And we're gonna take our cue from Feynman, one of the greatest quantum physicists ever, brilliant teacher, who introduces volume three of his lecture series with three experiments. The first of which is an experiment with waves. So think about water, or think about light if you prefer, but I'm thinking specifically about water here
07:21
where we've got something that's bobbing up and down on the water creating waves, or maybe it's a set of drips falling on the water creating waves. And we've got a barrier in front of it that's got a couple of holes in it, in a wall, and then we've got an absorber at the back. And what we know about waves, what hopefully everyone knows about waves, is that they exhibit interference. That is, if we have multiple drops falling on a puddle,
07:42
then the waves around the droplets spread out, but they just go through each other, right? They don't stop, they just go through each other. And at the point where they meet, they exhibit interference. So if two troughs come together, if two troughs, if two peaks come together like this, we get a bigger peak, and that's constructive interference.
08:01
And where a peak meets a trough, they cancel out, and we get destructive interference. No great surprises there. And exactly the same thing happens with light. If we have the same double-slit experiment, we make the slits really close together, then what we'll find is right behind the slits, we'll get this really bright point, because the paths are the same length.
08:22
But if we move a bit to the side, then one of the paths will be half a wavelength longer than the other path. So we'll get destructive interference, and they'll cancel out. And we get this black band. And then when the difference in the path lengths is a whole wavelength again, we'll get back to positive interference and so forth.
08:41
So now we'll do an experiment with bullets. And it's exactly the same setup, just slightly larger. We've got a rattly machine gun. We're not gonna kill anyone, but don't worry, no threads will die in this presentation. Bullets do not exhibit interference. They might occasionally bounce off each other, and all sorts of crazy stuff will happen. But if, when one hole's open, this is the distribution
09:01
that we tend to see of bullets from this rattly machine gun, and this is the other, then we just add up those two distributions, and that's where it'll be. If the holes are close together, then actually the most likely place to find a bullet will be in the middle. Bullets do not exhibit interference. We know this. So now let's do it with electrons. Now you know what electrons are.
09:21
Electrons are like tiny little bullets. They're the things that hover around in a sort of cloud around the nucleus of atoms. They're incredibly small. In fact, they probably don't have any size. We're not certain, but they're kind of point-like. But we can do things with them. We can have electron guns that are just like, well, actually, they're not very light machine guns, but we can have electron guns that fire them out over some angle, and again, if we have slits that are close together, we can do the same experiment,
09:42
and the completely unbelievable thing is that electrons do exhibit interference exactly the same way all the other wave phenomena we know about, too. And you think, well, that doesn't really seem to make any sense. But maybe what's going on is maybe electrons are going through both slits,
10:01
and they're kind of bouncing off each other, just like we can imagine bullets hitting into each other, and somehow, they're causing this interference pattern. Not quite clear how that would happen, but maybe that's happening. But that's not what's happening. We know that's not what's happening because if you reduce the power of the electron gun so that there's only a single electron in the system at any time,
10:20
you still get the interference pattern, which I know makes no sense, but there's a lot of not making any sense in this talk. The electrons act as if they were being guided by waves and that the wave went through both slits and the wave actually caused interference, and the position of the electron that we measure is only the places where there's positive interference
10:43
and where it cancels out, we never find the electron. And in fact, that's exactly how quantum mechanics describes the world. So Schrodinger's equation and all the more modern versions like Dirac's equation describe the path of electrons, the evolution of electrons and all other particles
11:01
in the form of a wave function, a complex wave function, whose amplitude at any point is a complex number and if you square that amplitude by taking the complex conjugate times itself, that gives you the probability of finding the particle at that position in space-time. So the wave function can be
11:21
in what we call a superposition of states. That is a mixture of states. It can be partly in one state, partly in another. It doesn't have to be a 50-50 mixture. It can be any mixture we like. These coefficients can even be complex. Schrodinger's equation tells us, or the more modern versions, tell us how psi evolves over time. But whenever we actually measure where the particle is,
11:43
we always find it either went through the left slit or the right slit. If we measure the qubit, we always get a one or a zero out, not a mixture. But in between, when we're not looking at it, it's as if it's in both states partly. We talk about the wave function collapsing, or at least we used to talk about the wave function collapsing. Now, you're probably gonna say that still makes no sense
12:00
and again, I can't really argue with you. But what I don't want you to think is that means there's something bad about quantum mechanics because quantum mechanics is literally the best scientific theory we have and the best scientific theory we have ever had. It's one of the two great pillars of physics. There's relativity and they're quantum mechanics. There's this slight embarrassment that they're incompatible with each other,
12:20
but don't worry about that. Relativity's really good at the big scale. Quantum mechanics is really big at the small scale. And in fact, there has never been a reproducible experiment that disagreed with the results of quantum mechanics. Sometimes quantum mechanics gives us probabilistic answers. Sometimes it gives us definite answers. But we've actually verified those answers more accurately than we've ever verified anything else in the world,
12:41
like to almost a part in a billion. It is our best theory. It works if you use the equations unless you make mistakes. You always get the right answer. The only difficulty is understanding what an Earthly answer means, except in terms of the probability of where particles are. Now, in 1957, a guy called Hugh Everett was working as a student for John Archibald Wheeler
13:03
and he came up with a particular interpretation that is favored by almost everyone in quantum computing, probably largely because one of the fathers of quantum computing, David Deutsch, very strongly advocates this way. And it gives you a very nice way of thinking about things, as long as you don't mind your head hurting. What Everett suggested is that every time there are multiple possibilities
13:21
suggested by quantum theory, what happens is the universe literally splits and all the different possibilities are realized in different universes. It's sliding doors if you've seen that film. Both things happen. Or in fact, perhaps not even just both things, but if there are lots of possibilities, maybe as many as a continuous infinity of different things happen in different universes.
13:41
So that there are loads of all of you sitting here in different universes, just with very slightly quantum states, very slightly different quantum states. And what the many worlds interpretation says is that when we measure the outcome of an experiment, when we measure psi, what we're actually doing is we're finding out the value of psi in our universe, the particular universe we're in at that point. But the really interesting thing is that
14:02
the different universes interfere with each other. The wave functions from different universes interfere with each other. You get constructive and destructive interference, and they influence the probability of your finding yourself in any particular universe. So that when we look at the qubit, if it's in a superposition of states with alpha and beta then the probability of finding ourselves in a universe
14:22
where the answer's zero will be proportional to essentially alpha squared and the other possibility will be beta squared. And again, you say this doesn't make any sense. I know I'm getting a bit repetitive here, but this seems to work. Now, let's go back to stuff we all understand. Bytes and words and things like that. So obviously we don't only do bits in computing,
14:43
we do bytes as well. Bytes are eight bits. We look at longer words like 64-bit words and so forth because it's pretty difficult to do anything interesting with a single bit. We can do something similar with qubits. We can form quibytes if you like, or quwords if you like, quantum bytes and quantum words
15:00
using something called entanglement. And entanglement is a phenomenon whereby multiple qubits, which are discrete physical things, but multiple of them are described by a single wave function. And that means that we say that they're entangled. That is to say, if we worked with eight qubits,
15:20
then obviously that could represent up to 256 different values. They're up to 255, say. Then we can actually have a wave function that describes the state of this qubyte, if you like, as being partly zero, partly one, all the way up to partly 255. And we can hold that thing for a while
15:40
and do computations on it, which is weird. So probably you're getting where I'm going here, right? Why was I talking about all this parallel computing stuff? Why was I talking about really stupid algorithms that would work well if we could run over multiple universes? Presumably what I'm saying is that quantum computing is like this free parallel computing where we just run our computation
16:03
across loads of different universes and one of them finds the answer and we're done. That's how, I'm glad there are people wincing, that's how almost every pop article I've ever read about quantum computing makes it sound like quantum computing works. And there's a little bit of truth in it. We can set the computation up like this
16:22
and we can set up the superposition of the input state of the quantum computer so that every computer, every universe would do a different version of the computation. So it's kinda right, but there's a but. And it's quite a big but.
16:43
And the but is that it's not just the quantum computer and the qubits that spread across the universes, it's us as well. And so if we do set up the computation this way, the trouble is we don't get to choose which universe we lock up the answer in. So almost certainly in this case, we're gonna find, you know, we tried three times four and it didn't equal 15
17:01
and the game's a bogey. So if you take nothing else out of this talk, even if you think all this bullshit, I'm sorry, I wasn't gonna swear, even if you think all this stuff I've said about quantum computing can't possibly be right, quantum mechanics stuff, this guy clearly doesn't know what he's talking about, what you should remember is this is not how quantum computers work. It's not free parallel computing across universes.
17:23
It's a lot more complicated than that. Thank you very much. Ah, but I'm not done because I'm gonna try and tell you how it does work kind of sorta, which is a bit harder. It's quite a lot harder actually.
17:41
So to benefit from quantum computing, what we need to do is we need to find algorithms in which all the bad bits of the computation, all the dead ends, all the bits we don't want exhibit destructive interference. If we think of this multiple universe model, which I think is nonsense by the way, it's not the interpretation I favor. I like one that goes backwards in time instead, it's much better.
18:02
We have to arrange things so that all the bad solutions, all the things we're not interested in, destructively interfere and kill each other. And the good solution, the thing we're actually looking for exhibits constructive interference, ideally so that we get with probability one, the answer that we're actually looking for. And if we do that, then it doesn't matter
18:21
which universe we're in because all the universes produce the same answer. We're looking for global properties of the computation or the system that we're doing. And if you've got a problem, if you've heard of Shaw's algorithm, which is the main way that this semi-prime factorization problem is tackled, that's what Shaw's algorithm is. It's an algorithm that distributes the work across a superposition of states such that the result
18:43
is a global property of the result of the computation. And Shaw's, so quantum computers have been built, not with all that many qubits, but they have been built and they do work and they have been tested and we have used Shaw's algorithm to factorize large-ish semi-prime numbers,
19:01
depending on your definition of large-ish. Biggest we've done so far is 21. But you know, it's a start. It's not like the smallest semi-prime number. There's another way, in fact, of doing quantum computing called adiabatic quantum computing and we've got up as high as 143 with that, so that's even more impressive.
19:22
And it turned out when they went back and analyzed what had happened, it had also factorized 56,153 just sort of as a little bonus, which they hadn't asked for. Quantum computing's a little bit weird. Now you know, these aren't incredibly impressive results numerically, because if I run my stupid little program, it factorizes 56,153 in about a 20th of a second,
19:42
almost all of which is Python startup time. And even my mind-crashingly stupid implementation does it in about a second. And that's running on a single core. If you gave me N squared cores, where N is like a really big number, guy would probably be slower. Anyway, so the summary before, I am gonna try and describe how Shor's algorithm works,
20:00
but I wanna do the summary first. I've got 10 minutes, yes. That should be easily enough to explain the whole of the rest of quantum physics. The physics behind quantum computing is 100% sound. As the president of the US might say, it's the best science we have. It really is literally the best science we have. It works.
20:21
The engineering is tricky. To build quantum computers, we typically have to operate at a few millikelvin, that is a few thousandths of a degree above absolute zero. And one of the problems is that we can't keep them, in order for quantum computation to work, the system has to remain coherent. What does that mean? It means that other stuff in the universe
20:41
mustn't interfere with its quantum state too much. And other stuff in this universe and all the other universes does have a bit of a habit after not very much time of causing the kind of interference we don't want that turns the whole system to a state that we can't do anything with. So it's very hard to keep the systems coherent, but we're getting better at this stuff.
21:01
There doesn't seem to be anything in physics that says we couldn't build a big one. We know that in principle the physics works, so everything's good. So it's probably coming. Now I'm gonna try and explain Shor's algorithm. So the way Shor's algorithm works is it doesn't tackle factorization of large semi-prime numbers head on.
21:21
It tackles it kind of sideways. It uses a number theoretic property, which is if we take almost any number x and we compute x modulo the number that we're trying to factorize, modulo n, that is the percent operator in Python, integer remainder division in Python. So if we compute x modulo n and then x squared modulo n
21:41
and then x cubed modulo n, we keep on doing that up to a really, really, really big power of x. And it helps, obviously, that Python is quite happy with arbitrarily large computations, but it also helps that we're doing it modulo n so it doesn't actually get out of control. That sequence will turn out in almost all circumstances to be periodic. That is, this remainder will repeat.
22:02
It might have a very long period, but it will repeat over some period that might be close to n, the number that we're trying to factorize. But it turns out if we can find that period of this sequence for x and we can find it for maybe half a dozen different x's, that's enough to allow us to figure out what the prime factors of n are.
22:20
I can't explain exactly why that is. You just have to take it, number theorists tell us that's good. And in fact, Euler worked this out in the 18th century. So if we can find the period of such sequences for a few values of x, we can find our two prime factors of n. So how do we do that? Well, what we do,
22:41
and this is another really crucial point, we don't allocate different possible factors across our, if we think of this multi-universe thing, we don't create a superposition over the possible factors, we create a superposition over the elements of this series that we're trying to compute. Okay, so we put x mod n on one, or rather x on one, x squared on the next one,
23:02
x cubed on the next one. We compute on each one what the remainder when we divide it by n is. And then we use a quantum Fourier transform to find the period. You probably all use quantum Fourier transforms. They're exactly like Fourier transforms except they're quantum. Because if you know nothing else about a Fourier transform,
23:20
you know that it finds frequencies and you know that frequencies are the inverse of periods. So if we could calculate a quantum Fourier transform, then we'd be done. And that's what we do. And the crucial thing is that the period that comes out is a property of the whole computation that we can look up in any universe, five minutes, yep. So I'm gonna now try to illustrate this using an idea by Scott Aaronson. But he didn't write the Python that wrote the JavaScript that does the demonstration,
23:42
so I'm gonna claim some credit. So what we have to suppose is that my circadian clock's a bit off and I'm not on a 24-hour cycle. I'm on some other cycle and we don't know what it is. But I'm very regular and I get up at the same time each day, the same time on whatever cycle I'm on. And what we've got here is we've got a set of clocks,
24:01
one of which goes around once every 21 hours, one of which goes around every 22 hours, one of which goes around every 23 hours, all the way up to 30 hours. And underneath each clock, it didn't start very well, did it? And underneath each clock, we've got a pinboard with a drawing pin that's stuck in the middle at the start. And what happens is every time I get up, I go to each of the clocks
24:20
and I move the pin under each clock in the direction that its hands are pointing. Its hands, it's only got one hand, it's our hand is pointing. And so what happens for most of the clocks is that they do this sort of little random walk and it never really gets very far because the period of the clock doesn't match up with my circadian cycle.
24:44
The direction the arrow's pointing is, the arrow that the hand is pointing is different at different times on different days, on different my whatever our days. But there's something special about the one that does have the same period as me and things that have multiples of that period and so on, which is that every time I get up on my clock,
25:05
the one that actually matches my cycle, then the hand's gonna be pointing in the same direction and therefore I'm gonna move the drawing pin in the same direction. And so if we watch this, what we should find happening, it's JavaScript, but it was JavaScript that was generated by Python, so I think it's kosher.
25:21
What we should find is that most of these are wandering around in a fairly haphazard way, but one of them, possibly the one for 26 hours, is trying to make a dash. I'm not gonna let it get out of the circle, don't worry. It's not gonna run right across the 27 or anything. But this is the one we can see quite clearly is in fact the cycle.
25:40
And in an incredibly hand-wavy way, that's how a quantum Fourier transform works. It arranges the computation in such a way that all the bad periods disappear because we get things that cancel out in this quantum sense. But the good period keeps reinforcing and therefore whatever universe we're in,
26:01
if you think of the multi-universe thing, or in fact you can forget thinking about multiple universes now, the result of our computation is a well-defined result, which in this case should be the 26, so we find our period. And then we reverse engineer that, we do that with a few different numbers, we find our p and q, we factorize it, we've broken the internet, and everything's good.
26:21
So that's pretty much what I had to say. Yeah, so you're welcome to email me about this stuff or stuff I actually know about. You can find me on Twitter, and I pop up in lots of universes.
26:51
Thank you very much. We've got time for one question if anyone has one. You do have to come up to the front. The microphones don't walk this time.
27:04
Oh, thank God it's not Peter Higgs. So one question. Given the fact that these famous algorithms usually require maximum fault tolerance, do you think there is any use of quantum computing before we achieve systems that require fault tolerance
27:22
to implement the quantum algorithm? No. So the question, sorry, if people didn't hear, I think the question was basically are quantum computers with short-lived, small numbers of quantum bits useful for anything practical? No, I don't think they are. I think they only become useful when we actually get long enough coherence times
27:40
and large enough numbers of qubits to do meaningful things. I don't think there are any algorithms that do anything. I mean, I could be wrong about this, but I'm not aware of any algorithms that do anything interesting in very small amounts of time with very small numbers of qubits. Okay, thanks. Sure.