The quantum theory of knowledge and computation (contd.)

0:00 After all, there is only a countable number of possible quantum computational networks that can be constructed from a finite set of gates, a finite set of different gates, whereas the number of different quantum computations is a continuum, so it can only ever be approximated to be new. But we don't know alpha exactly, for alpha has been machined with some level of accuracy but not perfect, so that successive ones of these will have slightly different alphas. Then the answer will be, just as in the classical, is error correction. Forming additional computations. After each computation is set, an error correction is set. I won't go into the theory of this, because you couldn't, by the way, build an error correction circuit out of these things alone. I'm just using this data as a schematic example of what happens generally in Europe. So we have error correction circuits. Now, the number of pages that we need depends on

2:30 Which computation we want to perform. And in general, if we want to perform computation in accuracy, epsilon, then we need of the order of one over epsilon consecutive gates. If there is an error, it will be multiplied by a factor. But the error correction first has to be performed one over epsilon times as well. And it itself must be performed by quantum. The error correction process has to be performed that number of times as well. And that error correction process will excel in the series. And in this case here, there is a lot more computation going on in the error correction than there is in the data itself. So it's not at all obvious, but just because I can construct a theoretical computation that would correct the errors. It's not at all obvious that if I translate that into gates and insert them into the network, the errors will be at all corrected. In fact, the situation is even worse than that is because, in the classical theory, errors which are corrected by means of multiple redundancies, having many copies of the same system take a majority vote, as they all say, and then setting them all to be the majority, that method... ...reduces the errors exponentially. That is, the errors go down exponentially with the number of redundant components. Or to put that another way, the errors go down exponentially with the number of components used in the error collections. In quantum theory, for example, the errors in general do not go down exponentially. They only go down polynomially. And therefore, one has to work out, through the actual organism, in the classical theory...

5:00 You can abstract away error correction and imagine an ideal machine that doesn't have errors because you know that the real physical system, simply by using error correction, you have to work out, for every type of computation, you have an error correction that goes with that kind of computation and you have to work out fresh what resources are required to make sure that that doesn't go up more rapidly than the resources required to perform the computation. There are several different kinds of errors. The entire error I was interested in was an error in machining the original components. The algebra was not what we thought it was. It was not the same in successive cases. So what would happen there is that even if you put the bit in perfectly in the original state, by the time it came out it would be very perfectly loaded with one group of components. Well, that's a matter of engineering in general. You could measure alpha. If you had a component, you could measure alpha with any desired degree of accuracy by performing successive measurements and then if it had the property that alpha remained constant and didn't depend on the temperature or the atmospheric pressure or how old the component was or whatever, then alpha would be the same. In general, errors have been produced by that and by thermal fluctuations. There are thermal fluctuations in the properties of the gate. There are also thermal fluctuations in the actual state of the system. One bonus that occurs, I said that error correction is much harder, one bonus that occurs is that although errors are only corrected polynomially, they also only grow linearly.

7:30 That is because of a difference between, the simplest way I can demonstrate this is, The only error is in the preparations in which we supposedly want to prepare it. Since the components are correct and achieved, instead of being S of something, it's going to be S of something. The probability of detecting the state is, in fact, of something, not something, is the modulus of the inner product, S of that. Sorry, one minus. In other words, and this, the error in the initial state. Error in the final state is equal to the error in the initial state. It has no tendency to grow at all. Similarly, if there are errors introduced in every step of the computation, they simply add together. This is quite different from classical mathematics where errors grow exponentially. You get partial compensation, that would be correct. So, complexity theory and error correction and what can be computed.

10:00 These are all interlinked and have to be studied together. And the idealization of having a model in which whatever is computable is exactly is not available. I'll just say a few words about knowledge. What does knowledge correspond to in physics? It has a number of attributes, we know. From philosophical theories, for instance, knowledge is that which survives. That's one property. It survives criticism. It survives error correction. Another property of it is that it's complex. So, for instance, a crystal, although it survives, retains its form, is not complex. And so we don't consider a crystal as sophisticated information as much as complexity. Classical physical definitions of knowledge, classical complexity, disadvantage, purely random contains the most complexity and they are, it's notoriously difficult to find a definition of complexity, of knowledge, desire of complexity that assigns a very high complexity to like a DNA strand.

12:30 And it also assigns a low complexity both to a crystal and to a strand of DNA with purely random base pairs. In the quantum theory of computation, that isn't at all so. Quantum theory has exactly the property that I've just said, the quantum theory of complexity. We define complexity as the number of steps, sorry, as the length of the shortest program that will generate the given state. Disordered, highly complex, generated by random numbering, with bone fluctuations in the same values. The complex data generated by some error collection, then is assigned a high algorithmic energy for knowledge. Let me just end with two roles in using applications of scientific economics.

15:00 About two years ago, Frank Tibbler was another owner of the evidence interpretation. Rhodes' paper on economics, which is a very unusual thing, it's even more unusual than it should be about quantum mechanics, it was about monopolies, and it analyzed the question, why do monopolies not raise their prices to infinity? I'm not speaking here about artificial monopolies imposed by government, but natural monopolies. The answer is, in everyday terms, because they are afraid. But if they make the price too high, their competitors will start up and will undercut their price. So the highest level of price they can allow themselves is just below the level at which competitors would start up. What do you mean by a monopoly in this case? A monopoly is just a single producer of an item where there are no competitors. Well, this common sense way of explaining it is rather odd because it refers to counterfactuals. It refers to these companies which would spring into existence. What Tippler does in this paper is rather nicely recast this analysis in terms of companies which actually exist but in other universes.

17:30 This monopoly is competing with companies that exist in other universes, other universes where it has erroneously set its price. The nice thing about this is that the theory of competition then applies absolutely uniformly. To the case of a monopoly, to the case where there is a, well another application, I put this forward as one of many examples of the way in which quantum ontology, especially when it comes to talking about knowledge, is superior to classical knowledge. Another, it is, the definition of market fluctuations is very elusive in classical economics. When we look at a stock market, we find that it goes up and down. Now, again, speaking anecdotally, we know that the variations in science consist of two types. There is a secular trend owing to some underlying changes in science, and there are fluctuations of outcome, but any particular change, you can't tell whether it's one or which one it is, and in fact, classically, it's very hard. To think of a definition that would separate the secular trend of fluctuations. The analogous problem occurs in statistical mechanics, and what they do there is to introduce an ensemble where you redefine the fluctuations. The fluctuations in ordinary language means going up and down, but in thermodynamics it doesn't mean anything like that.

20:00 Fluctuations in thermodynamics mean that different elements of the ensemble have different values and how to articulate them. Notorious classical problem of thermodynamics. Why are these two things related? The problem of the time average and the ensemble average, for instance. Why are they related? And they really can't be related. They aren't related classically. They are only related in the quantum. In the quantum theory, the ensemble, or rather, an ensemble, state of, is not represented by, is represented by a number of things, a very large number, represented in one diagram. Thin indicates prices where very few of the universes, where at that particular price, at that time, you had either a few or a few of these big choices or three or many. Well, now we can, this price is now quantum observable, and we can define the expectation of it. And the secular trend in the price is simply the expectation pattern of the price, that is the average price taken over all the universes. If we can obtain a value which is above that average, then what is objectively happening in our particular branch of the many universes is that the price has fluctuated above its true value, or it has fluctuated below its true value. But the average is not measurable in itself. All we can measure is specific easiness. So the average is a theoretical quantity only.

22:30 Suppose for now that this represents the fluctuations in the price, and I'm interested now in the effect of a particular intervention in the market. So I could work out in principle, that couldn't work out, but in principle there exists a function that represents, a wave function, that represents the fluctuations in the price. And equally, if I intervene in this market as a speculator, Then you can work out the change that the speculator causes in the sense that of what would have happened if the speculator was not there, and it does, the speculator is there. Anecdotally, it's very easy to see, if the speculator systematically buys when things are expensive and sells when they're cheap, then he will increase the size of the fluctuations and also make a loss. And if he systematically buys when they're cheap and sells when they're expensive, he will reduce the size of the fluctuations, yet incidentally make a profit. You can prove that the maximum possible profit that the speculators can make is the expectation value of the change between the mean square fluctuations and the awake factor of the supply and demand. Maximum profit. Now, from the point of view of knowledge, what has happened here? The speculator makes a profit which is bound by success. The speculator makes a maximum profit which is proportional to the amount by which he has reduced the fluctuations.

25:00 Now think of what that means in terms of my previous description of complexity and knowledge. Allocation of resources represented by a particular price is very complex. A different price represents a different complex allocation of resources. A large fluctuation in the price corresponds to a large fluctuation in this complex allocation of resources. And so, where there's a large fluctuation, in different worlds, resources are allocated differently, and so the knowledge in the resource allocation is low. If the fluctuations have been removed, the amount of knowledge in the source allocation has been increased, computation required, the amount of selection, the amount of error collection that have gone into producing this great computer program, the quantum computers that have been required to produce that information from scratch is long gone. There are many ways in which quantum versions of mathematics exist. That's a fantastic, interesting talk. I'm one of the at least two believers of the Edwardian Deplication in this room. I'm bold to think of this world, and I'm trained in the close neighborhood of this one, in which I didn't come. However, I'm inspired by what you said about the classical versus the quantum definitions of realities.

27:30 Now, it seems to me that the quantum-flexical versions are played by slightly different rules. I take it that what one's trying to do in classical complexity theory is to try to define randomness independently of the sequence, independently, or where the information content, or sequence, independently of problems. Now, it seems to me that, and that's why... And finally, we have a random sequence of what we would normally think was random, which comes out as a mayhem, I acknowledge, because basically the best program is simply one that draws out the digits one by one. But now, it seems to me that were one to apply the same strictures in the case of quantum theory, you get exactly the same result. Here you've got a random sequence. And of course, in quantum, the quantum field is genuinely random. Now, you say, well, that's fine. A quantum mechanical random number generator is a pretty simple thing, so a very short program can generate that sequence. But on the other hand, if you were playing by the same rules as in the classical case, you'd have to say, what's the shortest program which can be guaranteed to produce exactly that sequence? And of course the answer is that a quantum mechanical random number generator can't be guaranteed to produce this at all. Of course, there's a finite probability that any quantum mechanical number generator could have produced exactly that sequence. So what you have to do is apply a different criteria, which in effect is to look at the neighborhood quantities as well with respect to very close possible worlds. And so that a genuinely random sequence is one where... Where the sequences produced in the very close neighborhood of the possible world produce this one, produce radically different sequences, whereas when it's knowledge, worlds in the close neighborhood, well this is what you tell me, but that is to define knowledge of information in terms of prominence, and it seems to me that were one allowed to appeal to prominence in the classical theory, one could also, and genuinely. It's true, though, that one plays by different rules. The thing is, one is allowed. There is a new kind of object. You can look at the randomness of knowledge in a sequence, look at the randomness of knowledge in a state, in the state of the physical.

30:00 In the classical theory, those two are the same. State is simply, sequence is simply a label for a state. But in the quantum theory, more than one sequence enters in different ways. So there's a new thing to look at and look at if we have a number of, let's say, complex sequences, then one thing you can do is look at the knowledge or information in sequences. This only occurs in individual branches. The universe as a whole is not represented by any one sequence. So if you look at one sequence in one universe, in one branch. They can apply the classical definition in the quantum theory if they wish, and it gives the answer, again, that it's very hard to distinguish a random sequence from the knowledgable one. But in the quantum theory you can go on from there. You can say, well that is probably because knowledge isn't a property of sequences at all. It's a property of the state. It's a property of all the worlds together. Classically, we can grope towards it by thinking of the history, but that's not satisfactory in action because, again, the problem of counterfactuality.

32:30 Only one history actually happens, and considering other histories besides physics. In quantum theory, you can now say, well, one measure is the measure of the world, but another thing you can ask is, how much computation would it require to produce this entire state? There we can get radically different answers to worlds separately. You said that quantum theory can generate real random numbers. Well, real random numbers is something that only exists in quantum theory. What it means is the possible number of universes that see each other. Why is that so? Well, a random number generator for one bit, a very simple quantum random number generator, is a quantum computer program. All of these terms have a property. It's actually the square root of naught. It's a single one-bit random number. It takes zero and one. So if we start, let's say, with state zero, and apply this process, and we now make a measurement, we get zero and one. We have set it to a random number between zero and one. If we apply this process to n bits, we obtain the n-fold product of state at this time, At the end, multiplying out this vector, we obtain all possible sequences of physics.

35:00 This is the state obtained by applying a random number generator to n bits. Applying a random number generator to n bits produces a state the vast majority of whose components are very highly complex, almost all of them. Because it was generated by a very short program, it only takes a very short program to apply the square root of not to each of the set of n bits. It scores very low in quantum effects, the lowest possible. Professor, if I could just pick up Michael's point again. I think it is the same kind of problem here. If I have a coin and I place it in the right conditions, I can produce a pretty good random sequence of knots and wands. Well, in a sense. I'm likewise giving you a very short program, which will generate a very complex programming sequence. What's the difference between specifying the state, the chaotic state, or just take chaotic systems in general? The answer to your question is different, according to whether they're speaking about classical or classical. Well, that's just the thing, because I'd like to understand better why it is that... The specification of the quantum state is so different from the specification of a classical chaotic situation, which produces a very good random sequence. But just to extend the question a little bit, even if you adopt the unitary evolution, the unitary dynamics of quantum theory, the many-worlds interpretation, as you do, for example, in the bone universe, you really, your choice, you've still got a choice of interpretation between a bone single-world interpretation The quantum potential plays a big explanatory role, but still allows for quantum computations, or the many worlds interpretation, where the many separate worlds actually kind of hyper-distance play a role in the explanation. Now, in the Bohm theory, of course, the quantum randomness is rather like classical randomness, in the sense that you get out quantum statistics because you put in a mixture of the hidden variables,

37:30 The mixture that you would get in say thermal fluctuations. So really, even though you're using exactly the same dynamics and two theories are empirically indistinguishable, in your interpretation, quantum statistics comes out of intrinsic formulas, whereas in the Bohm interpretation, quantum statistics comes out of something that's rather like, and you've got exactly the same predictions. So isn't there an escape route for anyone who, I mean, how would you argue against the Bohm interpretation? Which seems to have undercut this fundamental distinction in the world. Well, there are various versions of the moment. That interpretation of the table is logical. Quantum potential can perform computations, can carry knowledge, people, algorithms, and things. There's no sense in which you absolve it from how much reality it represents. I think that there are three different kinds of theory. There's classical theory, as in classical physics. There's classical stochastic theory. And there's quantum theory. And the answer to your question was different. In the case of ordinary classical physics, it's untrue that when you toss a coin, you produce a random sequence. Tossing a coin involves executing some program, a definite outcome. And even though you might find it hard to infer what the program was from the outcome, because it's a trap for a function, nevertheless, it was a very simple function that caused the outcome. And in fact, the sequence isn't random. Almost all sequences. This is the classical because all the sequences have to be obtained by simple algorithms and therefore almost all of them will not be obtained that way. They will all have a low classical algorithm. In the case of classical stochastic theory, well this is, classical stochastic theory I think is just a mistake. It's good as a method of approximating answers.

40:00 But to think that this is a theory of the world, I think it's just a... Can I just interject here? I mean, if we just do accept that it makes sense, that it's metaphysics in its own right, and that it's consistent with the soul... Classical stochastic theory. Classical stochastic theory. Then I'd be interested in your answer to Harbour's question, because it does seem that one could, in a way, develop quantum computing, but with classical stochastic theory. Same thing doesn't make sense. It wasn't my answer. It was just a side note. Right. The point there is that you always have to, assuming they're classical dynamics again, it always has to be as in Bohm's theory that the statistics is put in ab initio because there's no room for stochastic evolution in classical physics. You could invent a new physics which has stochastic input every ten minutes but the dynamics change stochasticly, but there's no fundamental difference there. Information in a state, and the knowledge in a state, is not entirely determined by physical methods. Part of it is determined by this thing described, not by physicists, but just as possibly by Theans, as being random. Well, certainly if we know, if we're told in the theory, in the model, that a certain number is random, that is, if we label it as random, then functions of that will also be random, and so on. But there's nothing in the actual world that distinguishes that from an identical number that had been produced by a way that was not labeled. And so our assignment of the word random to this sequence is purely arbitrary. Nothing in physics would be changed if we assigned some other word to it or even said it wasn't random. Objective meanings for things like knowledge and randomness, we can't possibly find them in a stochastic theory because the information in a sequence in a stochastic theory is put in by something which by hypothesis is not described by physics.

42:30 Professor, it seems to me that you might have a methodological argument against this sort of theory along your lines, but in a way that would be independent of the question of could you have a classical analogue of your quantum computing. I mean, you may say, yes, you could, but it was up in the drawbacks. But would you say that, in fact? I mean, would you say, okay, there are these drawbacks. If we put these, Suresh's first slide, would you then admit that they would be a classical analogue to quantum computing? There is, well... Which would seem to me to be really analogue computing. I mean, I always feel that... No, no, no. Well, several questions. Right, yeah. Classical analogue to definition of quantum complexity. Classical analogue is possible. Classical analogue as a theory of quantum computation in general isn't possible. It's not the same as analogue. Let me answer those two separately. You can always redefine the quantum theory of the finite process as a stochastic in a larger space, as in Bowden. Classical? Well, yes, classical. As long as you're not interested in how the computation is done, but only in the relation between the inputs and the outputs, you can always have an equivalent and that's true not only for the theory of computation, but for the S-matrix theory of anything, well, that leads us into the interpretation of quantum mechanics and so on, but that's... So yes, the answer is you could, and actually, at all satisfying my present definition of quantum complexity, actually doesn't draw on... The full, it only draws on the weights of complex amplitudes. Therefore it has a direct theory of ensembles. The fact that I don't like those for other reasons is, as you say, not good.

45:00 I think that the full definition of the definition of the world, ultimately abstracting away the actual computation, The classical theory of a system, of a one-bit system, is already an idealization. That is, it's quite hard to build electronic components that have only states zero and one. Actually, they have all the intermediate states as well. We choose three volts as being one and zero volts as being zero, but what is 1.5 volts? Well, we engineer it so that 1.5 volts is very unlikely to occur, that it occurs only for a microsecond and it will take a huge thermal fluctuation to allow that value to be actually measured. You can have classical computers that have more than two states. You can have three states, ten states. Whatever. And in the limit, you can have an analogue. So why can't you just build a classical computer where things change as they used to?