Logics of Quantum Computation

0:00 It's less or equal than the probability of square root of nought. In classical case, these are, of course, equivalent if you replace the square root of nought with a nought. It is a simple counterposition law. But not here with the square root of nought, which is a half negation. It's not a real negation. So, we have our structure, the reversible quantum computation of structure, the set of all density operators, this is a pre-order, this is not partial order, this is a pre-order. Then we have the end, based on top poly, the non-negation, the square root, the true property, the true property, the false property, and the completely undetermined property, the one-half, the one-half, the semi-transportant effect. Now, we have a lot of ingredients here. We have too many, maybe too many density operators, because we have the set of all possible density operators for any possible finite dimension at the universe. So, what happens? Is it possible, from a logical point of view, to confine our self, limit our self, only to C2? This would be a very important improvement from a computational point of view, because if you can dispense from the whole possible set of density operators and restrict our self only to C2, which is a very elementary structure, it would be very nice. What is the leading idea? The leading idea is the following. Classical logic, the first of classical logic is irreversible, because the operators, the connectors are clearly irreversible. But as well known, every bar, every Boolean irreversible function can be be transformed into a reversible one at the cost of increasing the complexity, of course. You have to pay cost. For example, in this case, an end, if you take end here, which is reversible, you can transform into a reversible function just by the top of it, and what emerges, the top of it. Now, in the case of quantum computational logic, we have the opposite way.

2:30 Because quantum computational logic, the birth of quantum computational logic is on a reversible framework, unitary framework. Now the question, is it possible to base quantum computational logic only on the set of all density operators based on C2? In other words, is C2 the quantum analog of the zero, or the classical zero, one? In other words, is there any logical difference from the logic based on the whole class of density operators and the logic based on the irreversible part, C2? The answer is no. And that's a very interesting result. So, first of all, now we are in an irreversible framework. Now we are confined to C2, so everything is easy. We have the block, Pointer is clear, so everything is measurable. What is the I, I mean C2 stands for irreversible end, on two, on two, uh, of two-density operables. It's just rho, it's a mixture, with the index of the product of the two probabilities. So we have the two probabilities, you compute the two probabilities, you have a number, and then you construct rho of that number, that means of the mixture, of the, uh, of the convex combination. Sorry. Sorry, what is the probability of the density of P, P, no, P tau, no word, yes, P tau is just the trace of tau, P zero, P one, where P one is the projection of the set of, okay? So we have here the probability, the product, the product of the two probabilities. Now, how can we reach this theory? The goal, the aim is to prove that the logic of the whole, of the whole set of density operators is just the logic of C2, which is a very economical way to represent a counter-configuration of logic.

5:00 So I skip here the details of, you know that every density operators in C2 can be represented as a matrix representation whose basic ingredients are three numbers, the real numbers R1, R2, R3, such that the sum of the square of these three numbers is less than one. So, they are completely, every density operator is completely, uniquely determined by a triple of real numbers satisfying this condition. In other words, we have that Q-mixes, statistical operators, are one-to-one correspondence with the points of the Poincaré Sphere, or the Bloch, and the so-called Bloch Poincaré Sphere already one. This is very well known. Pure density operators, proper mixtures, are proper, correspond to points of, sorry, pure density operators, pure space, correspond to the points of the surface of the block-marker sphere, and proper mixtures, proper statistical operators, are in the point of the block monotonic sphere. So, now we can take statistical operators just as a triple, R1, R2, R3, the triple which uniquely determines the statistical operator. So, what is the probability, now, what we define that way, of rho? Rho, which is defined, determined by this triple, is 1 minus R3 over 2. So in other words, the probability of Rho strictly univocally depends on the last element of the triple. The probability of the square root of naught depends uniquely on the second element of the square. So, as I mentioned, the order is defined in terms of probability of rho and probability of a square root of rho, so you can quotient and ignore the first element, because the first element will result in nothing from a probabilistic point of view. Because the probability of rho is determined by R3 and the probability of a square root

7:30 of rho is determined by a second element. So we can, in an algebraic mode, you can say that you can say the quotient, you can say the equivalent classes, modulo, modulo, class, okay. So we have what I call the planetary quantum computational structure. It's just the same structure that the OD restricted whose universe, whose universe now is the set For all that, since you're courageous. Professor? Yeah? Surely you could also define, instead of your root of not, a root of x not with a different choice of i and minus i, those coefficients, and then r1 wouldn't be better. If you just restricted yourself to what you might call z preferred and y was used to talk about your root of not. You mean, when I define the rho lambda, this is just a way to define the convex combination. It's just the usual definition of a convex combination. You have the truth property, in the givens, and the potential property. n, then you have in the computational basis, okay, then you have, you fix a lambda, a real number is zero, one, then you take a convex combination of the true and false, okay, in in the way that rho index by lambda is just lambda is 0 and plus 1 minus lambda is 1. This is the usual way to define a convex combination. And if you have lambda equals to one half, you have the unpolarized state, the semi-transparent effect, one half of the identity. This is just the usual way to define. Now, I'm not satisfied about C2 because it's still complex for the logician. So, we'd like to have the complex numbers.

10:00 So, is it possible to develop quantum computation at large, only with a segment of complex numbers? As you can see, in C2, we ignore the first element. So we have only the second and the third element, which are real numbers, that is complex numbers. So complex numbers are just pairs of real numbers. And in this way, we can fix now a particular set of complex numbers. set of pairs, A and B, which satisfy this condition. This condition essentially means that A and B have to incorporate the probability, the rho 2 and rho 3. Okay, so the probability of rho is just 1 minus R3 up 2. The probability of the square would 2 and over 2. So if you confine to these pairs of elements, you have to impose a restriction on the pairs of real numbers in such a way that you obtain the probability. So you don't have, you have to remain inside the probability. So, now, so this you can see, you have the row, the last two elements, you define A and B, which A and B depends on R2 and R3 in such a way that the conditions are satisfied, the statistical conditions are satisfied. Then you have the three, what are the three privileged elements? One, zero, the zero element, which is a complex number 0, 1, 1, 1. So, the first element of the pair represents the statistical part of the mixture, the second element. The third part, the second part of the pair represents the statistical value of the square root of naught. Okay? Now, you can define now operations on complex numbers.

12:30 On complex numbers. Now, for example, what is the NOT? The NOT is just the negation applies to both the members of the pair. One minus, one minus. Notice that the semi-classical, the semi-classical operators do not reverse the order of the pair. So, it's quite interesting from a mathematical point of view that if you have quantum, you are reversing on the pair. So, quantum operation, square root of knot, look, reverse the order. So, the square root of knot of the pair A1, A2 is just A2, the first element, and 1 minus A1. If you apply twice this operation, you obtain the negation. You have to think, square root of square root of A1, A2 is just the negation. It's just an image. But notice here the reversing of the volume. This is very important, very important. Okay, of course we can define a natural order on the pairs of complex number component-wise, this is a usual rule. So, we have now the third structure, the complex quantum computational algebra, based on complex numbers and these operations of irreversible operations of quantum experiments. What about, what about now logic? of five minutes. So, what about logic? Logic, we have a language, of course, we have a negation, we have a symbol for this new connecting, the square root of the negation, and then we have a ternary conjunction. Let me put ternary conjunction, if you remember, the of Toffoli, fixing as a third element the cat zero. So it's quite useful to have the conjunction as a ternary, as a ternary, but it's just a better linguistic definition.

15:00 So this is the minimum quantum length without the plus, okay, you see. So only reversible. And now we can find in the usual way the interpretation, the valuation of this language, if you say in a movement in algebraic form, as a normomorphism from the free algebra of the language into a quantum computational algebra, interpreting the connected in the obvious way, the negation into the knot, the end, into the end, the gate, and so on, and so on. So, nothing new. The usual definition of evaluation of a motor. And now, you have the definition of consequence. When you say that beta is the logical consequence of alpha, if Q, Q means the valuation, the valuation of alpha into the algebra, is less or equal than Q beta. I recall that the less or equal definition is based on the probability. The probability of the first is less or equal than the probability of the second and the probability of the square root. Okay, so what is quantum computational logic is the logic which is characterized from a semantical point of view by this consequence relation based on the set of all possible elements. You can define similarly, now, you can expand the language, L, and you can add the Ucasiewicz sum. So, you have the irreversible structure, based on the set of old density operations C2, and you have the usual definition of evaluation of model in this structure. This is the IQCL logic, that is, the logic, the irreversible function computational logic based on C2, and the Ukashevich quantum computational logic based on C2 plus the operation sum. So, this is a fragment of the whole of the whole logic.

17:30 What is the main result? The main result is that these logic are the same. So, the logic, IQCL, the reversible part of computational logic, based on C2, and the logic, reversible, based on the set of all possible density operators, are exactly the same logic. So, from a logical point, which is the philosophical point of view, the conclusion we can go, that there is no logical way to distinguish reversibility from reversibility here in this context. Because the two models have the same logical power. So there is no way to discriminate the reversibility here from the logic point of view. But our IQCL, our logic based on C2, can be more simply, simpler, characterized by our complex algebra. So, what is the conclusion? The conclusion is that the big logic is the same logic of the small logic here, which which is the same logic of this very small logic based on the complex numbers. So, you can dispense from the whole set of all possible density operators, from the set of all density operators of C2, and confine yourself to complex numbers. It's a complex number, I'm just thinking. So, what is the problem now from a logical point of view? Is this logic axiomatizable? Are these logic axiomatizable? Is there any set of axioms such that you can prove the soundness and complete theorems with respect to these concrete structures? My conjecture is yes. We have some partial results, fragments of the language, in which we have the square root of naught and the sum, which we can prove and complete, we have an axiomatization with respect to the, to C2. So we can prove that alpha is a, sorry, A is a theorem in a syntactical way if and only if

20:00 alpha is valid in the concrete model. I think that this is an interesting example because this is the first example, this should be the first example of a quantum logic arising from Hilbert space for which you can prove a completeness theory. So far, we know that orthodox quantum logic, automodular quantum logic is not complete at all with respect to the class of all Hilbert space, Hilbert space model, because there are some equations that call the concrete model parfait in some particular monoliths. And nobody knows that quantum logic is finite axioplasm. Nobody knows. Here, we have very good indication, we have some repartients on, very good indication that our logic, this computational logic, is completely characterized by the standard complete complex model. So, thank you very much. So my thoughts are running on similar lines to Jeremy's. So I take it that the square root of knot operation is the pi by 2 rotation about the x-axis. So that raises the following two questions. One, does the theorem that you quoted about there being no... Does the theorem you quoted at the beginning about there being no classical animal with the square root of knot other rotations around axes in the xy plane I suspect it would be really weird if it didn't and the second point which is what Jeremy was saying is that your compression down to just using the complex numbers is based on just using the square root of knots but that's picking out you know one axis has been the preferred one in the block sphere but you could equally well take all the other possible axes in the xy plane and that would give you a different choice between these three parameters, and that raises the thought that in the quantum computer, to capture the power of quantum computation, you need to have access to the full state space. So if you only restrict yourself to having access to a very small amount of state space, then this is just intuitive, but it seems to me that you wouldn't really be grasping what's really going to be characteristic of quantum computers. No, here, there is a preferred basis, which is the so-called computational basis.

22:30 Of course, you can define also any unitary, you can transform any unitary, any autonormal basis into another autonormal basis. This is just because you take the computational basis just because you want that your input be a number. This is the reason why. You want to use quantum computer to compute. And so, if you want that a number, a binary number, be interpreted as a string of 0 and why you take this preferred basis, the computational basis. But in any case, if you define our example concerning the impossibility of the square root not function, it's completely independent of the basis. Because the theorem says that there is no continuous function in 0, 1, such that applying twice the function, 1 minus x. So, this is completely independent of the basis. But the point remains that there are continuous number of squares, not operations. Yes, there are continuous. There are continuous mean gates, in this case, because you have the possibility to rotate. We just choose, we have to choose this gate, because it has a clear logical and physical impact. But you get the same logical impact from all this variety of different physical operations. Yes, for example, the Hadamard, the Hadamard, yes, for example, the Hadamard, Hadamard In this one, I'm not quite sure. Maybe this is the other one. That's self-inverse, but it's not going to be a square root of a knot. No, it's not a square root of a knot. This is the other one I gave. It's the square root of the identity. This is the square root of the identity. This means that if you apply twice, for example, the square root of the identity to the square root of the identity to each side, you get a chocolate color.

25:00 This is a partial statement, not a partial, this is a half statement. You try to state psi, you don't succeed, then you again act on psi and you get psi. So, you have an unaccountably many possibilities, of course. The problem is finding some physical interesting and logical gates, not different systems. But is not the point that your logic is so simple that you only assume these few operations and, in fact, in quantum computation on the least of great, there are more gates than beans. And so, any attempt to, um, UNG-16, enacting the kind of simple model is, I don't see how it's at all relevant to the logic of quantum computation. I don't think so. I think it's just a question to ask sometimes. There is no problem, because I can define, for example, every gate you want, you want to store the control nut, I can add it, no problem. So, if you want the phase shifter, the phase shifter, okay, I can add the phase shifter. And the model is completely unsensitive to adding, so this is the strength. Sorry, I didn't emphasize this point. So, if you have this structure, I only treat this three times because otherwise it's too complex. But there is no problem in adding other gates to our structure, and to have a corresponding model is not so simple, maybe not complex number, but real complex numbers, or triples or complex numbers. But in principle there is no difficult problem. But in principle, in practice, one doesn't know which gates are going to be of use yet. And so if you keep adding gates as time goes on, you keep changing your models. You keep having to change the system to the X in the practice. It seems to me that you're never going to know where to start. Otherwise, you have to have a logic in the counter-many

27:30 solutions that you have a logic in the counter-many objectives. Ok, you can. Start from... Since we know that quantum gates are counter-many, if you want to be exhaustive from the very beginning, you have to have the logic in the counter-many quantum gates. No problem. It's quite unopensable to start from a logical point of view. in this way, to have, how can you, how can you hope to obtain an axiomatization of a logical It's not simply an uncommon number two, because there are relations between these gates, there is an algebra of operators in human space, and so you don't start with just an arbitrary very uncountable set of connected, you've got these algebraic relations, which you've got to have different relations involved. But there are, from a truly mathematical point of view, there are uncountably many. In my own case, of course, because you've just made units, just units, present in C2, you have uncountably many, so you're going to be many with constraints. Yes, of course, but I'm counting the many. So, what is the, for example, what, how do you develop a logical structure? What is your idea, for example? Take the set of more possible, but this is a possibility. I emphasize the fact is an unorthodox way to be lost. I just anticipate that once you incorporate the whole structure upon a computation theory, that the marble event of it is no longer some finite pencil product involved seed, which would be a deep object with a whole lot of constraints in relation. We can end up in a number of times, this is the way to create a different space, and we also have a different space. We can develop, for example, all these machines in infinite dimensions. For example, you can prevent the treatment of these obstacles. If you want to, for example, if you want to avoid the fact that you have the sound, the union of all possible guests, you can take the sets, you can take all spaces.

30:00 That's a simple question. You have three logics. Yes, I wonder if you could use, for instance, instead of the complex, it's just a two-dimensional Yes, for the last logic, yes, it's a complex number, an ascension, an ascension, in the end of it, it's just two linear numbers of the two dimensional vector space. Yes, exactly, this, this is, you are right in the sense, for example, since here you have only the square root of North, and you don't have, for example, the square root of the other man, which I defined previously, you can ignore the first component, or the triple, or the triple. If you take into account a logic, for example, in which you have the square root of not and the square root of the identity, then you cannot ignore the first component of the triple of the triple in the block sphere determines just the probability of this fracture. So, this is true. This is simple because it's quite simple, the starting point, because you have only these two or three elements. But we know that if you have a control now, thoughtfully, you can do a lot of computation. Yeah, one further question. I didn't quite follow this, but there seems to be a similar theorem, a relevant theorem,

32:30 ... that establishes that if you restrict your, for instance, of case use, to the control not, and the Pali and the Hara, they have a new person, no, they are not a new person, they can actually do business projects. Yes, they have a new person in an absolute way. Oh no, no, no, no, they are not a new person. So, they allow you to preserve a lot of interesting things, but they are not good for both computations. Because they can be efficient in classes, they are very simple, right? So, they are not good for both computations. Now, you said something, I don't quite see what they are, but they look suspiciously, right? by similarly small descriptions. So, it makes me suspicious that I think that maybe these as well could be, how much of those could be efficiently simulated by another curve at the point of the time. Simulation here does not mean logical implications of course. Simulation is a different logic. Here, you have a logic which has all the features of what you can call logic as a model. It is not classical logic, because there is, for example, it's not classical logic, because just because you have that, that the end is not as important, because the product is not as important. So we have a logic which has a model, which is not, which is a sublogical, classical logic. It's a different question of whether then the gates can be simulated. This is just, the results they present are only in, they have only a logical impact. They say that you have two logics. It's a logic which is not classical logic, which is greater than classical logic. witness theory, is there an axiomotrification of this logic, which is somewhat complete with respect to the complete model, as it happens in real logic, or in partial logic with respect to the 0.1. So, this is quite, there are two different aspects.

35:00 What's the relation of quantum computation, the interesting question is where you can But I know that, for example, there are lots of places, I don't know, I hope that we are not wasting their time trying to find, for example, an implementation of the top-polligate or the control-control knot, which is not so easy because if you use, for example, some Some algorithms are based on the XOR and the other one, I think. For example, if I'm not wrong, for example, the Deutsche... Josh's algorithm is based on the XOR, the control knot, and the other one. So, if you are able to implement from a physical point of view a XOR, a control knot, I think is a good advance in quantum computation. Because Deutsch-Geosha algorithm is a spin-up of the usual classical algorithm. But then we're going to find it, no? Yeah, it's the same. We have a little late in time, but one short question after one short remark to that. I'll try to ask Uribe's question again. I think we're just asking you, I mean, it's fascinating as much, but if you were going to a theoretical quantum computation conference, and you were talking to people involved in error correction, is there something in your general set of ideas which you can tell them to help with their problems? For example, I can tell them that this is a quite economical way to represent, I have no time to explain, to represent quantum circuits. So, and which is a very economical way to, which is not clear at all for a logician, for example, when I see the nature of the treatment of the circuit as a generalization of the Bunyan Circus, for example, here, just the form of the quantum computational language

37:30 are in an intuitive and economical way to represent quantum circuits, and, oh, I cannot Now I need 10, for example, by means of, it's a nice and elegant treatment of a tree, of a tree, of a formula, for example, of a circuit. A circuit can be thought of as a tree, as a tree, having at the bottom the formula to evaluate where you have the atoms of the other formula, Well, the occurrence of the atom is important here, because you take into account also the number of occurrences of the same atom. So, you have input information, output information, you have an elegant treatment of a franco circuit as a form of a gs. And then, we try to, also to, as I mentioned, to, I know, as far as I know, it's not easy to implement also this simple, this simple, for example, the control-not, from a physical point of view. There are a lot of efforts to try to implement because you have some non-linear effect. They try, for example, with a Kerr effect, with the Binscript and Kerr effect. But when you have a control gate, then you have to take into account some non-linear condition. So, I think that realizing, also realizing XOR-8, a control-not-8, and a top-folio-8, which is a control-control-not-8, which is based on XOR, is not an easy task at all. So, it's been done. It's been done. It's been done, but not... It has been done, for example, in Australia, or in New Zealand, or in the United States. What are some limits? For example, there are different ways to approach the problem.

40:00 There's a photonic wave with photons or with electrons and so on. We try to... and there's a problem of scalability. Because the scalability is a problem that you can realize a gate, but if you want to have an efficient way to compute, you have to put together a lot of gates. So, there is some problems in the scalability of, I know that, the scalability of this was French. You're much closer than the Eastbrook, right? Eastbrook, Eastbrook, yes. The principle scale of this is hard. Yeah. It's hard. Yeah. I guess we have to stop here. Ten minutes. Copyright. This is not a sufficient condition to have a new process as opposed to having a senior and a senior, but it's definitely to do, and that's when he's looking out for this. Yes, I've been in the first one. An old tracing computer rotation is a very easy thing to figure out why this is more around the system. Do you have any sort of union? Do you want to use this principle? Yes, okay. Thank you.