 Ie, hi, yn ffnwyshwyr. Rwm eich gweithio am ddedig i. Dleidio'r ffilm gyda'r gweithwyr. Mae'r wybodaeth gyda'r gweithwyr. First, fel ydych chi'n ddu i'r bwysig i'r dweud gyddiadau mewn a'r rhagleniadau i f demişio o'n gweithio dylai wahanol i'r gweithio. Rwy'n i, fod eich adeg. Fyddwch gyrddwch ar y Cymru Gweithgwyr Lund, yn ddechrau Dan Brown, am enghreifft y gweithwyr yng nghymchwylliau, wrth gwrs, yn ymwneud o gael ymddangos ymwneud ar gyfer amlwg cymdeithasgwyr, amgylcheddol i'r cystalogrefiaethau cwysig, a'r cystalogrefiaethau cystalogrefiaethau cwysig, oedd yn cyflawni'r Unedig. Felly, mae'r cystalogrefiaethau cynserf yn ôl ymgylch yn ddweud yma'r pethau cystalogrefiaethau cystalogrefiaethau cystalogrefiaethau. Felly, rwy'n ei ddod i'n penderfyniad. First of all, the people involved. The collaboration is between two so-called CCPs, or Collaborative Computational Projects. I'm attached to CCPQC for quantum computing, and the other people involved from CCPQC are at the bottom, and some of them are here today, obviously. The people at the top are part of CCP4, and they are the crystallography researchers. Now on to the substance of this talk. So now on to the substance of this talk. What is macromolecular crystallography? So macromolecular crystallography is concerned with understanding the structure of biological molecules such as proteins or viruses, DNA, et cetera, and this has many important applications including in biological research, understanding how the machinery of nature works, and in medical science understanding the mechanisms of viruses and other diseases and how to tackle them. And it's also key to the drug development pipeline in the pharmaceutical industry. However, given the small size of these molecules, it can be difficult to study their structure directly. However, under specific conditions, many of these molecules of interest can be forced to crystallise into a regular structure, and then shining x-ray beams at the sample from a scientific light source, such as the diamond light source that we have in the UK, can produce a diffraction pattern, and this diffraction pattern encodes information about the crystal structure. Now much of the early work in macromolecular crystallography was done in the early to mid 20th century by researchers including Dorothy Hodgkin who studied proteins, including insulin, and by Rosalind Franklin who looked at RNA and at DNA, and obviously by Quicken Watson who studied DNA. Today there is the protein data bank which contains more than 180,000 molecular structures, many of which were found via macromolecular crystallography. And this resource has enabled many modern accomplishments including the recent alpha fold model from DeepMind, which is now being used to protect the structure of proteins. So to be specific, the goal of macromolecular crystallography is this. So we want to perform that x-ray diffraction experiment and collect the diffraction pattern, and from the information contained in that diffraction pattern we want to construct a map of the electron density for the target molecule which tells us what its structure is. However, there is a particular problem with doing this, which is called the phase problem. The equation that relates the electron density to the x-ray reflections looks like this. The row on the left hand side there is the electron density at real space coordinates x, y and z, and the measured x-ray diffraction intensities go into this equation via the amplitude term here, which I've shown here in blue, and those hkl numbers are the reciprocal space coordinates for the particular reflection. However, this is not the full story. There are also these phases, these phase terms that appear in the equation, and the diffraction pattern doesn't provide any information directly about these, so how can we find them? So classically there are a number of approaches that are used to find these phases. Firstly there are these so-called direct methods, which involve exploiting phase value properties coming from the relationship between diffracted intensities, and this is sort of the approach we're taking with the quantum annealing, so I'll come back to this in a bit. Secondly, there are experimental methods that exploit phase signals from naturally occurring or even deliberately introduced heavy metals or other large atoms in the structure. Thirdly, there is the so-called molecular replacement. It so happens that proteins that are evolutionarily related to each other and which perform similar tasks will have a similar structure, so if a structure of such a related molecule is known, it can be used to give an initial approximation to the target phases and they can then be refined. While these methods are all useful, they all have their limitations. In many cases they can take a very large amount of computational or experimental effort to perform, and sometimes they may not even be possible to do at all. It turns out that the phases are the most important term in the calculation of the electron density map. However, it's also the case that their estimation tolerates quite a high degree of inaccuracy. So for example, even reducing the precision to the point that you only have two options for the phase values can still produce a map that's interpretable. So these two figures here demonstrate this. So the top one shows a map produced by the correct phases for some molecule, where the blue regions in that top figure are the electron density. The bottom figure shows the same structure, but with the phases fixed to the nearest of two allowed values. As you can see, the shape of the electron density regions are changed somewhat, but overall it convincingly shows the same structure. This lack of required precision led to the idea that this could potentially be an application for near-term quantum devices, including anelys. So the reflection phases are not all independent of each other. They are related through the scattering from the crystallised structure, and in particular there are specific triplets of phases that have an approximate phase relationship. So for triplets of reflections whose reciprocal space coordinates are related in the way shown here, basically if the reciprocal space factors add to zero, then those three phases should add approximately to zero. This relationship holds best when some conditions are met, so it holds best when the amplitudes are large, that is to say when the reflections are strong, and it also holds best when the reflections mean that the atoms are fully resolved with a resolution of around one angstrom or better. With that, I can come to the approach that we are trying, so we are trying to formulate this triplet rule constraint in such a way as to be suitable for quantum annealing. To do quantum annealing, we need to formulate our problem as anising Hamiltonian so that we can program it onto the annealer. In general, that looks like this. It's an equation that's been in most of the talks. We have these quadratic two-body terms with coupling coefficients j, jk, and the single-body terms with filled coefficients h, hj. Then the first step for writing the triplet rule constraint in this form is to use the qubits to represent the angles. We discretise the angles and represent each one in binary form using some number n of qubits for each. Then we can write an angle variable operator like this. This operator has two to the n, evenly spaced eigenvalues basically going from minus pi to pi. A problem that sometimes arises with such a binary encoding is that the coefficients can sometimes get quite large and can saturate the dynamic range of the annealer. But given the lack of the required precision, there's no reason to think that this number of qubits per variable n would need to scale much with the problem size, so it will hopefully be okay. Anyway, now that we have our variable encoding, we can get a quadratic Hamiltonian for each suitable triplet by simply adding the three variables in that triplet and squaring them, and then to get the full Hamiltonian we can just sum those. That was an overview of how the Hamiltonian is constructed, but there are some more details that are worth mentioning. It's the case that not all possible triplets should be used. As I said before, those triplets with the strongest reflections are the best ones to use, as these are the ones for which the triplet rule should be best satisfied, so we can set a threshold and use only the strongest triplets. In fact, it's choosing which triplets to use based on these reflection strengths that is unique to the particular structure. Without this, we wouldn't actually be using the fraction data in constructing the Hamiltonian. The values don't explicitly appear, as I've presented. It's also true that not all phases are independent. For example, phases which have oppositely pointing reciprocal space vectors should have opposite phases. There are some other symmetries that exist for some molecules, but we've not considered this in much depth yet. The final structure will also have a three-dimensional real space translational freedom. The crystal structure is still regular, regardless of where you put the origin, so we can also try fixing three independent phases in order to set that origin. Going back to what I said about the triplets with the strongest reflections being the most important, even after choosing the triplets based on the threshold, we can also then weight the triplet terms that we are using according to the reflection strength. We could do this just by including the product of the strengths of reflections as a weight term when we sum up the triplet Hamiltonian. On to checking whether the problem of Hamiltonian works. So far, we've considered a small molecule with some fake diffraction data generated numerically from the real structure. Then we've used classical simulated annealing to test the Hamiltonian to answer the question, do low energy states work well in terms of giving phases that translate into sensible structures? Occasionally, it turns out that some of the low energy states do, so these two pictures here demonstrate this. On the left is a picture of the true structure. The picture on the right shows some low energy outputs, so the blue regions in that picture on the right are the electron density of the real structure, so basically the same as is in the figure on the left, but the red and yellow regions show electron densities calculated from low energy outputs from the simulated annealing. It can be seen that although there are some erroneous regions, there is quite a lot of overlap between the productive structures and the real structures. On to what we hope to do next. Firstly, it seems that not all of the low energy outputs give sensible structures, so this begs the question of can we bring more physics into the cost Hamiltonian to maybe rule some of these out. An alternative to this, which I'm convinced could be an alternative based on discussions with the crystallography collaborators. Perhaps these are sensible structures in disguise, so perhaps some of the substructures are correct, but they could be wrong relative to each other, and if that's the case, then they could maybe be used to bootstrap classical algorithms. Secondly, we should try different levels of discretisation, so in the example I showed, we used n equals four qubits per reflection, but perhaps we can get away with using fewer or maybe we need to use a few more to get the results better. That's something to investigate. Then, of course, we want to try it in practice, for example, on a de-wave device. It's not entirely clear whether the small examples that we looked at so far, whether they might be a bit too large still for the current devices given the need for the minor embedding and depending on what level of discretisation we need to use, but if that's the case, then we could obviously still try some of the hybrid solvers that are available. We could also try creating some toy model structures that will fit on the de-wave more cleanly in order to test this a bit more thoroughly. That's everything I want to talk about regarding this, so thank you for your attention, and I'm happy to take any questions. Good. Sins. Could you go back to the slide where you defined the problem in the problem Hamiltonian, like this? The computational problem that you're trying to solve. Yeah, well, so the computational problem is to take that diffraction data and turn it into an electron density structure map, and an approach that may work here is through this triplet rule, and this is an approximate relationship, and we're using this as the basis for our optimisation. I'm not a crystallographer, so I don't really understand the physics of this. What I'm trying to understand is what is computationally hard about this, and why do we need quantum computers to do this? The crystallographers tell me that it's computationally hard. They approach it as they use, work well for small molecules, but then, unless they can find some initial guess, like maybe through those molecular replacement things that I mentioned at the beginning, if they can't find any information about that, then sometimes their procedures just don't work at all. Right, but if you try to map this to some well-defined computational problem that, you know, is this going to be hard? Can we associate a complexity class with it? So, I don't know the answer to that. As far as I know, they haven't tried the exact analogue of this classically ever before, so we're thinking that maybe if this doesn't work, then at least we could maybe come up with some kind of applied algorithm or something, but I don't know what the complexity would be. The form of the how it only turns into looks like it may well be an NP-hard problem in the same way as many other combinatorial optimization problems are, but yeah, obviously in its native form it's not a discrete problem, so that could change things there. So yeah, those are my thoughts. The short answer is I don't know. I don't know the answer to that. Thanks for the talk. It seems to me that to map the problem into a Google, you need to discretise the angles, but then you use SA. You could have used SA directly in your own union, right? The reason we used SA was because it was an easier way to investigate the low energy states of the Hamiltonian that we intend eventually to apply on a quantum annealer, so that's why we used the discretised Hamiltonian form of it, but yes, certainly applying the continuous version with a classical optimiser would be a good idea, and as far as I know, that approach hasn't been tried classically yet, so that's another avenue, yes. Thanks. Any other question? Actually similar question to what Daniel asked, so what you're writing here is a linear question, right? So if you are trying to solve linear question on quantum annealer, then it's definitely a misuse, but maybe you had the slide where you described how you embed this problem on a quantum annealer, so I actually didn't understand what is in this question more than the linear question. Could you clarify? I mean, so it does look like a linear question, I mean, because the values, because this sum can be positive or negative, we square it in order to put it on the device and it becomes quadratic. That's correct, but the question phi t1 plus phi t2 plus phi t3 approximately equal to zero is a linear question, right? Yes, I agree. I mean, so you're suggesting that? Then probably we need to talk with the crystallographers to figure out what problem is actually hard in that regard, because this problem does not look like a hard problem. Yeah, I mean, obviously there's a bunch of overlapping constraints here, so these same terms appear in multiple different... But at the end, is there a system of linear questions in model or arithmetic model or two pi, right? Sure, yeah. Yeah, okay, yeah, that's a good point, and I should probably talk to the crystallographers about that. If no more questions are there, let's thank the speaker once more.