 Can people hear me OK? Louder. All right, we'll move closer to the mic. All right, cool. OK, yeah, hello. So first of all, thank you to the conference organizers for this lovely conference and for having me. My name's Natasha. I'm a PhD student at UCL, supervised by Paul Warburton. And I'm going to be talking about some of the work we've done, looking at the effects of XX catalysts on annealing spectra with perturbative crossings. So if you were here for his talk on Tuesday, Paul's already talked a little about some of this. But I want to clarify before I start, just so there's no confusion. While his talk was focusing more on the applications of the user's votes to diabetic quantum annealing, I'm going to be focusing more on gap enhancement for adiabatic evolution. So I'm going to start with a brief introduction to the use of catalysts in quantum annealing. I'm then going to describe what a perturbative crossing is, how they can form, and how we can consider removing one with a catalyst. I'm then going to outline our problem setting without a catalyst in which we produce a tunable perturbative crossing. And we'll then look at the effects of introducing a targeted catalyst to these systems, which has been designed to enhance the minimum spectral gap. And we'll see that while in certain cases it's able to do this. We'll also see how in subtly different settings the same catalyst can result in closing gaps in the annealing spectrum. So a catalyst in quantum annealing is an additional Hamiltonian that we introduce in order to change the path that the anneal takes from the driver to the problem Hamiltonian. And this path change has the capacity to enhance the minimum spectral gap if done correctly, thereby suppressing excitations out of the ground state during the anneal. And there are already a number of examples in the literature where catalysts have been shown to be able to achieve this. However, in order to implement this as a strategy more generally, we need to understand how different catalysts affect annealing spectra in different settings. So one particular source of closing gaps in annealing spectra are perturbative crossings, which conform when we have a problem Hamiltonian with excited states that are close in energy to the ground state. When this is the case, small perturbations from the driver can result in these energies crossing. And the result of this is an avoided level crossing towards the end of the anneal where we can expect the gap size to close exponentially with the problem size. Whether or not these crossings form will depend on the perturbations to the problem energies, and this will depend on the couplings that are introduced by the driver between the problem states. Specifically, it can be shown that for the very lowest energy problem states, those that are coupled to generally lower energy states will receive a greater negative perturbation from the driver. And so in the setting that we're illustrating here, we have a problem Hamiltonian with a first excited state that is close in energy to the ground state. And this first excited state is coupled to generally lower energy states than the ground state is, meaning it gets a greater negative perturbation and two energies cross. We can understand a catalyst's ability to remove a perturbative crossing from the fact that it will introduce additional couplings between problem states. And so again, what we're doing here is illustrating a hypothetical introduction of a catalyst which couples the ground state to a significantly lower energy state than it couples the first excited state to, thereby hypothetically removing the crossing. This strategy has been shown to be able to work by, and this work by Vicky Choi, introducing a stochastic catalyst that is a real catalyst that has been introduced with the same sign as the driver so that all the off-diagonal elements in the total Hamiltonian are real and non-positive. The catalyst that I'm going to be talking about can be thought of intuitively as a complementary approach to this in that they are introduced with the opposite sign to the driver, such that the total Hamiltonian is non-stochastic. And the catalyst is chosen such that it couples the first excited state to a low energy state while coupling the ground state to a significantly higher energy state. And as well as being sort of intuitive reversal of the stochastic approach where we're coupling the ground state to a low energy state. There are also a number of examples of numerical results in the literature that can be used to motivate this catalyst as well as some analytical arguments made in a more recent work by Vicky Choi. So more specifically, our catalyst contains a single XX coupling between qubits introduced with the opposite sign to the driver, as I say, so that it's non-stochastic. And this coupling is chosen to couple the first excited state to a low energy state. And so our aims in this work were to examine the effects of this catalyst in a stripped-back problem setting in order to determine, A, how well does this catalyst work? And B, how sensitive is the effect that the catalyst has to small changes in the problem Hamiltonian? All right, so now that I've sort of introduced the background and what our aims were, I'm gonna now talk about the problem setting that we used to investigate this. So as Paul talked about on Tuesday, our problems are instances of the max-weighted independent set problem, and we're gonna make use of some nice properties introduced by Vicky Choi in their construction. So the max-weighted independent set problem is, takes as its input, a undirected weighted graph, and the aim is then to find the set of vertices with the largest weight for which no two vertices are connected by an edge. Specific problems were on complete bipartite graphs such that the problem has two local optima. So for this five spin or five vertex example, one of these local optima corresponds to selecting the two blue vertices, and one will correspond to selecting the three orange vertices. And by allocating a total weight to each of the vertices, we can decide which of these is gonna be the global optimum, which will then correspond to the ground state of the problem Hamiltonian, while the other will correspond to the first excited state of the problem Hamiltonian. So in its encoding into a problem Hamiltonian, each vertex is represented by a spin, with spin up representing a vertex that is selected to be in the set and down representing a vertex that is not. So by selecting the weight on the blue vertices to be larger than that on the orange vertices or the total weight on the blue, larger than that of the total weight on the orange, our ground state is given by this spin configuration. So the two blue spins pointing up and the three orange pointing down, while the first excited state has the opposite spin configuration. So we're now in a position to discuss what states the problem ground of first excited state become coupled to by the driver. We take as our driver the standard choice of the homogeneous local X field, such that each state is gonna get coupled to states that are one spin flip apart. So for this ground state, it's gonna get coupled to two states where one of the blue spins is flipped to the down position and three states, one of the orange spins is flipped to the up position. This means it's gonna be coupled to two states corresponding to independent sets and three states corresponding to dependent sets as illustrated by the red vertices, oh, red edges. And then similarly or opposingly to this, the first excited state is gonna be coupled to three states corresponding to dependent sets and two states corresponding to independent sets. And by the way, the problem is encoded. The dependent sets are gonna have significantly higher energy than the independent, the states corresponding to independent sets. And as a result, we now have a situation where we have a first excited state that is coupled to more or generally lower energy states than the ground state. And by the reasoning that we gave previously, we expect this to create a perturbed of crossing. It's then straightforward to scale up this problem graph by simply adding more vertices to each subset of vertices. And so long as we have more vertices in the orange set, we expect the same logic to hold and a perturbed of crossing to form. So I'm now gonna present some numerical results to confirm that we do indeed get this perturbed of crossing in the catalyst free instance. So what we have here is a gap spectrum for the anneal on the five vertex graph. And we can clearly see the appearance of the gap minimum between the instantaneous ground and first excited states towards the end of the anneal. We can also look at the ground state evolution over the course of the anneal, which we do by plotting the overlap of the vector with the different problem states, highlighting the overlap with the problem ground and first excited state with blue and orange respectively. And we can see that over the course of the anneal, the instantaneous ground state evolves towards the problem first excited state before there is a sharp exchange into the problem ground state at the same location as the gap minimum. And this is what we expect to see at our perturbed of crossing. Lastly, we can also look at the scaling of the gap minimum with the system size. And we can see here from this plot, the gap appears to close exponentially, which again is what we expect. So now that we've established the presence of the perturbed of crossing, we can think about how we can tune its properties. So to do this, we make use of some parameters in encoding of the problem, namely the edge penalty and the precise weight difference between the two subgraphs. And as long as we keep these parameters within a certain regime, we can use them to tune the relative energy gaps within the problem Hamiltonian without actually changing the ordering of the states. And so the problem to be solved remains identical. And by doing this, we can change the properties of the perturbed of crossing. So here in the middle here, we have the same plot that we saw before for the ground state evolution. And then to the right of it, we have another plot for the ground state evolution, again for the five spin example, and again for the catalyst free case, but with these slightly different problem parameters. And we can see that by changing the problem parameters, we have significantly softened the crossing in that the instantaneous ground state is significantly more mixed before and after the transition. And also the exchange between the problem states that occurs is not as sharp as it was in the previous parameter setting. We can also compare the gap scaling in these two cases. So here in the darker purple, we have the gap scaling that was associated with our previous parameter setting. And now here in the lighter purple, we are showing the gap scaling corresponding to the new parameter setting. And we can see that while the gap still appears to close exponentially, the exponent is significantly milder in this case. Okay, so now that we've gone through the system without a catalyst, we're now gonna look at the results when we introduce a targeted catalyst to these two settings and we're gonna refer to them as the weaker and stronger gap scaling cases. So the excess coupling that we introduce for our catalyst is between two vertices in the orange subgraph. And this gives the couples the ground state to a dependent set while coupling the first excited state to an independent set. And so as we discussed before, this is a catalyst that is giving, that is coupling the first excited state with significantly lower energy state than it is coupling the ground state to. So we're first gonna look at the results for the weaker gap scaling case. So here we have a plot showing the gap size at the perturbative crossing with increasing catalyst strength for our five spin example. We can clearly see that the catalyst does indeed produce some gap enhancement with the gap size reaching a maximum at around a catalyst strength of 1.3. We can also look at the effect of the catalyst on the gap scaling here. And we can see from this plot that if we introduce the catalyst with the optimal strength for each system size, it's able to significantly suppress the closing of the gap, which was what we were hoping to achieve with this catalyst. So we can now look at the same results for the stronger gap scaling case. And so, yeah, here again, we have the gap size with increasing catalyst strength. And again, we can see that the catalyst is able to produce some gap enhancement, but we can also see that for a lower catalyst strength, there's actually initially a closing of the gap. And then if we look at the results for the gap scaling, we can see that while the catalyst is able to achieve some gap enhancement for each system size, in this setting, there isn't any noticeable change in the gap scaling, even if we introduce the catalyst with the optimal strength for each system size. We also observe another additional effect in this setting, which is the appearance of a new gap minimum in the spectrum. So here we have four plots for the gap spectra over the course of the anneal for four different catalyst strengths increasing from left to right. And we can clearly see the closing and reopening of an additional gap minimum in the spectrum. So in order to understand the closing gaps that we observe in the stronger gap scaling setting, we can look at how the catalyst impacts the evolution of the ground state vector. So first, I'm gonna note that because we have a non-stochastic total Hamiltonian, our ground state vector can have negative components as well as positive now. I'll also note that for our particular system and choice of catalyst, the components corresponding to the problem ground state and first excited state take opposite signs at some point over the course of the anneal and that this point moves earlier in the anneal as the catalyst strength is increased. So here we have a plot showing the same data for the gap size at the perturbed of crossing for the stronger gap scaling case. But now we've also plotted the location of this gap minimum with a dashed gray line and the location of the sign change that I just talked about with a dotted gray line. And we can see that this gap closes when we have these two positions intersecting. We can also look at the same kind of plot but for the new gap minimum that forms in the spectrum. And so again, the purple line is showing the gap size of this additional gap minimum that forms. And then again, we have the position of the gap minimum in the annealing spectrum as well as the position of the sign change. And again, we can see that this gap closes when these two positions intersect. And so while we can associate these closing gaps with these intersecting positions, this isn't actually quite sufficient to differentiate between the two settings. So if we look at the same kind of plot for the weaker gap scaling case. So again, this purple line is showing the same data for the gap size in the weaker scaling case that we saw before. And now again, we have the dashed line and dotted gray lines showing the position of the gap minimum and the position of the sign change in the anneal. And we can see that while there is a catalyst strength which these two positions intersect, we don't actually observe a closing gap in this setting. So while it's not yet entirely clear to us why we're observing such different behaviors in these two subtly different settings, we do discuss this a little bit further in the pre-print you can find online trying to figure out what some of the key differences between these two settings could be. So to summarize, we've applied a targeted catalyst to two subtly different settings and found that its effect on the annealing spectrum is highly dependent on small changes to the problem Hamiltonian. So specifically when applying it to our weaker scaling case we found that the catalyst was able to achieve a significant reduction to the gap scaling with the system size, which is what we were hoping to achieve. However, when we applied the same catalyst to our stronger scaling case, we didn't observe the same reduction to the gap scaling. And we also observed an additional effect of the appearance of a new gap minimum in the spectrum. And so what's interesting about this, and this is more to do with what Paul talked about on Tuesday, is that while we weren't able to achieve the same gap suppression, closing of the gap in this setting, what we've actually ended up with is an annealing spectrum that appears amenable to a diabetic quantum anneal in that we could allow the system to transition out of the ground state at the first small gap and then back into the ground state at the second. And so potentially still having some speed up over the catalyst free case. And some initial dynamic system, dynamic simulations for closed systems have shown that this might be possible. Obviously all these results are for a very stripped back problem setting at the moment. And so it still remains to be seen how these results are gonna translate to larger system sizes and more realistic problem settings. With that, I'd like to thank everyone for listening. I'm happy to take any questions. And also I have got the code for our archive pre-print on the bottom right if you want to take a look. Good, thank you. If you look at the top left plot on a log log scale, does it look like it becomes power law with the catalyst? I haven't actually done that. I've said I would really like to go to larger system sizes before I try and make any comment on- Of course, but just out of speculatively. So it would be very interesting because it would effectively turn what looks like an exponential scaling into a polynomial scaling. And then that raises the question of how much effort is required computationally to find the optimal parameters that bring about this perhaps power law scaling? Yes, I definitely agree. It would be really looking really promising if it was actually a reduction to polynomial scaling. But I don't know if it is yet. And so yeah, I definitely want to go to high system sizes before. Okay, thanks. Another question? Yeah, I want to ask a related question. So do you have any intuition like when you go to larger actual problems beyond the bipartite? Do you have intuition regarding like where do you need to apply the XX catalyst? Yes, so the way I think about this is the XX coupling is sort of guiding the anneal away from the optimum or the local optimum that it would be heading towards and the one that you would get stuck in due to a perturbative crossing. And so if you had more local optima that were producing perturbative crossings, you might perhaps have to introduce a coupling or a kind of sub catalyst as it were to tackle each local optimum in turn. But that's again something that we'd want to numerically check by going to larger system sizes so we could have systems with many more local optima. Interesting, thanks. Any more question from here or online? So if there are no more questions, let's thank the speaker once more. The next talk is by Professor Paolo Zanetti. Recording stopped.