 to talk about signature open quantum systems. My work was in collaboration with those listed below. So let me just sort of get into what we did. So the main objective of our work was to be able to find a dynamic model that was able to fit data that we were measuring off of D-Wave hardware. And in order to do this, we had to sort of come up with a few different key ingredients that would allow us to find something interesting that the dynamical model could give us that like a Boltzmann distribution, for example, would not give us. So these first key ingredients would be one, a new annealing schedule which we used, which was able to give us some relatively interesting results. Second, we did some comparisons of open and closed system simulations. After that, we introduced this sort of new idea of something that I'll call longitudinal field noise. And then lastly, we need to be able to compare this simulation against the hardware and see what we're able to get out. So I'll go ahead and start with the novel annealing schedule. So let me give a little bit of an introduction on how annealing on D-Wave works. So you have your Hamiltonian shown here where you have your initial transfers field given by these Sigma X terms. And then you have your final or target Hamiltonian on the right side given by these Sigma Z turns. And these are controlled by your annealing schedule functions, which in the case of the Landl D-Wave machine looks something like this. We also have this annealing parameter S, which effectively gives you the percentage of the anneal which you've completed. And lastly, we also need to consider the total annealing time of the system in our simulations. So let me just sort of give a little bit of a discussion of some of the limitations we have when trying to perform a measurement on D-Wave. So we're not actually able to interrupt the system. We have to perform a measurement when S is equal to one. And the way that the documentation recommends that we work around this is by doing something called quenching. And this is where as we continue our anneal, once we want to perform our measurement, we rapidly increase the rate of the anneal. And this has a few drawbacks because if you're working with a relatively small system size, something that you could actually simulate on a classical computer, you end up having such a large minimum gap that on the time scales that we considered with D-Wave, we were getting relatively difficult to differentiate results. So that led us to introduce a sort of new annealing schedule, which we call the H-Kill protocol. So this was our alternative attempt to quench the system effectively and sort of measure these dynamics partway through the anneal. So in order to do this, we used D-Wave's H-Gain schedule parameter, which allows us to vary the strength of the longitudinal field, allowing us to effectively kill the field partway through the anneal. The overall form of it looks like this and we're able to go very quickly, much faster than we're able to complete even the fastest anneal you're allowed on the hardware. So the big downside of this though, is that there is only an H-Gain schedule parameter. There is no J-Gain schedule parameter. So we have to consider independent qubits with no couplings. So this doesn't end up actually being a limitation though for the work that we're wanting to do here. And there is one last thing which I sort of wanna mention, which is this residual transverse field which ends up playing a big part later on. So that gives a little bit of an introduction to this novel annealing schedule that we introduced. So now I'm just going to very briefly cover the open and closed simulations that we performed, effectively just giving an outline of the equations that we were implementing. So for the closed system case, we were considering the Louisville von Neumann equation as shown here. It's effectively the same as the Schrodinger equation only in this case you're working with density matrices. And then for the open system case, we were considering the adiabatic master equation with independent ohmic heat baths. So the adiabatic master equation effectively has this Louisville von Neumann term and then a correction term which is dependent on these ohmic heat bath terms given by gamma, which have a certain bath coupling strength hidden within it. So that's what I mean when I'm saying that we're doing closed system and open system simulations specifically. So let me now get into this longitudinal field noise that I was talking about earlier. So this ended up being something that we found was able to be almost like a secret sauce that would let us reproduce results that we were seeing on the hardware. We don't necessarily have the best intuition for why it would be there, but basically we find that there seems to be some sort of what we decided would be Gaussian distributed noise that is constant in each simulation shot. So effectively what we would say is, okay, we have our Hamiltonian. Now we're going to modify it by a term which before we do the simulation, we sample from this Gaussian, scale it and then run our simulation. So that's what this longitudinal field noise is. It ends up being very important for reproducing the dynamics that we observe later on. And now with that, let me get into our experimental setup for running our hardware simulations or hardware experiments rather, which we compare against our simulations. So the experimental setup was that we applied this H gain kill protocol across various different values of S. In this case, for what I'm showing in this presentation, we did not consider S equal to 1.0, but the results do end up being consistent with those. Instead we consider the open range between zero and one. And then in order to make sure that our predictions are consistent, we vary the program field H in and we vary the annealing time given by tau. So when I say that we did simulations at low values of tau or low values of H in or medium to high, this is what I mean by that. And in this case, the annealing time is in terms of microseconds. So what do we actually observe when we run this on the hardware? In the low annealing time regime, we get this relatively flat curve. And as we increase H in and our annealing time, we get some various effects. So we basically noticed that there were three main stages to the results that we were observing on the hardware. There was this first stage where we were getting what appeared to be a lot of mixing with the environment. And then effectively a 0.5 measurement probability of the ground state. Then we had this sort of transitionary stage where the probability was approaching the third stage, which we call the saturation stage. So what do the open and closed system models without longitudinal field noise end up predicting if we try running simulations of something like this? And the answer is something very surprising actually. So in the closed system case, by introducing this H-kill, we are effectively having our block vector start rotating around that residual transverse field term that I mentioned earlier. So because of this, we end up getting these really rapid oscillations which we don't actually observe on the hardware. And this is to be expected because nothing is a completely closed system. But what's more surprising is that the open system simulations, we're not able to reproduce what we were observing on the hardware either. Basically the adiabatic master equation was predicting that due to the large time scale after we turned off the longitudinal field, we would effectively have this same oscillation but it would be dampened by the bath interaction. And then as we approached one for our H-kill point, we got gradually increasing probabilities. So what happens if we end up introducing this longitudinal field noise? And this is actually really quite surprising. In both cases, it produces S curves which look relatively similar to what we observe on the hardware. So we need to be able to fit what happens here. And I hope that this open system model is turning a few heads because this is really quite surprising that by introducing this longitudinal field noise, we're getting effectively an amplification of results which we don't even expect to see in the open system model. I'll give my best guess as to what's actually occurring later on in this talk. So let's start by trying to fit the closed system with longitudinal field noise. So when we try to do this for very low annealing times, we were actually able to get relatively consistent results with what we were observing on the hardware. But as we increased our total annealing time, what we ended up seeing was that the transition would not match what we were seeing. So this effectively ruled out both the closed system with and without longitudinal field noise because the longitudinal field noise was effectively adding a degree of freedom to our simulations. So there was no way that the closed system without longitudinal field noise could reproduce the results either. So this might be a little bit of a spoiler for my conclusion, but when we ran the adiabatic master equation with without longitudinal field noise, we were not able to reproduce anything that we observed on the hardware. So this is where I think I should start explaining why we're able to get this amplification, especially considering these high annealing time and high longitudinal field plots on the right side of this slide where we're effectively seeing what should be completely boring dynamics. So in this case, we're considering a 125 microsecond anneal. So between each of these S points that we considered, there is about a three microsecond time span. So we expect that to be long enough for the heat bath to effectively kill our dynamics. However, when we introduce the longitudinal field noise term, then we're able to actually get simulations which match up almost perfectly with what we were observing on the D wave hardware. There's a little bit of a discrepancy on some of the saturation values we were observing, but the transition stage is actually matching quite nicely. So our best guess as to why this actually works is because effectively as we're performing our anneal, we have our block vector pointing towards the one state and then we have the annealing schedule terms effectively dominating both the bath interaction and the longitudinal field noise. And then we introduce the H-kill protocol that removes the longitudinal field noise. However, pardon me, we remove the longitudinal field and the longitudinal field noise is now dominating the bath couplings. So we have this sort of biased vector which now that we introduce the noise, it's not getting as affected by the bath coupling because there's this other term which is ending up controlling the dynamics. And that's the best explanation we could come up with for why this is actually working. So with that, I think I'll go ahead and continue to my overall contributions and conclude. So in this work, we've proposed a new way of trying to quench a D wave system and finding a way in order to measure both the strength of the bath coupling and the longitudinal field noise term, which we consider. We also have demonstrated that it's necessary to include this longitudinal field noise term. And lastly, we've shown a difference for open and closed system dynamic models using only a single qubit. And with that, I'll open the floor for questions. Okay. Questions?