 Yeah. Yeah, you're not saying, so go, yeah, go and zoom. Yeah, yeah, yeah, zoom and record as well, is it okay if I... Sure, I think so. Recording in progress. Share screen. And then I guess I can share. Yeah. All right. And you go on full screen now. Yep. Let's see if it works sometimes. Oh, fantastic. Okay, everything works. Okay. I don't know. Yeah, I guess. Yeah, I had the meeting. Now the other one, the other one. The other one. I had the floating meeting controls. I had floating meeting controls. Okay. Okay. Very good. So the last talk, the very last talk, is by a Nicholas Chancellor, and the title is, Quantum Managing Solves Problem. Yep. First, I'd like to thank the organizers both for inviting me and for organizing a really great conference, sort of first AQC in person for a while. And I am aware that I'm the only thing standing between you and sitting on a nice beach, so I'll try to keep that in mind when I'm giving the talk. So before I get into the contents of my talk, I want to make people aware of two UK projects that myself and Jim Kenden are involved in that may be relevant. They don't explicitly have international collaborations in them. However, we would be willing to explore options. One CCPQC, which is collaborative computational projects looking at academic applications of quantum computing, heavy emphasis on quantum annealing, even if it isn't obvious from the webpage, and Excalibur or Quebec, which is looking at exascale, so very large compute applications of quantum computing, and Viv Kenden's actually the PI on both, so she's probably who you'd want to talk about if you're interested in learning more about those projects. The CCPQC is actually what has funded the work Adam talked about. So what this talk is about, I've decided since it's been a rather long pandemic, it would probably be good to review several bits of work that we've done that kind of fit into a common theme, and that common theme is the title of my talk, How Quantum Annealing Solves Problems. So first I'll present a bit of theoretical work about how to understand anneals when you're very far from the adiabatic limit, and then some bits of theory about both how dissipation-driven annealing works and how the encoding of optimization problem affects dynamics. These works were actually done in different places, particularly these two were done in my academic role, this last one was done with a company called Quantum Computing Inc. who I work with quite a bit now. So let's start with the traditional picture of adiabatic quantum computing and we're thinking about actually solving the problem not just about getting an approximate solution, although a lot of these arguments will carry over. So here you can imagine you map an NP-hard optimization problem to a Hamiltonian, there's an unknown ground state as a solution, you've slowly changed your Hamiltonian, of course you all know this, and the adiabatic theorem of quantum mechanics guarantees 100% success probability if you run long enough, and of course this long enough where all the nastiness is hidden, there's no guarantee on how long that will be in general or how short. So there's advantages and disadvantages of this picture and its extensions. First off, it is very theoretically satisfying, right? It's effectively deterministic, it always succeeds, that's nice. And there's an intuitive picture involving only the first few states in a gap, sort of simple mathematical quantities, but if we make some sort of standard science assumptions, if we assume that P is not equal to NP, then if this succeeds roughly 100% of your time, your needle time had better scale exponentially. Otherwise, you've proven something very surprising in computer science that most people don't think is true. I will want to just point out that there are more sophisticated approaches, this Cresson and Lidar review, give some very nice reviews of that. That's not the topic of this talk. The topic of this talk is thinking about this in different ways that don't involve anything that even looks like an adiabatic theorem. So there's questions of what can be done. Well, there's really two options. One, you can just say, again, this is if you're solving the problem exactly, but a lot of this will carry over to getting a good approximation. You could just say, well, I'm going to run for a very long time, so I'd better just do some error correction or have low temperature dissipation that's just at a very, very low temperature and keeps me in a low energy state. But this is going to be very, very hard work. You could also do shortcuts to adiabaticity, but somewhere in there, unless, again, something surprising happens in computer science, you're going to have to do a lot of work and probably you're going to have to run for an exponentially long time. The other approach, which I think is often implied in a lot of experimental work but isn't considered much theoretically because it's hard to get a good theoretical handle on, is to just run many, many times and you're going to be succeeding with exponentially low probability, which seems like a disadvantage, but if you believe P equals NP, in some ways is an advantage because it gives you a get out where you can run in time that scales mildly rather than scaling exponentially. So this is conceptually very unsatisfying as well. You know, here you're feeling like you're making your money by doing hard work, here you feel like you're making your money by playing the lottery, but I'd argue that a lot of the, let's say disadvantages with this approach are actually more sort of psychological and how you think about these things more than something that would actually come up practically. So, let's think about rapid quenches. What can we say about this? Sorry, this is a very dense slide. The details of this can be found in the paper, but basically if we think of a quantum walk, so we just have fixed parameters, we know the energy will be conserved just because a time-independent system, this is a closed system will conserve energy. For this first part, I'm only going to be thinking about closed systems. You can imagine trotterizing the time evolution and then all you have to think about is a series of these quantum walks with a series of steps where you reduce the ratio of these A and B's. If I assume I do this monotonically where A is always decreasing relative to B, I can rescale the time a bit and rescale it to a Hamiltonian where I have a gamma here. A B prime and then a problem which doesn't have anything in front of it and the gamma always decreases. So then in this rescale problem I can look at what if I start in the ground state of this driver Hamiltonian, what happens when I reduce the gamma. If you look at it, you'll always have some residual energy in this part and so you'll actually be reducing your total energy. So you have steps that conserve total energy and steps that, at least from where you started, reduce your total energy. And once you've reduced this energy a bit you can never actually get back. So your energy expectation with respect to your problem Hamiltonian if you do a bit of math will actually always be lower than, in this case I've called it zero for an Ising model, in a more general case it'll always be lower and for an unbiased one it will always be lower than what you'd get from just measuring your initial condition. Lower or equal I guess strictly speaking. If you do biasing you can actually get less than zero as I'll talk to you later. Again, this is just to give a flavor if you really want the details please look at this paper. So the advantage here even though it's not a very strong result is that this is a very general result. To be better than random guessing on average all you have to do is be monotonic, start in the ground state of this driver Hamiltonian and there's a technical consideration that's not really a concern in real life that your driver should probably be gapless otherwise bad things might or not gapless otherwise bad things might happen. So what is allowed there's no limit on how fast your algorithm runs including even discontinuous changes in this gamma these considerations I've had I've never assumed that this changes in continuous way and I've not made any assumptions about this driver Hamiltonian being diagonal on an orthogonal basis to the problem Hamiltonian I'll come back to this later you can actually do a bias starting state. One thing this doesn't cover is a reverse annealing type protocol where you increase this gamma because then this argument of you're always pulling energy out relative to where you started won't apply during this phase but you can do crazy discontinuous things pauses all of that so let's do some numerics and see an example of how this works so let's consider a very simple schedule we take gamma at a higher value instantaneously reduce it take it to a lower value so we can look at energy expectation here for the two parts of the Hamiltonian the driver and the problem you can see we start out there's some fluctuations the driver has nowhere to go but up increases total energy is conserved so the expectation with respect to the problem has to go down by energy conservation then our total energy stays fixed in this regime it can only change here it goes down this is gold line and when that happens if you look closely this driver Hamiltonian energy expectation goes down and then it goes back up because it's pulling energy back in when it goes back up it needs to be transferred from the problem Hamiltonian there are no guarantees here on finding your probability of finding your optimal state but in this small example I believe it was five cubits or something the point here is more pedagogical than to show how well this actually solves problems your success probability also goes up but there's no general theoretical guarantee of that and if you think of a non-instantaneous quench you can just imagine a stair step of these basically and you can imagine there's a limit of stair steps and in practice when you do exact diagonalization based simulations that's actually how you structure your simulation anyway so you just take a mental model that's exactly how you simulated anyway and you can imagine a bunch of these steps and they they do this so as I said before this is general but is rather weak because all I'm doing is saying well you do better than just measuring your initial state but what does it tell us? Does it give us anything useful and I'd argue that it does for one you now say well I know my dynamics are helpful I know my dynamics are helping me get better solutions in some sense so maybe if I'm designing protocols rather than thinking about doing something with a gap I can just say you know what I want some measure of my dynamics and I want to maximize it and I have a guarantee it'll be helpful and I'll give a simple example in terms of finding schedules later on as I mentioned before a bias search can already start from a very good guess and this gives you a mechanism of understanding dynamics very far from the adiabatic limit again some of the diabatic stuff can also give you this here I haven't made any assumptions about my spectrum though I haven't assumed there's a low line portion of the spectrum and a high line portion of the spectrum my spectrum can be whatever I want so the diabatic stuff is very interesting but this is sort of a complimentary view on it so one thing you can do the first thing I suggested is you can say well how do I quantify the amount of dynamics well one thing I can do is I can just sample from two state subspaces of my system basically looking at flipping individual bits if I'm doing a transverse field driver and I can't do all of these because there's exponentially many but I can sample and I can sample and I can say well I can get a measure of my dynamics how likely am I to flip and then looking at a conjugate basis I can get a measure of how much do I see the energy difference from the problem a sort of product of these two which is this green line the overall amount of dynamics because I don't want to be in the situation where either my off diagonal elements are small compared to my diagonal elements and so I never transfer between the two states or the opposite case is also bad because then I'm transferring between the two states but with no awareness of the problem what I really want is to be somewhere in the middle for most of these bit flips and so I can do this it's just a very simple measure of the dynamics and so one thing you note is as long as you're willing to accept some statistical error from sampling this will be efficient even for very large problems you can do it so let's look at a simple example of a Schoenck-Röck-Patrick model so this is much more of a proof of principle result than a you know rigorous demonstration of how well this works but if we look at this and we say alright let's see our change of the s as 1 over chi where chi I've actually changed notation here oh no not chi is this measure of the dynamics set it to the inverse of that and I can get a schedule that's not far from linear but does help me succeed a little bit better again this is a small example just to motivate the idea that this might be useful here it's probably the optimal schedule is pretty close to linear anyway so we don't get that impressive of the gain this isn't on a log scale so this really is just a 20% gain or something but it motivates it in principle and if you have a more complicated problem structure with constraints or something your optimal schedule may be something more important more complicated and you may be able to find something more sophisticated and it does perform a little bit better though so it does show that there might be something too this principle of maximizing the dynamics um so now if we think okay we have this mechanism can it be applied in situations where we have hybrid protocols well dissipated reverse annealing is implemented currently on the D wave is completely out because of this word dissipative everything I've done so far has been closed system so not suitable for coherent algorithms I'm thinking about even if you did it coherently you have this point where you're increasing the gamma that's not not going to be compatible with the theory we've done various coherent reverse annealing tends to involve turning on a Hamiltonian in the middle um again that's going to have you actually have two gammas but one of them will be increasing and again at least the way we've written our theory won't be compatible but you can look at a simple version where you've just biased your driver where you just say instead of being transverse I'm going to have a combination of transverse and longitudinal fields where the longitudinal field is pointing towards some guess there's been some interesting work on this this paper from Chinese physics letters and this work by Tobias Grass and this is compatible because remember what I said before none of this relies on being um in a sort of complimentary basis to the problem basis so here you can actually be completely compatible um and so this sort of concludes the first bit of this theory of how you would understand solving very rapid quantum annealing but in a closed system setting this given recent d-wave experiments is relatively relevant because they do seem to be actually approaching this regime where you can ignore open quantum system effects but only if you anneal very fast so trying to build more work understanding this limit would probably be very, very useful and we've taken some first steps that we hope people will build on so now moving on to a dissipative setting and here I'm going to be talking about experimental work um so in particular having eising models gives us ways where we can actually design energy landscapes the energy landscape is really a hyper cube but I can draw a cartoon on a line idea um the problem I'm actually looking at is here the details are in this paper but basically what you can have is you can have a 16 spin system where you have a starting state um you can flip four of the bits to get to a true minimum so it's fairly close to a true minimum and then you have to flip more bits to get to a false minimum but that's broader so these these cubits on the outside I've offset the fields and couplings so that they um these are free to fluctuate so if you do forward annealing and this isn't too much lower than this you will end up in this false minimum with a high high probability um if you do reverse annealing starting from here well maybe you land in your true minimum because you bias to something closer to this if you don't go backward too far um so this is interesting both to test the core idea of how the reverse annealing algorithm works but a lot of people have already tested that but we can also because there was a time when there were both higher and lower noise d-wave chips available really look at how does noise affect this picture um and that's what we're going to look at here so first off um we took a more and less noisy um version of the qpu um and we say okay well what happens well first if we look at so this gamma star is how far we've reversed it um because the schedules are different rather than comparing with s with the annealing parameter because those would actually be different we've just looked at this ratio of a and b that I call gamma um that gives us a more apples to apples comparison and I look on the left I have lower noise on the right I have higher noise um and I look at probability of finding this true solution this slightly lower energy state um versus this gamma and you can see for both cases if I'm don't reverse enough I spend a lot of time I'm just trapped in my initial state or I don't tunnel much if I reverse too much I go to the false minimum um and in between I can find this solution um now if you look at this um carefully um what you can find is for the same run time um more fluctuations are actually needed and it's kind of hard to say on this plot um for the lower noise device then for the higher um then for the higher uh higher noise um or yeah then for the higher noise device so this suggests that um something sort of I would say maybe not that surprising is happening that because this is mediated by dissipation the less dissipation I have the more quantum fluctuations I need um the advantage here is that it suggests that spin blast spin bath polarization that Paul was talking about is probably not the dominant effect here because then you'd see the opposite um then you'd probably see that what would actually happen is the opposite effect where having more noise would mean you'd need to reverse more because the spin bath polarization would be stopping you from tongue um that will of course still be an effect here just probably isn't the dominant one um so if we look at a more exciting difference we can then say well what happens if we look at the again probability of finding the true solution um for a higher and lower noise common theme this J.T. is basically the strength of the barrier between the two between the true minimum and the starting state and as I make this barrier higher I see that my um my lower noise system is less likely to find that is more likely to tunnel further to this false solution so in this case finding a false solution is actually indicating a longer range search um I could have reversed the differences in the energy it wouldn't have made much of a difference um but what I'm seeing here is I seem to be branching more into these um if I have a high barrier here so I can go further back I seem to be branching more into a further local minimum when I have lower noise so it's sort of suggesting a longer range tunneling um I can't get into the details here but we can build a more sophisticated model where we actually look at something where we can separate time scales and say okay let's imagine a situation where first we branch and then we tunnel between the true and false minimum and what we're really interested is the branching we can then do some fitting of the false minimum probability and look at what we've called a branching ratio basically the ratio of how much you go to the further minimum versus the closer minimum and the same result we find that when this well when the barrier is small we just tend to go to the true minimum um quite a lot for both when the branching is large we instead um the less noisy qpu we tunnel into this further minimum more um and you know this is just the fits we used there's not much dependence on the time for this branching ratio on how long we pause in the middle of the anneal um and it sort of confirms what we were thinking that lowering noise causes effectively a longer range search um we haven't done the open system modeling to understand this in detail um we think that would be very interesting um but it's probably an encouraging sign that something going on with this device is allowing you to search a longer range when you have um less noise um when you when people have engineered the device to um to reduce the noise um and this also may explain some results that have actually shown improved performance on lower noise device so now let's look at a different thing that can affect solving problems um which is looking at what the structure and encoding of your problem is now sort of traditionally the viewpoint is often to think of encoding as the computer science part of it of what you're doing and think of how the annealer runs solving the problem as physics and don't mix the two what we're going to do is we can think about what we're going to think about effects of problem structure and we're going to particularly think about this in the case of higher than binary problems and using domain wall versus one hot encoding but looking at the physics of what we can do and again investigating this experimentally again this was work done with quantum computing and you probably noticed the slide background has changed um so higher than binary problems show up all over the place for instance a truck can go down any of three roads a task can be scheduled at any of five times a component can be placed in seven places on a chip all of those are fundamentally not expressed as binary problems but are still going to show up in very important places all over the place so to do this we can define two index objects to describe each variable and build what we call a discrete quadratic model which is just products of these so it's a fairly straightforward generalization of a cubo but to higher variables still quadratic though we haven't allowed higher order interactions so there's three ways to do this I'll only really talk about one hot and domain wall but you can also just assign bit strings to configurations some problems are very naturally expressed in binary encodings the one we're looking at is not if one is then you're probably better off using a binary encoding to be honest however if they're not then you have ugly overheads because encoding arbitrary interactions between binary variables will require higher order terms if you have a device that only gives you quadratic terms mapping between those two will require a bunch of auxiliary variables we're not going to talk about it here but we actually looked in detail at the variable counting and what we really found is domain wall is where you win in terms of variable count unless you're doing trivial problems with a very low degree if you have to use auxiliary variables to map higher order interactions binary will lose to the other two just in terms of variable count you can look at the dynamics of that but we haven't because of many encodings you could consider and so we'll think of these two there was previous work presented earlier on the conference on domain wall in the setting of classical problems we're going to look at solving on an ineither so what we're going to look at is not quite a quadratic assignment problem but motivated by the quadratic assignment problem we're going to look at one weighted version of that so we're going to look at only solving the constraints so this is not a hard problem this is a problem that not only can you solve on a classical computer you can solve by hand it's literally you just have to give each one a unique label we encode that constraint on a quantum ineither since this is a device at finite temperature and has other errors if we encode a large enough constraint like this it will have trouble solving it even though it's not a computationally hard problem it has a very degenerate ground state and because it's an easy problem it's easy to thermally sample from we don't need to build huge Monte Carlo simulations to to get thermal samples from this which is part of why we picked this is we can do you know fairly simple simulated annealing running on a laptop to get a thermal distribution out of this and we checked that the details are in the paper so before I get to what happens when you use a very trivial problem I would like to just point out that this encoding has in other work, in this work on the bottom been shown to work quite well for non-trivial problems, in this case max3 coloring problems and here we just did a six way comparison between using one hot domain wall for 100 of these problems fail is where we've failed to embed on at least one of the problems and then we just look at win and loss versus meaning which gets a more optimal solution and then we count them and you can do statistical significance testing and you can see not how many times each one wins, doing a little bit of statistics you can say well what's the probability that happened by random as you can see for instance in this case where domain wall on 2000q got a better solution 97 times and one hot only got a better solution 2 times that's very very unlikely to happen randomly of course some of these lower ones are unlikely in these cases where it solved them all no one's better than the other so this five node one you just solve it both with both encodings an interesting thing here is using domain wall on 2000q so on the older processor versus using one hot on advantage you still do better so using a more clever encoding can actually give you more of an advantage than using a more connected processor of course you can do both you can run a domain wall encoding on advantage and do even better as you see here we also did this on K coloring problems and saw similar results so back to what we're looking at the problem we're actually looking at for the study which is unweighted assignment problem and so if we look at two encodings so this is actually a double one hot constraint what we do is we just replace one of the one hot constraints with the domain wall encoding the details of that can be found in some of the papers I've referenced earlier and we're also discussed in the previous talk I'll discuss later you can actually also download Python code to create these encodings if you don't want to do it yourself by hand but what we see is if we look at the number of feasible solutions and we have the total number of feasible solutions that exist how many we find for 100,000 reads at each side so 10 embeddings with 10,000 reads each what we find is up to size 6 which is find everything with both encodings then domain wall does a bit better one hot does a bit worse and these drop off until at some point your encoding is so large your sort of thermal effects mean you're not really solving it with either so if we then look at well what's the rate of feasible solutions meaning how many of my solutions are feasible how many of these follow all the one hot or domain wall constraints and have unique values of each of the variables and what we can see is this decreases exponentially or perhaps super exponentially but again in general on average domain wall is finding feasible solutions more often except for at very small sizes one hot does better we don't really quite understand why that is but it probably has something to do with the low energy states so one explanation for what's going on here is thermal excitations so this was actual device data with where you can see these crosses of 10 samples if we then do metropolis algorithm or just quickly sorry I said simulated annealing we actually can get away with even just a metropolis algorithm and we find that as we increase our temperature domain wall does better this is actually not surprising because if you have the same number of correct solutions and a one fewer qubit per variable as this domain wall gives you then your fraction of correct solutions is higher temperature limit then well in the infinite temperature limit your probability of success will just be your fraction of correct solutions that fraction will be higher if you have fewer qubits in the same number of correct solutions so it's kind of not surprising that in this high temperature limit your approach each of these you see the same cross over here so it suggests that this isn't necessarily a dynamical effect it's not but more of a thermal effect one effect we have to look at here is the fact that we've had to minor embed an interesting thing here is we've used the uniform torque compensation heuristic just to default it's actually choosing a slightly stronger chain strength for domain wall than one hot but there's not a huge difference and that difference gets smaller at larger sizes but this is something we need to take into account when we do our analysis so then if we again look at a model that's looking at thermal equilibrium and we just assume we're freezing at 5 gigahertz that's a very rough assumption but right now we're just looking qualitatively we again see the same behavior we've seen before now with m the size of the problem versus the probability of finding a feasible solution rather than temperature again we see the same crossover this point is left hollow because we don't actually have any experimental cases where we found any one hot any solutions using the one hot encoding here out of the number of samples we've had so it's not at the same location but it shows that what's sort of dominating here is probably the thermal effects but there's probably there is going to be some effect freeze time so we're assuming a frozen in thermal distribution we're thinking of a kibble ziric style approximation and we have a known physical temperature and we have our experimental success probabilities so from those two things we can back calculate this energy scale b the energy scale of the problem and therefore the freeze point and then I'm not going to talk about it here but then we have to go back and verify that are we really justified in at the freeze point ignoring quantum fluctuations are they small enough using again the known schedule and what we've found is maybe except for the very largest sizes at one hot we are I haven't actually included that plot in the talk but you can look at the paper if you're interested so now we can fit sort of effective temperature at the freezing point and freeze point and what we see is that not only has the domain long coding effectively um sample the lower temperature so here we have domain wall versus one hot encoding and we've fitted a what the effective temperature is um this is after we've removed the effects of minor embedding um so this um this is really looking at the effective model of the Hamiltonian after we've embedded it um one side note chain breaks are very rare in this experiment um probably because we didn't have that much frustration um and we've used fairly strong chain strengths um and so we see effectively lower temperature and if we go through and do a bit more math and look at the freeze point we see that indeed um domain wall is um effectively freezing later in the email um effectively using this success probability and you know this sort of approximation that we go from a thermal equilibrium to frozen um we're freezing earlier or freezing later if we've used this different encoding so encoding has a strong effect on the dynamics of how the problem is solved thinking about these as separately you have your computer science part and your physics part might not really be the way to do this um one motivation to do this is that a one hot value cannot be changed by flipping a single binary variable and a domain wall can um it's maybe a little bit more complicated here because we have additional constraints um but this intuition kind of agrees with what we've seen um in this case you're flipping minor embedding chains anyway but still the difference between flipping two chains and flipping one chain is probably going to be significant um potentially there could also be some effects on minor embedding chain length but I think the key message here is when you're thinking about how your problem's encoding encoded it's definitely worth thinking about how is this going to interact with the physics how is this going to interact with the physics of having you know a transverse field single qubit updates it would be very nice if we had multi qubit drivers with arbitrary number of qubits but we don't have that um and think about how does this work and one of the claims here is that domain wall seems encodings not only are they seem to embed more efficiently they also seem to get along more nicely with the underlying physics um and so just sort of as a plug for this encoding method um there is some open source python code um that was associated with the original paper on this that you can download from this repository here and play with if you don't want to deal with figuring out how to encode yourself um and I would encourage people to do that um so I think that's sort of the end of the talk and five minutes for questions would probably be good I'm not going to read through the key points because okay so let's thank the speaker questions thank you thank you nice presentation you you compare the performance of uh one hot encoding and domain wall encoding so the performance of binary encoding is worse than one hot and domain wall on d-wave machine um so it depends on the problem um if there's a problem that's naturally expressed in a binary encoding um then no you probably should use a binary encoding so problems that have a natural structure like multiplying numbers are very naturally expressed as a binary encoding um and I wouldn't encourage someone to try something like domain wall or one hot on those um however what happens when you have arbitrary quadratic interactions between the variables you can just do a degree of freedom counting argument and say okay how many quadratic interactions do I have versus how many degrees of freedom do we do I have and you find that that's a lot less um for binary so if you want an arbitrary interaction in binary you need auxiliary qubits now there'll be many ways to do this which is part of why we haven't tested this um but what you can do is you can just look at the counts of these and you can say let's say I need one auxiliary variable per higher order interaction and let's say I need an arbitrary interaction between binary variables um and if you do these counts you find that well one your counts now scale with the number of edges rather than the number of nodes because every interaction you'll need some of these auxiliaries so then the degree of the graph matters and you find that really anything beyond like a trivial three qubit problem you're using fewer variables with domain wall if you're encoding an arbitrary problem if you're encoding a not arbitrary problem then binary may very well be the way to go and in fact there's all kinds of nice problems um including a shortest vector problem that um Adam Callison looked at in one paper where the structure is the structure of a binary encoding and those you'd be silly to use domain wall encoding for um you'd very much want to use binary encoding but these kind of problems don't have at least I don't know of any nice structure for these kind of problems and in general there have to be problems that don't by just a degree of freedom counting argument you can't encode everything into binary without using some auxiliary variables in fact you can kind of argue well you can argue that domain wall is the fewest number of variables you can have without having to use auxiliary variables for um arbitrary interactions that's part of the paper but it's not something I've included in the talk but if you want to look at the sort of um archive paper that's kind of the um headline of this last part of the talk it very nicely goes through those degree of freedom counting arguments and shows that if you want to do arbitrary interactions you at least in terms of variable count you're not doing better than domain wall I have a question actually this was actually regarding the very first part of the talk so you spoke about approximations so finding approximate solutions so one thing is to find um the exact solution with a certain probability and another thing is to find an approximate solution with that probability however we know from the pcp theorem that even finding approximate solutions to some problems like SAT which are better than an infinite temperature approximation which you just throw the random it's still NP hard okay how does the adiabatic so this would be states in the spectrum of the problem I'm talking about which are pretty high up how does the adiabatic they have nothing to do with the gap do you think the adiabatic algorithm tackles these things if anything well I think you could take several approaches one would be to say well for any problem I know I have finite but maybe very small gaps so I can run long enough and then I know I'll find something and I mean if that's happening then you're going to have to have many very small gaps right not just one if you just had one then you'd find a pretty good approximation you'd find the first excited state but you probably have many very small gaps and so one approach would be to just say well I can run long enough I would suspect if you did a rapid quench though what would happen is good solutions ones that are significantly different from this sort of thermal average would be exponentially rare however I think one of the main points I want to get across here is there's I think sort of a idea that people have in the back of their head of if you have an exponentially low success probability that's it that's a bad method don't do it it depends on the exponent that's what you mean if the exponent is non-zero but still sufficiently smaller than log 2 then it's a good thing to do anyway if your probability of succeeding where maybe succeeding means just getting a good approximation is again significantly lower than random guessing then you can still win and so I think there's a I think the problem is this is kind of a nasty regime to do theory in you don't have a lot of theoretical handles we found one that's you know doesn't allow you to say a lot but at least it's something and hopefully it's a jumping off point to do something complementary to thinking about adiabatic theorems which you know there's great work I'm not saying people shouldn't do that I'm just saying having something else on top of that would probably be nice for this very rapid quench regime I'm for one more short question Daniel so actually I also have a question about this slide you write how to mitigate all errors for a very long time and emphasis on all why do you think we need to correct all errors that's not what we do in quantum error correction that's probably a valid point you probably I really should probably take that word out you'd need to mitigate the relevant errors and yeah you're absolutely right there would be some errors what I mean here is if you're really looking at a correct what I really mean here is that you'll have to if you're really looking at exactly solving the problem and you don't have open system effects one error that takes you out of your ground state you're not solving the problem anymore you're adiabatic following the first excited state let's say so yes probably should have worded that a little bit more carefully and there are some errors that won't be relevant errors on a certain basis will do nothing obviously okay so I think we can take the rest of the questions to lunch with us and let me take the speaker again