 Okay so the next speaker is going to be Professor Warburton who is going to speak to us about excited stays in quantum annealing and he's professor at UCLA. London not California. Good thank you Glenn and thank you to the conference organizers for giving me this opportunity to speak to you today just confirm that I guess you can hear me at the back. Okay thumbs are up excellent. Good so yeah my name is Paul Warburton I'm at University College London where I have a small group working on both experimental and theoretical aspects of quantum annealing and today I'll be talking to you about how we can exploit excited states in quantum annealing. This is work that has been sponsored by the agencies you can see on the screen here. Okay so here's the Hamiltonian that I guess everyone in the room is familiar with and in the conventional approach of quantum annealing you initialize the system in the ground state of the driver Hamiltonian. At the end you read out the ground state of the problem Hamiltonian in the computational basis and the goal I guess of adiabatic approaches is to stay in the instantaneous ground state at all times. For the purpose of today's presentation we're going to get rid of these three assumptions each in terms. So we don't initialize in the ground state necessarily we don't read out in the ground state necessarily we don't stay in the ground state at all times during the anneal. Okay so here's the format of my presentation today we're going to start with a couple of brief looks in sections one and two of initializing in the excited state of the problem Hamiltonian so this is what you might call reverse annealing I'll speak very briefly about that and then I'll move on to talk about reading excited states at the end of the problem Hamiltonian and talk briefly about some experiments we did a few years ago now on maximum entropy inference on the d-wave machine and for the bulk of my talk we'll be looking at section three which is the relaxation of staying in the ground state at all times and that's a technique which has been increasingly popular over the last couple of years called diabatic quantum annealing and I've got two two subsections of section three first of all we'll be looking at a something we call the locally suppressed transverse field method for diabetic quantum annealing which is heuristic which offers at least potential for speed up using existing superconducting device technology and in the second part of section three I'll be looking at using non-stochastic XX drivers as a way to give us the potential structure during the anneal which could be exploited for diabetic quantum annealing okay so first of all reverse annealing which is all about initializing in an excited state of problem Hamiltonian I have very little to say about this other than to to include it for completeness I guess so and there's been the last few years a number of quite impressive demonstrations of how you might use reverse annealing on the left hand side for computational problems is here's some data from collaboration between NASA and standard charted where the red data shows a combination of a greedy algorithm and a forward anneal and then reverse annealing to do a local exploration of states near near the final solution and they see some important some improvement in the performance and on the right hand side there's been a lot of work recently from Andrew King and collaborators in quantum simulation using a reverse quantum annealing and here's an example of a talk that I heard in Telluride last week where where Andrew King and colleagues were talking about using reverse annealing for studying magnetic monopoles so it looks like a really powerful tool and all I wanted to say today is just to draw your attention if you're not aware of this there's some potential drawbacks of this reverse annealing technique so first of all it requires clearly diabetic and or incoherent transitions because if you're starting in an excited state you need to transition out of that in some way and that requires a diabetic or incoherent transitions otherwise you just get back to where you started from in a reverse anneal and the secondly is really a warning that certainly as implemented on the D wave machine if not other implementations which may be available there's some memory of your initial classical state which is imprinted on on on the on the reverse anneal due to this phenomenon called spin bath polarization so here's some data which my student Daniel O'Connor has taken just to illustrate the point can I get a marker one second so here on the right hand side we're initializing the classical state either in up or down depending on where whether we're blue or orange and then we reverse a needle back to some value of s the annealing parameter and then go back again without pausing and measure the classical state and the point is that if we kneel back to zero in the annealing parameter we should completely erase memory of our initial classical state and that's obviously not the case in the experimental data if we run the anneal in order of magnitude faster we can minimize this effect but we can't get rid of it and so this is just a sort of caution really and I should point out as Andrew King pointed out to me last week there's a nice paper by a Trevor Lanting and colleagues which analyzes this effect in detail and does at least suggest there are some regimes of operation where you can effectively beat the time scales of the spin bath polarization effect to eliminate this effect that's really just a health warning about doing reverse annealing though it does seem to be very powerful technique second thing I'm going to talk about today is reading out excited states at the end of the anneal excited states of the problem Hamiltonian this is something I worked on a number of years ago with a Nick Chancellor who is here and it's what we call maximum entropy inference so we looked at the inference problem and the inference problem in a nutshell is you have some data it's corrupted by noise in some way and you want to extract the original uncorrupted data from that noisy data set and in very general ways there are two ways of doing this there's maximum likelihood inference and that corresponds to looking at the ground state of the inferred information and the ground state turns out to be the solution which maximizes the likelihood of getting the correct overall solution but there's also maximum entropy inference and in this case you're looking at distribution over excited states in order to maximize the entropy of the decoded information and that corresponds to getting the Boltzmann distribution over the excited states and what we showed is that as long as you have some prior information about the noise distribution that's corrupting your data in certain instances maximum entropy inference can outperform maximum likelihood inference and we published a paper in this the citation at the top of the page here in red where we looked at decoding of classical error correcting codes as an example of a maximum entropy inference on a quantum annealer and there's lots of details in the paper but the key result is here so here we're plotting on the y axis the bit error rate of our decoded information at some temperature t as a function of the ground state sorry normalized with respect to the ground state bit error rate that's the maximum likelihood decoding we're plotting that on the x-axis as a function of the Nishimori temperature which is basically a measurement of the noise which is corrupting your data in this case is the noise on a communication channel and so if you look at the top right we can see we've we've sort of used this usual proxy for temperature by having a factor alpha in front of the magnitude of the problem Hamiltonian terms and as we increase alpha which is equivalent to decreasing the effective temperature we can see that there are some regions of operation where this maximum entropy decoding outperforms the maximum likelihood decoding and what that is saying is that on the D wave machine that the excited states contain useful information about the problem you're trying to solve and I should also point out there's reference to this paper by the Los Alamos group where they're looking at using the D wave machine for for Gibbs or Boltzmann sampling from from Hamiltonian so that's a very similar sort of a technique that these guys are using okay so that's a very brief look at initializing your Hamiltonian in excited state and also at reading out to the end of the anneal excited states in your Hamiltonian and for the rest of the talk I'm going to look at diabetic quantum annealing so this is transitioning during your neal through excited states as I said we have two techniques and first of all I'm going to talk about something we published a few months ago now which is this locally suppressed transverse field heuristic for diabetic quantum annealing so there's been a bit of activity in the last few years on diabetic quantum annealing and you might think of that as basically a shortcut to adiabatic dasticity in other words you have some small gap in your annealing schedule if you want to stay in the ground state as you go through that schedule of course you have to go slow so that's what we call quantum slowdown if you can arrange let's say a potential structure with two avoided crossings during the anneal there's a possibility of moving fast through both of those transitions start in the ground state go to an excited state during the needle and return to the ground state so this gets rid of the speed limit for quantum annealing for adiabatic quantum annealing and this goes back I guess to 2012 in this paper by Orlando sommer and colleagues where they showed there was a provable speed up at least in an irracula setting for this specific permutation symmetric problem called the glued trees more recently Vicky Choi has published a very nice paper on how you might use xx interactions in your Hamiltonian in order to effectively engineer a structure which gives you the required avoided level crossings and I should point out also that somewhere this is sort of in the same spirit as this proposal by Elliot Capit and colleagues where by applying RF fields during the anneal you can effectively mix the ground state and the excited state effectively so there's a lot of published work and I refer to this review here by Elizabeth Cross on Daniel Nadar for more information so here's the basic idea here's a typical problem that we might look at I'm showing the ground state as the dashed line first excited state blue and second excited state orange and we have two small gaps least invisible at this scale in between the ground state and the first excited state so the idea is that by annealing fast you have these two Landazina transitions which you start in the ground state and end in the ground state okay so the question is how do we reliably and inverted commas create this additional gap minimum ideally using hardware that we have available to us today and in this paper by Louis Freiburio and colleagues we achieve this at least heuristically by creating this quasi degeneracy in the spectrum at some value of time during the anneal which we called TX and the way we do that is by suppressing the transverse field on a single target qubit that's part of your problem and then we pre anneal the target qubit so that it's longitudinal field at this time TX is zero so let me just explain that with a cartoon or two so here we've got a multiple qubit problem but I'm only showing you in this particular graph the single qubit that we call the target qubit so we have a longitudinal field ramp for example could be linear that's not critical but the key point is the transverse field on the target qubit is zero throughout the anneal so for times less than sx we assume our target qubit is in the ground state which I'm calling spin up as I go through this point where the longitudinal field changes sign because there is no transverse field there's no mechanism for the target qubit to flip so the target qubit stays in the spin up state but now that's the first excited state then for the other qubits in the problem we have as shown in red here a rather conventional anneal it doesn't start until this normalize annealing time sx so we turn down the transverse field on all the other qubits and we turn up the longitudinal field and examples I give it's linear but again that's not critical I don't think okay so how does this do we've tried to at least show the feasibility of this by generating some random examples so we have is a specific example on seven qubits and again the details of all the parameters are shown there are various local fields and all the couplers are ferromagnetic with half a gigahertz if we look at the standard linear annealing schedule we have this small gap at about 0.8 s which is going to cause a problem but if we do this suppressed transverse field approach locally suppressed transverse field approach and apply it just to qubit one in this case which is our target qubit where this apply to then we get this second avoided level crossing at s equals 0.2 which in principle at least might allow us to exploit this DQA approach so now we need to generalize this so we've looked at a bunch of different seven qubit examples there are some technical details of that so we have between six and sixteen edges in our seven qubit problem we assign randomly selected local fields from a Gaussian distribution then we rescale for the maximum local field is always one gigahertz we study the closed system dynamics on the q-tip that's sort of shredding a solver we set the anneal time to a hundred nanoseconds of course when we're doing this locally suppressed transverse field we need to select one target qubit to be the qubit with the zero transverse field and we have a choice of seven so we basically do it seven times in this case and select the highest probability of the seven sorry the highest ground state probability of the seven options that we choose and yeah I guess that's all the technical details okay so here's some data this is again simulation data in a closed system dynamics on a shredding a solver q-tip specifically and on the y-axis we've got the ground state probability using this longitudinally spliced transverse field called diabetic quantum annealing technique on the x-axis we've got the ground state probability using a standard linear anneal and so we have seven qubits and in this particular instance we've got between six and eight edges of these randomly generated problems I think there are 200 or so instances here plotted and there's various things going on some of these problems have a large minimum gaps in the adiabatic approach some of them have small as defined on the previous slide but you can see there's clustering of much many of the instances so let's look at the various different possibilities so first of all looking at these guys down at the bottom of the graph which I've circled in purple this is kind of bad news for the DQA technique where the using DQA our ground state probability is zero or close to zero where we had a finite probability with the standard linear anneal so what's going on here as shown in this beautiful hand drawing up in the right-hand side is that so I here I've plotted the ground state in red first excited state in blue second excited state in green as a function of the annealing parameter s and what seems to be happening here is that with our diabetic quantum annealing technique we're transitioning out of the ground and first excited state and ending up in the second excited state because of some small crossing between the first excited state and the second excited state and all higher levels so we end up in higher excited states in those particular instances next cluster to look at is these problems that I've circled in blue where we seem to have a very similar probability of being in the ground state for the DQA technique and the standard adiabatic anneal and these instances we have a perturbative transition or a preservative crossing I should say somewhere near the end of the anneal something we commonly observe in this in this field where in the in the in the DQA diabetic technique we excite to the first excited state and then back to the ground state so as we approach the end of the anneal where we have this perturbative crossing doesn't really matter whether I'm using this linear anneal or the diabetic approach so we get very similar ground state probabilities because effectively we have an even number of land now sorry I should say a total odd number of land asena transitions to or from the first excited state so here there's no gain for the DQA technique the final cluster of data is what you might call the good news for the DQA technique where we have a very high probability using the DQA technique irrespective of what's going on and that in that case we've engineered the the spectrum to do exactly what we want to do in other words we haven't an even number of land our transition land asena transitions between the ground state and the first excited state so that we end up in the ground state at the end of the anneal even there we may have this avoided crossing in the adiabatic approach there's a complicated array of possible transitions we might undergo some of which are good news for DQA some of which are not such good news if we change the problem a little by adding more edges we basically add more constraints which which removes this low energy transition towards the end of the anneal or increases the the gap in the problem Hamiltonian and that seems to suggest that we move to a regime as we increase the edges where the diabatic approach is more favorable okay so we should also look at the anneal time dependence so this is again just a single instance with seven qubits and eight edges where we're plotting the ground state probability as a function of the total annealing time and if you do a linear anneal here's the adiabatic bump which you can get which leads to high probabilities at very short annealing times followed by this increase here as we approach the diabetic adiabatic limit use me and you can see that for all anneal times at least shown here this locally suppressed transverse field approach gives you a higher ground state probability if your preferred metric is time to solution us on the right-hand side here so here's the effective the number of iterations you require to achieve a probability of finding the ground state of 99% as a function of the total annealing time and at least in the range of timescale shown here for this particular problem we see a couple of orders of magnitude improvement for the for the DQA technique so all what I said so far is in the closed system case for this DQA method we had a brief look at open system dynamics as well so here we restrict ourself to the two qubit case and we're using this adiabatic master equation technique using this nice hoax tool which is available online I believe and we have as of his the probability of the ground state using again it just in a two qubit case using one of these we suppress the transverse field on one of the qubits we're plotting that as a function of the anneal time in nanoseconds and we have two different types of system bath coupling independently on the two qubits one via Sigma Z and one via Sigma X in orange and blue respectively if we have this defacing decoherence we get a result that's negligibly different from the closed system case but interestingly as you might expect if you have an elastic scattering coupling via this Sigma X term in the system bath coupling that effectively promotes transitions between the first exciting relaxation between the first excite in the state in the ground state which means that you can end up in the wrong state at the end of the anneal in other words that because of this Sigma X system bath coupling you have relaxation from the first excited state to the ground state after the first level crossing and so you end up in the first excited state at the end and furthermore if you look at the details in the paper here what we've shown is that you can generalize this to a larger number of qubits and it's always the relaxation of your target qubit that dominates this suppression of the ground state probability in the locally suppressed transverse field DQA technique so that we think gives you a potential for studying the role of coherence in this DQA technique and furthermore you can extend it to a multiple qubit system and look at the decoherence effects qubit by qubit if you like okay so that's all I wanted to say about this locally suppressed transverse field heuristic one thing in which I perhaps I should point out is of course in this in this method we're only using the standard X Z and Z Z interactions that are available on the D wave machine what I'm going to talk about now largely inspired by work which was done by Vicky Choi is looking at how we might use a non-stochastic X X driver for implementing the sort of schedules that require for diabetic quantum annealing okay so this is work which is done by my student Natasha Feinstein who is in the room and she's been looking at Hamiltonians with perturbative crossings and thinking about what happens as you add a catalyst term to the Hamiltonian so just for reference we have a function of time we have a problem Hamiltonian in the computational basis which increases we have a driver Hamiltonian which is this local X field which decreases and to this we add in green this catalyst term which comes on during the needle and goes away again in this form a particular catalyst we've been looking at is X X catalysts and Natasha will be giving a talk at this conference on Thursday and we'll tell you much more about this in the stochastic and non-stochastic cases I'm just going to look only at the non-stochastic case and the effect it has on diabetic quantum annealing and again inspired by the work that Vicky Choi has done we're going to look at the effect of these drivers on the weighted maximum independent set problem so I'll talk a bit about the problem where we have this bipartite graph where we're looking to create an independent set i.e. a set where no members of that set are directly coupled by the black lines and we give each variable a weight and we want to maximize the weight within the set so you know the obvious solution here in this particular example is a set containing the two blue variables which is independent because there's no direct coupling between the two blue variables and that has a weight of four and in particular the ground state is shown here so we're going to use a spin-up to encode the variables which are in the identified set and spin down for the other ones so in this particular example as a ground state which is these two blues and the first excited state is these three oranges so just a reminder the spin-up encodes the identified set of the maximum independent set following again the approach of Vicky Choi we need to introduce the concept of neighbors so neighbors are those states which are directly coupled to for example the ground state by the X terms in the driver Hamiltonian so in example if you look at the ground state of this maximum independent set problem we have these two spin-ups here and the neighbors consist of either flipping a spin within the set and there's some two of these for example this one here or flipping one of the spins which is outside of the set in which case we have this threefold symmetry here and you know clearly in this case we no longer have an independent set because we have direct couplings between members of this set so this is an excited state but it's a neighbor of the ground state and similarly one can define neighbors of the first excited state we need to make this a hard problem in some sense so we have a tunable hardness parameter which is basically set by the weights so if we have this parameter delta w and make it small enough we can find the weights of the blue set and the orange set are rather similar and that basically makes the problem harder as you might expect so we have a tunable hardness parameter we also need some method of scaling to larger problem sizes which is shown on the right hand side where we have a generalized n-cubit problem where we scale the weights according to the expressions you can see here so that defines our problem let's look at what happens in the annealing case if we don't have a catalyst so in other words we just have a uniform x driver local x driver and here we are looking at a five-spin problem where we have delta w is very small which makes the problem hard and we have this nasty looking avoided level crossing so here i'm plotting in blue the overlap between the instantaneous ground state and the problem ground state in orange the overlap between the instantaneous ground state and the problem first excited state and in the gray plots you can see in the background here the overlap between the instantaneous ground state and all the other states that we have in the system plotting as a function of annealing time and you can see that basically the system has a strong overlap with the first excited state until we get to this avoided level crossing here towards the end of the anneal and then we have what i might loosely describe as a phase transition although not in the thermodynamic limit here of course where we have a sudden transition of weight to the ground state at the end of the anneal and getting the system through this is what causes the problem hardness so that's what happens again this is a closed system where we're just doing exact diagonalization because it's a nice small system that's what happens when we have a uniform x driver with no catalyst and as we scale this to a larger system size we plot here the minimum gap between the ground state and the first excited state as a function of system size and we have an exponential suppression of the gap so this is a nasty looking problem as we scale up to bigger sizes okay so what happens as we add a single non-stochastic xx coupler as a catalyst so we're going to put the xx coupler in this location between the two of the spins that are not in the maximum weight-independent set and effectively that creates new neighbors so because we now flip two spins simultaneously via this xx driver we now have new neighbors to the ground state so our new neighbors for the ground state consist of flipping these two spins here that gives us again a non-independent set as shown by the green line here and we also have a new neighbor for the first excited state where we flip these two spins which give us an apparent independent set which again is not the ground state shown here and the real question is what effect does the presence of these neighbors have on the state spectrum during the anneal okay quite a lot going on this slide so we'll start in the top left this is the gap between the ground state and the first excited state as a function of the strength of this non-stochastic catalyst again it's a single non-stochastic catalyst shown here and first thing we see is it's a non-monotonic so as you start increasing the catalyst it looks like bad news the gap size decreases and then the gap size starts increasing and then goes down again and I guess I hope I'm not saying this incorrectly Natasha will talk more about the effect of these catalysts on the gap size on Thursday and of course I should point out that there's this paper by Elizabeth Cross on colleagues where they've looked at the effect of xx catalysts on general problems and I guess this is consistent with that work sometimes it helps sometimes it doesn't and in general as Elizabeth showed in that paper the addition of the xx catalyst does not help and if you look at the okay so now we say well maybe we can make the best of this by sitting at this point here which is the optimum catalyst strength and here we're plotting the magnitude of the gap between the ground and first excited states in the two cases where you have no catalyst in black and the optimized catalyst strength in other words finding this maximum as a function of the system size and you do you see some small increase in the gap but again the scaling looks pretty bad one good thing to come from this is at least apparently with the sizes of systems we're looking at so far is the optimum value of this catalyst strength this single xx non-stochastic coupler does seem to saturate at high values of n suggesting that you know in general if you have a large problem of this kind you can just dial up that value of catalyst and you may get some improvement and we're currently looking at extending this to larger system sizes right now so the scaling of the gap minimum doesn't look too healthy but we notice something else going on which in the context of diabetic quantum annealing could be exploitable so here we're looking at the ground state along the bottom and plotting the first excited state here during the anneal as we increase the catalyst strength from left to right in this system shown on the left here and we can see there's a clear appearance of the second gap at some critical value of the catalyst strength and the second minimum gap at least might be exploitable for diabetic quantum annealing so where is this second gap coming from what it's certainly related to the non-stock elasticity in some sense so the the one of the key features of non-stock elasticity of course is that you can have negative amplitudes on the ground state and that seems to be what's going on here this value of the xx interaction which gives us this minimum in this additional gap that's appearing the state spectrum is shown on the right hand side and again the blue line is the weight of the instantaneous ground state on the problem ground state and there is a sign transition I've left off crucial zero on the on the axis here but this is the zero and there is a sign transition as you go through this first avoided level crossing here in the weight on the ground state and so this you know the the the the negativity of this weight which is only open to you in non-stochastic Hamiltonians seems to be playing a key role in giving us this second avoided level crossing okay I think I've been reasonably conservative on time so I'll reach my conclusions I've shown in various different contexts that you can you can relax your adherence to the ground state in in many applications in various different ways I've shown briefly at least at the start of this presentation that excited states certainly on the d-wave machine can contain useful information about the problem you're trying to solve and we showed that in the context of maximum entropy inference and in terms of diabetic quantum annealing we've come up with two methods of doing that so the first method was by surpassing the transverse field on a single qubit this seems to be a heuristic method but certainly a heuristic method which might give you performance improvement some other time the question is how do we generalize that I mean that's always the question in terms of this diabetic quantum annealing given accessible information about the problem you're trying to solve how do you engineer your driver and your catalyst in a way which which gives you the required state transitions that I guess is the million-dollar question for dqa and what we've shown is there are some hints of how you might do that at least in the context of this weighted maximum independent set program problem using a non-stequastic xx driver so I should acknowledge the people who did the most of the work I've spoken about today Nick Chancellor who did this maximum entropy stuff Louis Freiburio who did the suppress transverse field approach and Natasha Feinstein who did the xx coupling and we'll be speaking on Thursday and Daniel O'Connor who did the reverse annealing thanks to a las alamos on the ISI for giving access to the d-wave machine and these people along the bottom did the funding and finally I might abuse my time here by giving an advertisement for this new international network on quantum annealing so this is funded by the United Kingdom EPSRC funding agency part of the UK government that's really part of the brexit agenda if I'm honest about giving scientists mechanisms for reaching it's time since scientists in the UK mechanisms for reaching out to the rest of the world we're still open for business in science in the UK and this international network on quantum annealing gives us an opportunity to do so so what we're doing is having weekly online seminars in fact we're starting fortnightly to ramp up and we hope to go weekly in the fall semester those will start on Tuesday July the 12th we'll have funding for exchange visits if you want to make an international exchange trip and we'll have an annual conference penciled in currently for November this year and the network really brings together the existing international collaborations that we have in quantum annealing in the Americas in Europe and the UK and in Japan so if you'd like to get involved and I would certainly encourage you to get involved if you're interested I suggest you go to his website or take a picture of the QR code that's technology that's beyond me and we look forward to interacting with you through Inca thank you very much for your attention okay thank you for this talk I think we can proceed with the questions so yes thanks I really enjoyed the talk um so quick question uh I saw that for your xx you know it was spatially localized so do you think that this is a critical aspect of you know utilizing non-steal causticity that you kind of have to you know restrict it to certain eyes and jays and then um second question is have you studied this uh this in a reverse anneal I know early in the talk you talked about reverse and now this and I think that it potentially could shine in that context and so I was curious whether you've looked at it okay yes so so you're correct in saying that this this xx driver that we're adding is local we apply it just a single pair of qubits and that that we find interesting because it seems to as I pointed out earlier on scale to much much larger systems which is perhaps a little surprising that that you just you know uh tickle if you like two qubits in your entire multi qubit system and it has this radical effect on on uh the dynamics I guess in that sense it's not so very different from the first approach I mentioned where we really just change the field on a single qubit so uh whether that is uh essential for getting these dqa transitions I don't know I think is the honest answer I mean you know essentially I'm I'm I'm I'm resulting to conjecture now so everything I say comes without health warning um you might believe that the argument goes something like the following the you're adding a single additional avoided level crossing when you're trying to create this pair of avoided level crossings you have one already you want one more and to do that you might believe that you just need to affect a single thing I mean that's slightly hand-waving but it seems believable uh second question you had was about reverse annealing no we we haven't looked at that yet that's an interesting suggestion one that perhaps we can think about and discuss thanks David I actually have a question you ask yeah go ahead Edith um so um there was um so there was this mini gap and there was the second mini gap and I did not really understand how come the system's energy is drastically reducing to the ground state like those uh blue or orange cobs that was one of the stuff was on the last two last slide no this one yes as well this one uh so I don't quite know what your question is what now I have this so so whenever you're taking this this um inner product between this e0s and e1s and e0s with e0s whenever there is this first like this first mini gap and there is the second mini gap and there is this drastic fall so how is the these things are happening in this orange golf and in this blue golf okay well let me perhaps I can just explain it in the in the more straightforward case here yeah so this is this is if you like um I mean imagine extending this to the thermodynamic limit this is this is a quantum phase transition uh and uh as you know when you transition from one equilibrium to another equilibrium your fluctuations are maximized and this is an example of this you have a very strong fluctuation from the ground state to the first excited state does that answer your question um yeah but uh for the orange line there is uh this but the orange line is actually so the one that is actually an excited state it is actually so this inner product is actually falling so this is also a result of this quantum phase transition yeah so you're transitioning weight from the first excited state to the ground state and the system must do that somewhere because there's a lot of weight in the first excited state to the left of this transition and we know at least in closed system dynamics you must end up in the ground state by the adiabatic theorem uh and so uh things have to change rapidly this is what makes the problem hard so for example if my problem or say artificially somehow I create many many many gaps and uh there is a quantum phase transition at every mini gap I mean it generally happens um so after so what we have observed in our problem that after the very small exponential mini gap the system actually transitions to a localized state that is this many body localization state and after that after you have this the only phase transition that one can have is only many body phase transition so from transition from one localized state to another localized state or are you are you uh transitioning from many body localized to delocalized state in the subsequent mini gaps yeah I haven't thought about this in the context of many body localization I have to be honest with you but that sounds like an interesting discussion uh something for us to think about okay maybe we can sit over sit with my supervisor Dimitri sometimes give me the conference sounds good yes maybe you guys can continue later thank you yes hi actually I'm Dimitri Bagretz yeah later I told you try to talk to you further so there's many gaps which appear in the end of the spectrum yeah they typically happen with the states which have a yeah huge humming distance actually of the order of system size now the question to this diabetic type of annealing is it do you have some understanding if it works then it works only when by chance you had the states which were close in humming distance and that's why by acting on a single qubit you were able to somehow resolve your problem and these cases which didn't work they actually correspond to the situation when they were far away and your approach was unsuccessful so I'm I'm now trying to outline pessimistic picture or you are more optimist that just by acting on a single qubit you will be able to also yeah crack this problem when they are very far away do you did you did this analysis for your random instances when it works and when it not yeah the short answer is no we have not looked at the hamming weight during these processes and that's an interesting suggestion or something because in in-bill phase yeah they are very far away yeah and this is yeah yeah no I agree with you so so if we have single x drivers and only xx drivers then anything that's hamming weight three away will cause problems so so that's an interesting suggestion yeah thanks could you please again show the other plot like that where you had the two crossings yeah could it be that the first like did you look at the at the wave functions at the first crossing could it be that they're like bimodal in that regime and that basically these this tradition occurs when in the ground set the the sine flips between those two modes in the in the wave function I mean sine flips are playing a key role again let me just reiterate that this is the origin which I forgot to label so so yes I agree with you I don't quite know what you mean by bimodal I guess as in it's yeah the look less more less in two locations basically and the wave function is one case one mode plus the other mode and then you have one other state that's one mode minus the other mode and that those those types of states tend to be very close in energy and there are there are other states going on in gray in the background here so it's not not as if we have no weight on on on on other problem states okay so those are all the these gray lines sorry how often how do I read those gray lines then gray lines are the weights of the instantaneous ground state on the problem second third fourth excited state okay yeah thank you so in in order for a dqa to succeed not only do we want to have the second minimum gap appear but we also need to maintain a large gap to higher excited states as indeed you showed in the suppressed transit field case yeah so what what is the situation in regards to that issue with the xx story here so I mean as yet we haven't done any dynamical modeling of the process which is computationally intensive but we shall try and do it so so I guess I don't have an answer for that specific question I agree with your general observation that only looking at the first excited state is is perhaps overly simplistic but yeah agreed agreed but is is the the formalism that you're relying on so the work by vicky choy and and what you've done here does it give us insight into the gap to the second excited state I don't think that it does but that's that's I agree a key question yeah okay and another related question to this so you are arguing convincingly that the non-stochastic term is responsible for the sign change in the overlap but what happens if you change the xx to a negative xx have you checked that does it actually fail to help uh yeah so so and again Natasha will speak in much more detail about this on Thursday so we looked at this stochastic version of this which does give you um a gap enhancement and certain regimes but so far at least in our exploration of this particular problem we don't see this appearance of the um additional minimal gap because the argument that you that you made relies on on this picture with the sign flip of the the overlap but it also relied on um I think it was one of the previous slides um where you introduce two neighbor new neighbors uh due to the double flipping so you changed the hamming weight by two instead of just by one and that also played an important role and that would not be dependent on having non stochastic xx it would also work with stochastic xx so so we've looked at uh let me see if I can remember this there are four possible things you can do you can add a new neighbor to the ground state of the first excited state uh you'll do that and you can uh um either the ground state is is lower in energy in which case it'll dominate or the first excited state is lower in energy I mean the new neighbor to the ground state or the new neighbor to the first excited state is lower in energy in which case it dominates the effect and you can so that's two things you can do and you can do that either with uh positive xx or negative xx and the one uh let's hope I state this correctly the the one of those four possibilities that give us the second avoided crossing at least as far as we've seen so far with this particular problem is the uh non-stochastic interaction applied to the uh first excited state uh of the maximum independent set problem so other possibilities did not give us the second avoided level crossing and Natasha nodding I'm glad to see that good are the other questions okay if not we can thank Paul for the talk again thank you Glen