 We have now the talk by Gris Wilson and please have a seat and let's start with the Gris talk. Thanks. Good afternoon, everyone. My name is Chris Wilson, so I'm at the Institute for Quantum Computing. Obscured there at the University of Waterloo. I would like to join everyone in thanking the speakers for having me to this lovely location for this nice conference. Break the pandemic drought. Yeah, so I'll talk to you today about some of the work we've been doing over the last couple of years on analog quantum assimilation of topological models with parametric cavities. So we'll kind of touch on a couple of topics of the conference, which is microwave photons and then hopefully getting towards many body physics. So these are some of, so Jamal, Dima and Jimmy are students in postdocs who've done most of the experimental work. Ibrahim is our fab guy. Sanbo Chang, who's now with Nakamura in Tokyo, helped develop the platform. And Zheng Xi is a theorist who helps a lot with the latter part of the talk that we'll see. And then also kind of the first experiment I'll talk about. We got help from Enrique Solano's group and particularly Enrique Enrique. Since I know it's not, you know, it's not a quantum computing audience necessarily. I want to define these words of saying I'm doing analog quantum simulation. So what does that mean, right? So analog simulation actually has a long history in electronics and even before that with mechanical analog simulators. But in the 60s and 70s, electrical analog computers were used quite heavily before digital computers could do anything useful. So here's some screenshots from a brochure from the 60s for a desktop computer. And you can read it says analog simulation is a dynamic method of solving the problems that confront the engineer and scientist daily, right? And then they're talking about problems in aerospace and chemistry and biomedicine. So it's all kind of the same thing, right? So what does it mean? So what does analog simulation mean? So what this thing is is it's basically a bunch of op-amps configured as integrators and adders. And then you have punch cables, punch cords, like an old operator so you can connect them together to make a certain circuit. And then you have a bunch of potentiometers so you can set time constants and things like that, right? So you create a circuit that has equations of motion that are the same as a system of interest that you want to study. And then you just let the thing run, right? And then you measure the time evolution and that's your simulation of the circuit or that's your way of solving the problem. So, you know, that's how we went to the moon, by the way, right? And so what is analog quantum simulation? It's basically the same idea. I'm going to build now a quantum device. In our case it's going to be a circuit, a superconducting circuit, but it could be atoms and all kinds of things. But then we build a quantum device with the same now Heisenberg equations of motion of the system that I'm interested in. And then I'm just going to let the thing run and measure it and that's going to be my simulation, my way of calculating what the system does. And there are many ways of doing quantum simulation. The one I'm going to talk about is a kind of a very broad platform called Photonic Lattices. These are just a couple of papers that say lots of people think about it in different ways. So this could be optics, it can be microwaves, it can be atoms. I guess Photonic is more optics, but anyways, you know, but the basic idea is that I have, you know, a bunch of nodes, which are some, you know, could be cavities, resonators, something like that that can hold a photon. And then I have some kind of connection or some graph of links between these nodes, you know, and this lattice defines my simulation. And, you know, this can be mapped to a lot of different lattice models in, you know, fundamental field theory or condensed matter and everything like that, you know, lattice models are very popular. An interesting feature, you know, for instance, pointed out by this paper, I don't know that it's the first, is that in a lot of these platforms I can do something interesting. So in general, I'm adding some kind of tunnel coupling between these nodes. But if I can make that tunnel coupling complex or if I can give it a phase, right, then I can do interesting things like simulate gauge fields, right. And the way you can think about that is if I can make the right pattern of phases here, now my photon hopping through this lattice has a path dependent phase. And if this phase has these phases have the right pattern, I could think of that phase as coming from some fictitious gauge field, right, or the line integral of a fictitious gauge field. And then I'll also show in this talk that with these phases we can, you know, study topological and chiral dynamics and lots of interesting things. So on to, huh, now there's a lag. Okay, woo. Okay, I got too close to the mouse. So now I'll talk about our particular system, which is a photonic lattice, which as we'll see is programmable in situ, which I think is good for a simulator. And we're doing it with what I call a parametric cavity. So a superconducting microwave circuit. So what is the basic parametric cavity? So, you know, the basic system is something I've been working on for a long time, starting from when I worked in Paredelsing's group at Chalmers. So going back to 2006, 2007. And the basic idea is we have an on-chip transmission line cavity. So we have an on-chip transmission line. And then we cut it with a capacitor, which allows us to do input output and also forms one boundary condition. And on the other end, we shunt it to ground with a squid. And what we need to know right now is that the squid basically looks like a tunable inductance. It's a tunable boundary condition. And by putting a flux through the squid, I can change the boundary condition, which changes the resonance frequencies of the modes in the cavity, right? And importantly, you know, squids, I can modulate the frequency from DC to tens of gigahertz, right? So I have a lot of flexibility. Now, we've made amplifiers out of these things, but for these purposes, for the photonic lattice, we want to have lots of modes. So we just make the cavity very long. So the fundamental frequency is low. This is kind of an old device where it's kind of two gigahertz, but we've done now like 100 megahertz or something like that. So you can put lots of modes into your measurement bandwidth. So I have lots of modes to play with. And these modes are then going to become the nodes of my photonic lattice. And then I'm going to couple them, and I'll explain the details in a second, with parametric pumping of this squid, so parametric pumping of the cavity. And just to say, you know, as I said, we've been working on this for a long time. So at Chalmers, we did our first fast tunable cavity. We did the dynamical castamere effect as a crossover, one of para-students working in my lab. We did multi-mode JPA, and recently we showed that we could generate tripartite entanglement with this scheme, with multi-mode pumping. And I will say, I think at this point, there's too many other references to list in detail, but lots of good work at Chalmers and Paris and Helsinki and Yale and NIST and lots of other places. So we're not the only people working on this kind of thing. So to get into a little bit more detail of what we can do with this system and the kind of interactions we can create, so we just write down the Hamiltonian of the squid at the end of our resonator, right? And the basic idea is that we have, you know, a tunable, it's like effectively an EJ that's tunable with the pump flux. So this is the flux we use to modulate it, and this is multiplied by then the flux of the signals in the cavity, which is just the sum of all the modes. And then what I can do to lowest order is I have a little flux bias, so I get a linear term here. I further say the pump is parametric, so I say it's a classical field at this point. So these are alphas now. And then, you know, and then I'll expand the cosine here just to second order. And, you know, for concreteness, I put in three modes here, but we can have many more, right? And then the basic idea is that, you know, when I, when I square this thing, I get many, many quadratic terms of kind of all the possible, you know, all the possible interactions or combinations of A and A dagger, quadratic combinations for all the modes, right? So it's kind of a mess. But when I, when I pump, I mean, the idea is that each of those quadratic terms in the appropriate interaction picture will have a time dependence, right? And so by then picking a pump frequency which matches that time dependence, I can make a rotating wave approximation that picks out that interaction and all the other ones go away, right? So for instance, if I pick the sum frequency of two modes or twice the frequency, I pick out terms like this, which is a parametric down conversion term, or if I pump at a difference frequency, I can, I get a beam-switter coupling or coherence swap between the two modes, right? And what we've demonstrated over the last several years is we can put in many of these tones at the same time and generate many of these interactions. And so in the language of quantum simulation, you know, the way I would, I can think about this is, you know, this term here, this beam-splitter, you know, this is hopping or coherent tunneling. And this term looks like some kind of pairing potential, like I would have in superconductivity, right? And another important point is I have some, you know, I have some raw G-naught up here, but then this little G here is actually depends on the pump amplitude. So how hard I drive increases the effect of coupling in the rotating wave, and it also depends on the pump phase. So I really get to make these couplings complex. Okay, so I hope you can kind of see now where I'm going. So the idea is now I have all of these modes of my cavity, which become nodes of my photonic lattice. And by turning on different pumps, I essentially define a graph of connections between these things. So what I make is a lattice in synthetic dimensions, or maybe even multiple synthetic dimensions. It can be more than 1D. And just by picking pumps, I can simulate different things. I can do a triangular lattice, a square lattice, a linear array, all in one device, right? All in one cool-down. So that's kind of the introduction, and I want to talk about a few different models that we've simulated. I mean, kind of sum in exactly the same device, sum in sister devices, but the point is it's, you know, these are all done in basically the same device. So the first one we looked at is something called the Bosonic-Kreuzlatter. So the Kreuzlatter is a quasi 1D lattice model that has a cross-linked lattice structure. So I have kind of two 1D lattices, but there's cross-linking between them. And then I also imagine that there's a magnetic field that threads the latter. And this arose kind of in the 80s as an early toy model of chiral fermions in quantum field theory. And then, you know, so despite its simplicity, there's lots of interesting things going on. So in particular, when I have a phi equal to, I have a flux of phi equal to pi, there are various topological states in the system. So first of all, I can have states in the bulk which become localized, which is related to a phenomenon I'll explain more called Aronoff-Bohm caging. And then I also end up with localized zero modes at the end or end states or edge states. So we're going to make, you know, a small version of this. So we have four nodes, which is kind of like two plaquettes. But I want to show you, I want to start to introduce you to the type of data that we can take here, right? So the first thing we can do is, so imagine, you know, we have a few modes of arcavity, which are becoming our nodes. And I just do normal, you know, reflection measurement with my VNA, right? So I measure around one cavity node and I see some, you know, resonance curve, which is asymmetric because anyone who's done cryogenic measurements knows that they're often asymmetric. And then we can see, so this is with the lattice off, so no pumps, right? And then I turn on, you know, pumps in this way, so in this lattice. And you can see now basically this splits into four modes, right? So I'm coupling these four nodes together and I get four normal modes, right? With some splitting. So basically the idea is doing reflection off of one node gives me access to the energy spectrum of the system, which does actually still depend on the position in the synthetic dimension. So we can see a different spectrum at each node, which has to do with the spatial support across the lattice, right? But, you know, I said we also have phases here, so we have a synthetic flux. So we can take this data now as a function of flux and this is what it looks like. So here, this is now our network analyzer frequency where we saw the multiple resonances. The color scale is the reflection and this is the lattice phase. So because it's a closed loop, only the phase around the loop matters, so we can just change one phase, right? And this is what we see. And already here we see there's kind of a lot of interesting things going on, right? So I see kind of zero, maybe zero modes forming. You know, I see bands kind of merging together, but there's kind of a lot of structure going on as a function of flux, right? So in addition to using the reflection measurements to measure the spectrum, we can also do transport measurements to measure essentially map out the wave function across the different nodes, right? So that's the idea. So, you know, here again I'm saying I'm putting in a signal at A and I'm measuring what comes back at A, right? And this is our reflection, so we get dips, right? Yeah? Also different modes of one cavity? Yes, so it's different modes of one cavity. So there are different frequencies and so for these transport measurements we have to do frequency conversion measurements or buy a fancy network analyzer that has multiple sources, both of which we've done. So, you know, and then I can say I can now do a transport measurement. So I put in a signal, I sweep around Omega A and I see what it comes out at Omega B, right? So now I get peaks because there's nothing when, you know, I'm not, when there is in transport and I can do that for kind of all the combinations. I can measure A to C and A to D, right? So I can map out the transport across the lattice, which we take as a measurement of, you know, as a proxy for kind of the structure of the wave function across the lattice. So, you know, this state and we can, which we frequency resolve, you know, say in this case, well, and then we can, yeah, do this all as a function of phase, right? So we're measuring these curves, reflection and transport as a function of the phase again and see, we see structure like this. Yeah? Yeah. So you have to spend a lot of time testing different microwave generators and finding which ones have really good phase coherence. So it turns out, you know, phase locking at 10 megahertz isn't phase locking, it's frequency locking. So, you know, you have to, you have to play with it. But this is the right generator, which is not, it's SGS from Rhoda Schwartz. You know, you lock them at 1 gigahertz instead of 10 megahertz and then you just change the phase on the front panel. Or if there's no pump, I turn off the pump. I just turn off the output. Whenever you sum, you just set the zero. Yeah, if I have, yeah. Yeah, or, you know, basically the way I get zero phase here is from looking at the pattern. So, I mean, you turn everything on and there's some phase, right? And now, and then I sweep the phase and I say, okay, based on what I know of the system, you know, based on kind of fitting, but I mean, it's pretty obvious fitting. I call this phase zero, right? So, I, like I said, so we can do this, so we can measure all of the reflection coefficients as a function of phase and all of the scattering coefficients, all the transmission as a function of phase and map out the response of the whole system, right? And so, you know, this is, the left here is the experiment and the right is the theory. So, this is just a simple four-mode scattering matrix, you know, for parametrically coupled modes. So, from the NIST guys that we just adapted, you know, and you have to deal with little microwave stuff, but we can fit things very well. And, you know, did I walk too hard and move the mouse again? And there we go. So, you can see, for instance, already here that we see things like non-reciprocal transport, right? So, this is kind of, you see the transpose elements here. What's red here is blue here and what's blue here is red there, right? So, these are actually, the off diagonals are compliments of each other, which tells me that I have very non-reciprocal transport in the system, right? And, you know, which we can connect, you know, to time reversal symmetry breaking of our synthetic flux. But I also want to look at some of the modes that we see emerging here, right? Which we can connect, you know, this tiny system, topology is about big systems, but you start to see hints of things coming out, right? So, the first mode, so one mode we can see is at phi equal to pi, right? So, these are the lines down here, right? And so, we see, you know, we have, this is where our four states merge into two, kind of symmetric about zero, so I have degeneracies. But you can see here, you know, this state now doesn't transport very much to B. This is the structure of the lattice. And you can see that here is kind of the transport is pinching off here at this point, right? But then there's lots, there's plenty of transport or much more to C and A. So, I have some kind of state that's kind of living on the corner here, right? And I have four of these corner states we can map out. But how do we connect this to the quartz ladder? So, we can kind of think about this as being this trapezoidal plaquettes or part of the lattice of the quartz ladder. And what's the idea, you know, so if I put an excitation here, you know, it can get to B two ways, right? It can go this way or it can go that way. But there's a pi phase shift on the lattice, right? So these two paths just interfere destructively, right? So the system can't get here. And that's the heart of Aranoff-Bohm caging, right? And that's what microscopy kind of leads, well, we connect this to flat bands or these, yeah, or the, you know, the localization in the system. So we see at least the precursor of that. Then the next state, so if we go to pi equals zero here, you know, we see we have kind of two modes that collapse to kind of zero energy in the rotating frame. We see there, and you see again that, well, this, this transports a lot to, you know, this mode, but much less to these two modes, right? So again, it's kind of pinching off there. Now on our map of the lattice, this is what this, this is what we had. So it's transporting well to B, which is kind of the far corner. And we also have the C and D. Now, at the same time, I told you, remember, that it was phi equals pi, that's interesting, not phi equals zero. But if I think about it, you know, I can draw these connections how I want. So if I twist this plockette, right? You know, what I see is that, you know, if I put a flux through the top loop and the bottom loop cancel because they're twisted, right? And this happens in the quite slatter. So the flux here is zero even for whatever flux I apply, right? The net flux. And so then I can, and when I do this twisting, the modes that are captured here are now these modes on the ends instead of being across. So this is starting to look like our little edge modes or something. Okay, so that was the quite slatter. That was published in PRL last year. And so another model that we've looked at, and so again, this is getting, we have a programmable system, right? The SSH model. So the SSH model Sue Schriefer-Hieger, and the same Schriefer in BCS. So this was, so this is a, it's a tight bond binding or hopping model. So I have sites and hopping in between them with alternating coupling strength. So I have strong coupling week, strong week. So originally it was a model of molecular dimers. So I have dimers that are strongly coupled that are then weakly coupled to each other. And again, despite its simplicity, this has become one of the kind of paradigmatic models of topological features or non-trivial topology. So there's a lot of interesting phenomenology that you can explore in this model. So first of all, to see anything interesting, we have to assume what we call, or what is called chiral symmetry, which means there's no energy difference between the two locations within the dimer. If they're like particle and antiparticle, they're symmetric. And then the feature is for a general tight binding model, I don't necessarily have a gap, but when I add alternating coupling, I open up an energy gap. So now with the alternating coupling, I have a gapped model or like an insulator. And then I have interesting things that depend on then whether I have an odd or even chain. And this goes for however big it is because of the chiral symmetry. So if I have an odd chain, I have one zero mode in the gap. And that's because, yeah, I have one zero mode in the gap, and that lives on one edge. And this isn't actually topological, it's just due to the chiral symmetry at this point. But then if I have an even chain, if my even chain ends with closed dimers, this is the topologically trivial phase. There's nothing interesting, but if it ends with the even chain, if the even chain ends with open dimers, then I'm in the topologically non-trivial phase. So this is a plot from kind of an online court tutorial on topology. So what I'm plotting is, they're kind of just calculating the energies of like a 20 site model. And then this is the coupling strength of the kind of the edge band compared to the not edge or coupling. So on this side, on this side I'm in the trivial phase and on this side I'm in the topological phase. So I'm changing the coupling. So in the trivial phase, I just have a gap model. As the couplings go towards tight binding, they're equal, the gap kind of gets smaller and smaller. But then on this side, this is where I enter the non-trivial topological model. And what happens is two of these modes split off from the band gap, they're split off from the bulk and kind of crash towards zero. So these become approximately zero modes that are localized on the edges. And these are plots of the wave functions of various modes. So you can see these two modes near zero kind of live on the edges. In an infinite chain, one would be on one end and one would be on the other. For a finite chain, they couple and I get even and odd superpositions. And then a random state in the bulk kind of spread out over the whole lattice. So we can look at this in our simulator. So we'll start now with five modes. So again, same device, but now I'm just putting in different pump frequencies. So I'm programming a different lattice into the system. And so I can start with five modes and actually just start with a tight binding model. So I tune it up. So each mode is coupled with five megahertz of coupling. And that's just adjusting the pumps. These numbers will always be now the coupling rates. And then I'm doing the same kind of scattering matrix measurements here. But now because there's a loop, there's no loop phase. So there's only one phase to measure at. So we just see one network analyzer trace. But it's the same basic idea. Now the diagonals give me the energy spectrum and the off diagonals give me the transport. And for instance, what we can see here is that because it's an odd number, it's an odd numbered lattice, there is a zero mode. But the zero mode is delocalized. See, it's just as strong in the center as it is on the edges. And this is what you would predict. And now we can go really to the SSH model. So now I just tune my pump strengths. So I make some weaker and I make some stronger. So I have now two and a half, ten, two and a half, ten. And what we can see is now that zero mode has localized to one edge, right? So I see the zero mode over here where I expect it because that's the weak coupling and there's no longer a zero mode here. And you can see it's kind of decaying and you can also see it in the transport, right? So there's little transport to here and almost none to there, right? So the strength of the wave function is decaying. We can re-tune things again and now our zero mode moves to the other side. And then we can also do things like make a defect. So we can break the pattern of the couplings and now it's like we have a domain wall in the center and you see now our zero mode gets pinned at the domain wall. And finally, yeah, so here we go to the actual topological version. So now I have a six site lattice where I programmed it so that we have the weak dimers on the end. So this is the topological phase and you see what we expect where we get now kind of two zero modes localized on each end. I mean, you can't tell so well that these are decoupled but this, the transport helps a little bit. So again, you can kind of see, you know, the transport dies off going, you know, for this one it dies off going this way and for this one it dies off going that way. So we can kind of see this zero mode is localized here and this zero mode is localized there. And then I had another model to talk about but I think I'm out of time. Is that correct? Am I out of time? I have some time? Okay. A few minutes? Okay. I'll dive in. So another topological model and this one is interesting because everything so far has been hopping but now we're at the pairing, right? So this was a model proposed by Oshklerx Group which is the Bosonic-Katayev chain. So it's a hopping model but now adding, you know, parametrically down conversion of a simulation as a pairing. So I called this the Bosonic-Katayev model so we could remind ourselves what's the Katayev model of Myerana fermions which is very famous now. So it's fermions but the Katayev model is basically, I have, you know, a single lattice that can host a single spin. I have hopping between these lattice sites so it's like tight binding but then I also have a pairing potential that exists between sites. So we have some superconductivity in the model. So this is the Katayev model, hopping, pairing and then Katayev showed that this could be solved in terms of Myerana fermions which are even an odd superposition of the electron operators. And, yeah, and, you know, there's been lots of excitement because it turns out it's topological and there are end states that people want to use on computing and are having lots of trouble finding. But, and also just to mention, you know, what's the topology in this model? Where's my donut? I keep always asking myself. So to see that we can transform that real space Hamiltonian into K space and so I've written it now in a compact form where now big C is just a vector of little C's, C and C dagger. And then I have, you know, polymatrices and then I have a kind of fictitious field which then encodes the actual information of the system and in this case this fictitious field has this form. So, right, I have, you know, no sigma x and I have sine and cosine and you can kind of see in your head as K varies through the baron zone so zero or minus pi to pi, you know, this vector winds around the origin, right, in the YZ plane and it's that winding is the winding number because the origin is a singular point for the model. So we have bosons. So can, you know, which are very different from fermions. Can we still find interesting things in a bosonic version? So how, first of all, how do we implement it? You know, we've talked a lot about hopping. So these are our beam-sweater couplings and then this is just down conversion, right? So I turn on some down conversion pump and I have something that looks like a pairing potential. And as it turns out, you know, well, and again, you know, if I transform to K space my fictitious field has this form. It's a little bit different but with the right choice of the free space parameters it still winds around the origin so I get a topological model. Now, I can again solve this topological model in terms of some kind of myerana which is a superposition of the creation and annihilation operators but anyone who does microwaves just recognize these as the quadratures of the field, right? So this becomes solved in terms of kind of myerana quadratures. And, you know, so it's topological, there's these myeranas, is there still anything interesting? And to see that, I mean, the easiest thing to do is to calculate the Heisenberg equation of motion of the system and this is what they are. And you see two interesting things, right? So first of all, X and P decouple. So the X and P quadrature decouple from each other which is, I didn't mention, the myeranas and the cataya model decouple from each other. And then you also see that the transport is very asymmetric. So going in one direction I have T plus delta, so tunneling plus pairing and the other one I have T minus delta. So at T minus delta this term goes away and the transport becomes completely asymmetric or completely chiral. So one quadrature moves in one direction and one quadrature moves in the other, right? So I get quadrature particles that propagate in a chiral fashion. So how do we measure this? Again, so we can tune up a simple system, a three model system. So we have pumps and down conversion. We didn't have another phase locked, well phase locked generator. So what we did is we inject a signal now in the center here where we just, it's a constant amplitude but we sweep the phase, right? And then we look at what comes out, the two ends, like the six and seven, the two ends of our model and we see that we really get kind of an alternating phase. So one quadrature phase is going left and one quadrature phase is going right. And we can do this, there's lots of pump phases. We can map these things out. So this is magnitude and phase, reflection, transport, experiment and theory. And basically what's going on is what we can see is that by changing, here we're changing the difference phase between T and delta or the beam splitter and the down conversion. And if those are, and when we do that basically we're kind of defining new generalized quadratures. I'm kind of changing my definition of X and P, right? And so each link has a different, slightly different definition of X and P and by tuning these things I can line them up. So if they don't line up, I don't get transport across it and if they do line up, then I get chiral transport across my whole chain. And I am now out of time but I'll just say, you know, we can also see fun things like in a closed chain we can see chiral instability and things like that but I'll stop there. Thank you very much for this interesting talk. And yeah, please. Interesting talk, thank you very much. I have a bunch of questions but let's start with the most important one. So first question is where are the limitations of your circuitry given the model that you want to simulate? Because at one point the circuitry is of course not an exact map of what you want to simulate. It's in certain parameter ranges and so to phrase it differently H junk that appeared during this conference several times what's the impact and where does it appear H junk? I mean there's a lot of things that happen so there's a lot of things hidden in the phrase tune things up. So when you pump these things we get lots of frequency shifts and things like that and you have to kind of deal with that. Now in these cases we're not really getting we're not putting, we're not getting down, we're not generating lots of photons so we're not getting like kerr shifts but you know when you pump you get shifts and things like that. Those are things we play with and I think you know at some point you will hit a wall. Am I going to lose this? At some point I'm putting all of these tones into one squid in the end and that squid will have a finite kind of dynamic range. So to me that will be the limit I imagine that will be the limit. I think we've gotten up to like 9 or 10 or 11 pumps so far and you know it's still working so my sense is that eventually this is nice because it's very hardware efficient. I have one cavity and I can do all this and one readout but I think at some point this could be a unit, one part of a bigger simulation or I mean there's no reason I can't do the individual cavity because I think there would be interesting things to do mixing cavities and trans bonds and things like that and then I can still use the parametric coupling I'll just have to have more lines or something like that and maybe more readout so I think there's lots of places to go. Chris thanks for interesting talk. I'm wondering how far can this thing can be really called quantum simulator because in fact you're sort of minuscaling it's single excitation manifold it's classical picture so can you do dublons for example in SSH model double excitation, localization of things probably not. Well I think at this level I mean you're right so this is all kind of the way I look at this is what we're seeing is the energy band that we're creating and so far all I put in here was coherent states so yeah I mean I do a mode diagonalization and that's a classical not hard problem but yeah so I mean that was my last slide I already quit. I mean the point is what do we have to do to make it fully quantum? Well first of all we can put it in quantum states it's a lattice and then we can load it so we haven't done it yet but we can throw in single photons it's very easy to throw in entangled states we can throw in two mode squeezing and we're already doing it so I think there's lots of quantum states of light we can load into our lattice that's interesting. I think the other thing is that you do want to start I mean what also can make things computationally hard to simulate is particle-particle interactions and so that speaks to having kind of more nonlinear resonators these are highly linear resonators so again if I went you know this is where I mean I think you could use this parametric coupling but now with kind of discrete resonators right and I think you know essentially I can go continuously with essentially the same Josephson junction plus capacitor I can go continuously from very linear resonator to qubit to nonlinear resonator which means you know photon interactions and things like that right so this exact platform is very linear I can still load non classical states but I think there's lots of things to do kind of in the same family I think with nonlinear driving so you know we had a paper where we did cubic parametric down conversion or three photon parametric down conversion and so like the how the HALC group showed that if I use that so if I use a nonlinear beam splitter so I couple one photon in one mode to two photons in another so I could use an auxiliary or linear mode but I use this nonlinear pumping I can split the mode of the you know I can split mode so I can use a nonlinear pump which we've demonstrated to turn on nonlinearity in the linear mode so I think that's another possibility right so I think there's lots of tools to continue exploring to make it more non classical or harder to simulate okay last question but please be brief I was just wondering whether you are limited in the size of the chain regarding the room temperature hardware you need to stabilize all the phases or are you limited by the saturation of the Josephson junction I mean so far I mean I mean I I mean so far we're not limited by any of those things or we weren't I mean maybe here we are limited by how many sources we had and now we bought a bunch more and we haven't seen problems with mode locking many of them yet so I think it will be maybe how much we can pump the junction that will limit us eventually I mean but what's limited so far and we're working on it is just as a tuning up problem like Frank talked about yesterday I have these frequency shifts and I'm trying to tune and I tune one and it affects the other so we're also working on better designs to reduce the frequency shifts and you know and things like that okay let's thank Chris again