 about the coding strategies to enhance parity quantum approximate optimization and by the way she's from Innsbruck University. Thank you so can you hear me clearly yeah good okay so yeah I will present you to you my work about decoding strategies to enhance parity quantum approximate optimization which I do during my PhD at the University in Innsbruck so here you can see our group so in the middle you see the head of our group Professor Wolfgang Lechner who is also involved in this project as well as our postdoc Glendon Bang which you can see here or just sitting right next to me and yeah as some people may know our group we work a lot with the LHSAT model so and to come to this I want to show you here shortly how to encode optimization problems so here you see Hamiltonian and the 2d qubit arrangement so the circles are the qubits and they're connected in line so the edges are the interactions where the interaction strength is j i j is where the problem is encoded and and on a physical device now one would have this problem to encode this long range interactions and this is actually now the main motivation behind the LHSAT model which implements these interactions as local fields model was introduced in 2015 by Wolfgang Lechner and Philipp Haukend Peter Zola so it's the LHSAT because it's just the first letters of those surnames and to show you how this works I want to demonstrate on this wall for a qubit example already see if they all the qubits are up and now in the LHSAT model I actually encode the parities between those qubits so if they're parallel they will get the value zero and if they're anti-parallel they get the value one in the LHSAT model and as I have six interactions I have here six qubits yeah each with a local field but as I have now I'm more qubits than on this logical model here on the left you need to implement constraints because now I can have configurations here in the physical model that do not translate back to the logical one so I want to show you this by flipping the top spin so here on top there's now a one now the question is how is the configuration on the left hand side in the logical model so I start with the decoding so we look at the first qubit here which tells me the parity between qubit zero and qubit one so it's supposed to be parallel so we can fix them on the left hand side to both up for example and then I look the second qubit it's also supposed to be parallel so it's also up but now the third qubit is supposed to be anti parallel to qubit zero so we've set it to down and if we go on now and look at the parity between qubit one and three which is supposed to be parallel in the physical model here but here you will see on the left they're already anti parallel the one is up and the one is down so we ran into a contradiction and yeah so we can't construct a valid logical state but as we can just throw this state away in the physical model it can give it an energy penalty because we are just interested in the ground state put is with the constraint so we have a constraint condition and the constraints you can see as this great triangle and this great diamond so the condition on this great diamond would be that the amount of zeros here of this four qubit should be even and as we have three so this is violated and yeah would get an energy shift to higher energies also if I would add one more qubit in the in the logical one this relates to like a full row in the let's add model so in fact the number of physical qubits square quadratically with the number of logical ones so you know also in the next slide so I see here the number of logical qubits on the x-axis and a number of possible states on the y-axis so we the green line corresponds now to the let's add model and the blue line to the logical one so we see that we have a great overhead in the physical model of this constrained violating states so states that never translate back to this logical one and of course we interested in the logical states because one of them is our ground state and this can actually now be a problem in QA OA which is short for quantum approximate optimization algorithm proposed by Farid Ali in 2014 just you can see here a schematic circuit how it works I mean we already heard it a few times today so we start with an initial state and then we apply our unitaries and P times so the unitaries you can see here on the right we have a driver term and a problem one unitary containing the problem Hamiltonian and yeah if would be would go to infinity we would reach the exact solution but yeah as we can't do that we get an approximate one then we can perform our measurements where the outcome depends on our first guess of this parameters better in gamma and with this we can evaluate our energy and hand it to the a classical optimizer which six to your minimize the energy so the optimizer can then return a new set of parameters to our quantum circuit and I repeat this till I reach a reach a come against criteria and now as I said here start with an initial state and in general this is a equal to a position of all the states but as I said in a physical model we will have a lot of this constraint violating states and as we can't apply this unitary infinite amount of time we will always have a high probability to measure this those states yeah which will never contribute anything to the solution so we will have a low performance and that's why we need a solution which I want to present to you now so here you see the adapted circuit like in particular always on qo and the let's set model we have one more unitary and this is because we split up our problem Hamiltonian into the local field term and into the constraint term but actually the new stuff now happens here at the measurement because here we also decode and we decode a physical state into logical state so we can hand a logical energy to the optimizer so we so to say good rid of our unphysical states but as I told you here we it's not easy to translate them back because we ran here between qubit and free already in this contradiction but actually before we read out qubit one free they actually had already a logical state and if we now look at this red line so the edges between those qubits we see that they actually cover all the nodes so all the qubits without making a loop so as soon as I made the loop I ran into this contradiction and that's actually the definition of a spanning tree and if I look at other spanning trees here we see that we can always decode a state so on the bottom you see the logical state where the arrows denote the spanning tree and on the top you see the physical model where the marked qubits translate and correspond to these interactions to the trees so I basically can just take those three qubits which corresponds to these lines and decode the state and in fact here for example on this first example on the left a fixed qubit number zero to up and look at the parities to yeah the other qubit so I can determine okay is the other qubit should it be parallel or not and like this I can decode and in a complete graph so in an ultra connected graph as I have here they're in total n to the power n minus two spanning trees possible so here I show four but as I have four qubits they're in total 16 so that would be other 12 spanning trees for the decoding also want you to see if we focus now again on this four big examples on the left that the decoded states they are not always the same so you see in the first one the third qubit number three has a downspin and the others two in the middle they always have upspins and this is because my physical state violates a constraint so if I have a constraint way violating state all these spanning trees could return and a different logical state but if all the constraint would be fulfilled all the spanning trees would give me the same logical state and yeah if this information I can show you how we construct now a new objective function on the bottom you see this physical state we want to decode and now we just consider physical qubits that correspond to a spanning tree in the logical model so we for example we take those three qubits and then we can construct our logical state like this but of course we can take more of them so for example we also take yeah a second one so we have two spanning trees and two logical states and then we just take the average energies of those states and to calculate them energy like this so in these examples we would just add those two energies and divide it by two and now I already want to show you our results so you can see here that on the x-axis we have the number of logical qubits and if I use if I have six logical qubits I also take six spanning trees for the decoding and on the y-axis see on the left plot we have the residual energy which is calculated like this so the measured energy minus the ground state energy and then the maximum minus the ground state energy like this and low energy value so the values go then from zero to one and a low value is preferable also on the right plot where I have one minus the success probability and the blue line you can see here corresponds to the parity QOA so you can see it really and rises fast to the value of one which is bad but with the decoding which is the orange line actually get really good results so we could improve it it's also better than the the green line which is the swap approach and I call it swap because in a logical model and it's done to implement this long range interactions with swap gates so I have to bring them together so I would swap those qubits so they are next to each other and then implement this interaction in this case it doesn't matter because I have an ideal circuit but now I introduce a deep polarizing noise to our two qubit gates so now it matters how many synods I use so because one swap gate consists of three synod gates here and in the LHZ model I have to implement my constraints with synod gates so a four qubit constraint would be implemented with six synods a three qubit constraint with four so I have more synod gates in my parity embedding and but still as you can see this orange line so the parity with the decoding already has a higher success probability that already with 10% noise it's as good as the one with the swap approach with zero noise but this is done with six logical qubits so 15 in the LHZ so exactly the size you can see here so I wanted to make a different comparison with modular parity QAOA so and this was actually published I think about two weeks ago from some people of our group and there you actually start from a constraint fulfilling state and then you apply a driver that only introduces transitions to other constraint fulfilling states so that I never leave my constraint fulfilling subspace to non-constrained fulfilling ones and they do this by introducing so-called driver lines so here we see the LHZ model and one line here that is driver line the purple one goes through all the qubits with the number of five and the orange one goes through all the qubits with four in it so that the lines and the driver term is just the sum of those lines like this our model is equivalent to the swap approach so this would all be equivalent that's why here with zero noise this bar here starts at the same point but as soon as I apply the noise we see the difference again I have more she-knots in the blue and approach in the parity then in the in the swap approach but I see when I use the decoding I'm very stable to noise so if this I can summarize it now I showed you how to do the decoding and to yeah construct the new objective function so we could improve parity QAOA with it and we also show better performance than the swap model at least with small system sizes but we are more also more stable and to noise if you use a modular parity QAOA so now I'm open for questions things actually have a question in slide number 10 can you please go to there this one yes so in the last place you are taking this EL alpha beta gamma and why did you come up with this formula what is the objective and because now we have because we have several logical state now that corresponds to one and physically one so then was the question which energy should we take so we could also take the minimum energy but as we just look at the small problem size we thought it's not such a nice comparison as well that's why decided first on this on the average and I actually also have another question that I actually had from yesterday why would somebody choose a parity qubit instead of a normal qubit and this is because as I am headed in the beginning that implementing these interactions especially this if qubits are far away and is not possible in a physical device so we are limited to the connectivity of this of the physical device and with a parity qubit I just need local fields and the qubits with the constraints I just can implement see not with with the neighbors so I'm very independent of the hardware use thank you yes so you explain how to how to get the energies from the many decoded states you get but how do you get the states you want out of the many decoded states and I can decide on the spanning trees I can take random ones I just have to fix an amount and if I have those as I said it on the spanning trees I have to decode them and get my collection so I just have to decide how which spanning trees they want and how many yes so how did you decide like which spanning tree you take as your solution and what I first did I have to like look at the performance for example on yeah how does it depend on how many spanning trees I take and then I chose when I have yeah six logical qubits then I chose actually the spanning trees as you can see here in this big example but just for a bigger with more qubits so I decided to because then I cover all the physical qubits and yeah I chose those spanning trees which we call also like the logic lines you read out the logic lines of it yeah okay so as you increase the number of spanning trees you ground state probability increases correct so as you increase the number of spanning trees you actually consider you get that you have more probability to actually get the ground state and yes but it will yeah stagnate at some point are there any other questions yes could you comment once again what is your initial state is it some pure state of your no I have an always an equal superposition of all the basis stage it's like like in quantum but is it a pure state of your more complicated Hamiltonian or it is some no of my of my physical device yeah of my physical configuration yeah and how does it look like it's do you have some formula how it looks like allowing all your constraints or it should not be just xxx like in a standard qa because you have constraints right yes yes but here I also consider the the constraint violating states I take all of them just not in the last slide I showed yes yeah but with this model on priority I can't tell you exactly so you start from some state which yeah it's which is not the which fulfills all the constraints yes mm-hmm okay are there any more questions if not we can thank