 Okay, good morning everyone or afternoon or evening wherever you are. I'm Daniel Adar. I'm chairing this session and we'll have a series of four 20-minute talks 15 minutes plus five minutes of questions and our first speaker is Micah Dreeb-Schoon on parity quantum optimization and coding constraints Thanks for the introduction. Can everyone hear me? Okay now Perfect. Okay Yeah, first Thanks for the possibility to do give a talk here at the AQC in Trieste I will present a work about how to encode optimizations problems which constraints and Therefore I will give one short overview about the talk We are interested in Constraint optimization problems. So namely optimization problems with polynomial equality constraints And we want to find an encoding which encodes both the optimization problem together with the constraint in an easy implementable way so for current hardware devices digital and analog I'll therefore start with the LHD model on which is our approach based on so We consider an ising spin encoding of this form here with two-body interaction into local fields and we Consider all-to-all connected problems. So for this example, we have a six qubit example and first of all, I will focus only on Problems without local fields where every product of logical Conceal or logical qubits Transforms to a single party qubit and it's called party qubit because it represents the party of the logical products But here so for example, if both are up or down we get a plus one and if they are of opposite direction we get the minus one and If you have considered now the Number of poverty qubits we get with it with this method We see that it much larger than the original problem and the degrees of freedom and this means we have to reduce our degrees of freedom and this do We with party constraints. So we have to find a certain number of constraints This can be calculated by the number here party qubits minus the number of the logical qubits plus the degeneracy of our graph. So for example, here we have a global spin flip degeneracy. So give one and How we find these 10 particular constraints here We search for all closed loops in the logical graph and we take out 10 of them such that all edges are covered and if we have a closer look on One of these closed loops you see that every closed loop includes an even number or zero number of logical qubits And this means the product out of it gives always one and this is what we use to construct our party constraints and therefore we find An LHL layout in which we have now the party qubits are physical qubits and Only local magnetic fields. So this is one part of our Map time alternative and the other part is a constraint part Which consists out of the products over all party qubits of each of these placets. So we have these Triangle placets here. These are local three-body interactions now and the four-body local Interactions are the squares and each of the placets has a weight as a constraint strength C which ensures during the Computation process that we always conserve this one here for an even product of logical qubits and adding the local fields Meaning here in this example that we need now 15 party constraints and we can simple add them as an extra row. So we get here the extra six physical qubits for with the five additional Party constraints now we can ask the question. What's about sparse connected graphs? So for this example We get some qubits here in Gray which has zero magnetic field and in principle they are not needed and This is why we introduce the party encoding. We want to have a more compact Encoding here and we see that we need for the simple example here only four party qubits and by having Look on all closed loop of our logical graph We see we have in total five and we pick out four of them in such a way that we can construct a compact party encoding now only with party qubits and only four body and three body local interactions now we come to the optimization problem with heart constraints and We focus here on Equality constraints of the kind that all coefficient coefficients here J G are only zero or one and we leave out inequality constraints Which can be mapped on a set of multiple overlapping equality constraints, but with coefficients different from zero and one There are already methods for Constraint optimization problems and the most common and traditional method is the penalty method for sure and it can encode all of these equality constraints and These by adding and to the cost function the square over the equality Constraint with an extra German front-alpha of it which leads to Extra additional k-body interaction which for a sparse connected model is not so good and in the worst case we get an all-to-all connected again and Additionally, we get these large energy scale by alpha here, which can be very large for some problems There was one approach the optimization based approach Which can solve single some a sum of a single logical qubits here as a quality Constraints and it was adjusted and optimized only for the camera graph and the d-wave systems and there was a constraint quantum annealing approach with also consider only sums of a single logical qubits and they use new driver Hamiltonian, so an exchange term which acts only own Neighbored logical qubits here and Therefore the Experimental engineering was a bit unclear, so not full clear and Now I present the constraint party quantum annealing approach, so We have here a sum over products of qubits over this means we have here polynomial Equality constraints and we combine the party encoding with the constant quantum annealing approach idea to get a more experimental visible Encoding and more of its platform Independent and we have only local interactions and we are always able to neighbor the constraints qubits and We can solve therefore constraints with Polynomial type of logical qubits and in the following I will present how we do this combination Therefore, let me give a short overview. We are able to encode now k-body Problem Hamiltonians together with polynomial constraints and these polynomial constraints become sums of a single physical qubits Which can be lay out? Neighbored always And this makes us possible to combine it with the constraint quantum annealing approach and for this example here with the two and three body interactions and local fields and The polynomial constraints here we get for example after the party and transformation To non-overlapping some constraints which have qubits which can be lay out neighbor in the encoding here and These question of neighboring yes, we are able to lay out Cubits neighbor even if they are not Neighbored by nature in the party layout this means we consider this simple example here where we have these Close loops here of the logical graph again, and we have this polynomial constraints with product terms Which are already in the problem example, but They cannot lay out Simple neighbors we need ancilla qubits to neighbor them but with ancilla qubits we are able to neighbor them and again we can apply exchange terms on it and This has the disadvantage of course that we have to introduce additional qubits and additional couplings for the new party constraints here Now I will come to the simple annealing example to explain how it works now for the computation with quantum anneal us and Therefore I consider here again the simple logical graph with these Hard constraints which has product term which are already present and optimization problem and can be lay out neighbor without any Ancilla qubits and on this constraint qubit. I apply an exchange Hamiltonian now which acts only on these constraint qubits and we have additionally the X driver with X as the single spin flips only on the unconstrained qubits here and We will initialize our system in a classical state which fulfills our constraints already This means we need an additional Hamiltonian which encodes all the Constraint qubits here in a way that the constraint can be fulfilled and the rest can be choose on randomly such that the ground state of the initial Hamiltonian is for sure a state which fulfills our constraint here and our Hamilton dynamics so the annealing protocol Can look like this we have here s which is driven from zero to one again and we drive the initial Hamiltonian with a function 1 minus s squared and the problem Hamiltonian here encoded and the party encoding is driven linear by As and then we have the single spin flip driver which acts only are unconstrained and the exchange with X only on Constraint qubit and this is driven by switching it on and off during the annealing is with with a bit reminds on reverse annealing and in the end these are two example of Our results what we get so what you see here are two energy spectra for two different examples with the same problem Example here, but with different constraint qubits and different constraints here in yellow and The first example So presents the case where the ground state of the optimization problem already Fulfills our constraint and the second is an example for the case where the Groundstand is not the fulfilling state, but What an higher excited state is the lowest lagging state which fulfills our constraints here and more over Here are colored lines and gray lines the colored lines Represent the energies corresponding to instantaneous Eigen states which fulfill our constraint and the gray on all Eigen states which do not fulfill our our constraint and This is because our our annealing here Is hasn't restricted evolution so the evolution of our annealing process is restricted to the subspace of Eigen states which fulfill all the constraint and the rest so we reach an open and superposition during the annealing process of all Eigen states which fulfill the constraint, but not more and not less and Finally, I mentioned that The final adiabatic running time now is not any longer given by the Minimal gap of the full energy spectrum, but more given by the gap between the lowest laying states Eigen states of the Optimization problem which fulfill the constraint more over we showed so more or less the research word Bakilian and I showed that We also able to encode it for digital quantum computer with the quantum approximate optimization algorithm and Yeah, now I will shortly summarize now. We have shown that we are able to encode logical optimization problems with polynomial Constraints into some constraints which can be lay out always as Neighbor to cubits and the party encoding together with the optimization problem and with the here presented annealing scheme we are able to Evolve the system restricted to the subspace of constraint fulfilling Eigen states In the end I want to thanks for the collaboration first of all this work was done in collaboration with Wolfgang Lechner the group the head of our group in Innsbruck and of party QC and By Kylian Ender and Eunice Yavan mad and I want to thank the whole work group and party QC for our fruitful Discussions for this work and other works. So thank you And I stalk do we have any questions? Somebody on zoom. Could you please mute your microphone? Oh, okay, please go ahead Again could you repeat that? The degeneracy Yes, if we have a degeneracy and the optimization problem only But not in the constraint optimization problem in place no role because the constraint problem Then it's not degeneracy, but if we have a degeneracy in the constraint problem So we have more eigenstate with the same energy and all fulfill our constraints Then we have a bias normally so a bias on on these Degenerated ground state and we can adjust the bias of the ground stands by choosing our party layout so namely the kind of Exchange Hamiltonian we act let act on so the neighboring of the cubits and we can adjust the bias additionally by Changing our party constraint slightly Okay, thank you for your nice presentation Could you go back to the slide so whose title is simple quantum example this one The next ten the next the next Yes, this one. So in this side, so you introduce the example of the hard constraint Which is a constraint of the so to body interaction of the logical cubit So I was wondering so how to deal with the so one hot for example one hot concentrate on the logical cubit So because so one hot example is a one-hot constraint is one of the so common example of the Constraint is a quantum annealing this constraint is a so to body interaction constraint So for example, so in this constraint so sigma 2 sigma 3 plus sigma 1 sigma 2 plus sigma 2 sigma 6 equals to some const variables So I was wondering so for example, so sigma 1 plus sigma 2 plus sigma 3 equal 1 Okay, you mean if I have other product terms then in the problem Hamiltonian yes for first and yes Yes, of sure. This is a very simple example, but in the paper we wrote we will give Some additional example for for these cases and yes, this is possible You have to pick out these products which are present in the constraint But not in the problem Hamiltonian and at them by the party con by the party transformation To the encoding and again lay out the neighbor and then they are in and they are solvable with them Thank you Other questions Thank you very much for your talk How difficult is it to find all these closed loops when you're doing this encoding? This is a good question. Do you ask how easy is it to find it always that they are neighbor layout, right? Yeah. Yes So for sure there are some examples so for the most of them it is not so hard but we cannot say what how hard is for the hardest and So there's even the open questions how many and Silla qubits I will need to encode every problem away that they are always layout and Yes, for sure to say it very precise. It's more kind of future research work But in the end with enough qubits and with enough couplings We are always able to lay out the neighbor. So it is possible, but I cannot say How much? Yeah, it's strongly problem dependent So for some problems one will get an advantage for other ones. It could be that there's some disadvantage The even for like The simple example here where you have to find these closed loops even that step. How difficult is that? How we find these closed loops? Yes, so if I had a generic graph like Is that a problem? This is already possible and this is what the work of party QC do So they they wrote a compiler which is able to find always party encoding for Problems so these compiler does exactly these That's exactly this we have they have an algorithm was transforms The optimization problem here on the left hand side to the party encoding Thank you very much question in the Access it on your computer speaker Okay For you in the chat Okay Yes, I can explain that could you read the question first the question is I will Repeat the question directly on the slide where I have it. I think the question was for For this example here in the end where We add the local fields here to the problem and the question is how we come to the number 21 minus 6 plus 0 So as I said case the number of parity qubit which we need in this case we have here 15 interactions on total and six logical crew visits six Single local fields when I add these local fields gives us additionally Six physical qubits means that we have 21 physical qubits here. Then we have the six Original qubits which we have to substrate and then we have a zero degeneracy because these local fields now break the degeneracy of our problem Which we have before and this is how I come to this formula and why we need 15 party constraints Okay, one final question So if there is no constraint does this OHC Formalism still work Is this Formalism totally about constraint optimization thing your question if it works for the constraint optimization or Works it also for other problems. Yeah, like if you remove any constraint so you can always any optimization problem versus given in the ising spin encoding so can be mapped on the party encoding and also on the LH that LHD model and That constraint optimization problems can be mapped as only an Extra so an add-on so you can say and we cannot solve in the moment Not all we cannot encode all constraint But only if how I showed only the ones with equality constraints, which are polynomial and have coefficients 0 or 1 So if there are no constraints We what we are the loops be like there will be all kinds of loops We are no constraints in and or do you mean the party constraints here? Do you these these party constraints are only needed to reduce the degrees of freedom here to have a valid map into the party encoding? So these are these plaquettes here and these plaquettes ensure I Can heuristically say they ensure that never when I remap it to the original logical problem One of the same spin can be up and down at the same time. So this would be an Invalid solution and these party constraints here Avoid that that I get and result which is not Okay, thank you very much further questions. Please ask Micah later and Thank you and our next speaker