 Mae'r cyffredin iawn. Hei, iawn am Pyre. Roeddwn i chi'n hoffi Dyrrwyng Wyrddol ac rwy'n gweithio'r gwrthodd y gweithio'r Llywodraeth o'r wahanol yn ymdeithasol ar y maes yma. Mae'r bwysig erioed yn yng nghymru nad oedd Adam i Llywodraeth, ac mae'r bwysig yn dal yn yng nghymru. Mae'r preprint o'r archif wedi'u bod yna i chi'n gweithio, Felly'r gweithio yw'r ystod yn ddiwedd yn fan gondol o改-dwyntau gwahanol ar gyfer gweithredu gwahanol i gwybodaeth a clasgol. Felly, mae'r problem yw'r gwahanol yn ymgyrch o'r problemau gwybodaeth yma, yn yma'r problemau gwybodaeth, ac mae'r problemau a'r problemau gwahanol e'n gwneud yn ymgyrch yn ymgyrch. ac mae'r ffordd maes i'w FFNP hod, mae'r ffordd maes i'w FFNP a FFNP yn oed yn gweithio i gael cyflosol clasig o'r problemau sy'n gweithio llwyddiadau yn y ffordd, sy'n gweithio'r ffordd, yn ymgyrch, yn ymgyrch, ac yn ymgyrch yn ymgyrch yn ymgyrch yn eu ddwylo'r problemau. Felly, yna'r cyflosol yma, mae'n gweithio'r ffordd yma'r cyflosol sy'n gweithio'r cyflosol by Crassane a her co-workers. So these instances were selected to be hard for particular type of quantum annealing. And we've also used randomly generated instances, which we call the typical instances. So why do we want to compare the hardness of problem instances for the different algorithms? Well there are two motivations, firstly we want to find out whether small sized problem instances are a good representation of larger problem instances in terms of hardness, and this is an important question to consider for many studies that numerically simulate these algorithms on small sizes. Secondly, we want to find out whether a simple hybrid approach that runs some combination of a quantum and or classical algorithm in parallel is able to perform well. So the intuition here is that if two different algorithms find different instances easier and harder, then one algorithm can speed up the instances that are hard for the other algorithm. So I'm going to talk a bit more about Max2Sat here. The input to Max2Sat is a Boolean formula phi in two conjunctive normal form, so I've given an example of such a formula here. In general it needs to be a conjunction of clauses where each clause is a disjunction of literals and a literal is either a Boolean variable or a negated Boolean variable. So conjunction, disjunction and negation are equivalent to the logical and or and not operators, which I've coloured in red, orange and purple respectively. So then the problem is to find an assignment to these Boolean variables such that the number of clauses that are satisfied, i.e. the number of clauses that are evaluated true, is maximised. So to solve Max2Sat using continuous time quantum computing methods, we need to map it to a problem Hamiltonian. So to do this we start off by encoding a single clause as a Hamiltonian term which is shown on the right. Here we use a notation where the literal li represents the variable xi if i is positive and if i is negative it represents the negated version of that variable. So this term adds an energy contribution of one if the assignment corresponding to the basis state does not satisfy the clause and if it does satisfy the clause it does not contribute any energy. And then the full problem Hamiltonian can be attained by taking a sum over these clause terms for every clause in the Boolean formula and the problem becomes to find the ground state of this Hamiltonian. So as you all know one method for finding the ground states of Hamiltonians in a continuous time quantum computing setting is AQC. So here we implement a time dependent Hamiltonian using the transverse field driver Hamiltonian and we start in its ground state which is the equal superposition state and using a linear schedule we turn off the driver Hamiltonian while slowly turning on the problem Hamiltonian and then so for our instances we have a unique ground state so then the success probability which is the probability of finding the ground state at the end of the sweep is given here and so our measure of hardness for an instance for AQC is the time required to attain a 99% success probability or T0.99. Another method in continuous time quantum computing is quantum walk computation so here the Hamiltonian is time independent and it's parameterised by a hopping rate gamma so we use a heuristic value for the hopping rate which is equal to the average energy spread of the problem Hamiltonians divided by 2n. We also use the same driver and problem Hamiltonians as for AQC. So then in quantum walk in general you don't have a probability of near unity for attaining the ground state for measuring the ground state so in general you need to take repeat measurements and so we define an average single run success probability which is shown here and this is for measurements that are taken uniformly at random within the interval of T and T plus delta T so for our simulations we set T equal to 0 and delta T equal to 100. So now going on to our results and I should mention that hardness for quantum walk is inversely related to this average single run success probability and going on to our results here we show the quantum walk average success probability at the top and the adiabatic duration at the bottom for instances separated by the hardness so in the top left plot we've separated that the instances by hardness for quantum walk where the lightest shade of green represents the 10th percentile instances for quantum walk hardness the darker shade of green represents the 90th percentile instances for quantum walk hardness and the the blue data points offer the 99th percentile instances and we plot the quantum walk success probability for these percentiles and this orange data point is for the is the median value for the instances that are hard for quantum annealing and to each of these percentiles we've given them a linear fit and the gradients so the scaling exponents are shown on the right so yeah so the fact that these lines are spreading indicates that as you increase the problem size the difference in hardness between the easiest and hardest instances is increasing and the placement of the orange point shows that the instances that are hard for quantum annealing are also hard for quantum walk however not as hard as they are for quantum annealing which indicates that there's while there's a correlation between quantum annealing and quantum walk hardness it's not perfect so potentially a hybrid strategy involving quantum walk and quantum annealing would be beneficial and in the bottom we show similar plots for AQC so this time we're plotting the AQC duration for AQC hardness percentiles and we get similar results with the main difference being that the 99th percentile instances scale considerably worse than the rest of the percentiles which is to be expected and the implication of this is that is that a hybrid strategy involving AQC would be particularly beneficial for solving these instances so in this slide we show similar plots but with a different comparison so we've swapped over the algorithms that are being used to categorize the hardness so in the top we're plotting quantum walk and median quantum walk success probabilities for AQC hardness desiles and in the bottom we're plotting median AQC durations for quantum walk hardness desiles and the fact that harder desiles correspond to harder instances shows that there's a correlation between quantum walk and AQC hardness and looking on the right the fact that this plot is decreasing throughout shows that this correlation does not only exist for the easier instances but also for the harder instances so to make a comparison with classical algorithms we've counted the number of times a classical algorithm called mix b and b accesses the problem specification and we call this the number of calls or n calls and in this histogram we show the approximate probability density of the logarithm of the number of calls for the typical instances in green and the instances that are hard for quantum annealing in orange so both of these distributions are double peaked which indicates that there's some separation in hardness for the instances for this particular classical algorithm and the instances which are hard for quantum annealing lie on the mostly lie on the hard tail of the distribution for the typical instances which implies that there's a correlation in hardness for between quantum annealing and this particular classical algorithm and in the bottom we show a table that shows the correlations in hardness between different algorithms so we can see that the correlation in hardness between the two quantum algorithms and the classical algorithm are lower than the correlation between quantum walk and aqc so this implies that a hybrid approach involving a classical algorithm and a quantum algorithm would be better than one involving quantum walk and aqc finally we also analysed satisfiable instances so this is because satisfiable instances are classically easy as twosat is in p there's a linear time classical algorithm for twosat and so we're looking at how hard they are for these for quantum walk and aqc here and we find that while for both quantum walk and aqc satisfiable instances are easier they appear for quantum walk the scaling appears to be exponential and for aqc the results aren't as conclusive so the what we can get from this is that a hybrid a hybrid algorithm involving a classical algorithm would be also be beneficial for solving these instances that are satisfiable efficiently so to summarise we found that there are correlations between quantum walk, aqc and quantum annealing hardness but these correlations aren't perfect so this implies that a hybrid strike a hybrid strategy involving these algorithms would likely be useful in real world computation and the proportion of hard instances grows as you increase the problem size and we found weaker correlations between the quantum and classical algorithms than between quantum walk and aqc and finally why we've identified certain weaknesses that can be overcome with a hybrid approach such as the very poor scaling of a small subset of instances for aqc and the fact that these quantum algorithms can't solve satisfiable instances efficiently so I'll end my talk there thank you for listening how did you run how did you run the aqc so we numerically integrated the Schrodinger equation with the linear schedule and transverse field driver Hamiltonian you can do it for 20 qubits sorry how much how much did it take for 20 qubits the microphone is on uh did you say for 20 qubits um so if we yeah I should have mentioned this but for quantum walk we went up to 20 qubits but for aqc up to 15 um so that's why there's no um orange data point on this graph as well hello is it on yeah question oh okay okay please ask the question uh can you see it up no I'll read it out um so the question says are the results for aqc and quantum annealing taken from simulations not quantum annealing machines um yeah so the we only have results for aqc which were um from simulations and the quantum annealing the instances that are hard for quantum annealing were taken from um the paper brought by croissant and their co-authors um and that was implementing um essentially the same thing as aqc but with a constant duration so for most of the instances that were generated they were in the adiabatic regime but for the instances that were selected to be hard the success probability was very low so that's why I called it hard for quantum annealing um but it was a simulation yeah questions yeah five minutes for his question any question online yeah question any idea on how to combine quantum walk and aqc uh yeah so um there are more sophisticated um so do you mean so here what we're looking at is just running them in parallel right um but there are also more sophisticated sophisticated hybrid strategies um that my co-workers have worked on um so you can have hybrid quantum walk and aqc and you can have rapid quenches uh so this is just the simplest case of a hybrid strategy where you just run both algorithms um maybe one after the other or in parallel or something like that any other question well if no questions let's thank this speaker again