Given a list of cities and the distances between each pair of cities, how do you determine the shortest route that visits each city exactly once and returns to the starting location? This famous problem is called the "traveling salesman problem" and is an example of a combinatorial optimization problem. Solving these problems using conventional supercomputers can be very time-consuming, and special devices called "quantum annealers" have been created for this purpose.

Quantum annealers are designed to find the lowest energy state (or "ground state") of what's known as an "Ising model." Such models are abstract representations of a quantum mechanical system involving interacting spins that are also influenced by external magnetic fields. In the late 90s, scientists found that combinatorial optimization problems could be formulated as Ising models, which in turn could be physically implemented in quantum annealers. To obtain the solution to a combinatorial optimization problem, one simply has to observe the ground state reached in its associated quantum annealer after a short time. {module INSIDE STORY}

One of the biggest challenges in this process is the transformation of the "logical" Ising model into a physically implementable Ising model suitable for quantum annealing. Sometimes, the numerical values of the spin interactions or the external magnetic fields require many bits to represent them (bit width) too large for a physical system. This severely limits the versatility and applicability of quantum annealers to real-world problems. Fortunately, in a recent study published in IEEE Transactions on Computers, scientists from Japan have tackled this issue. Based purely on mathematical theory, they developed a method by which a given logical Ising model can be transformed into an equivalent model with the desired bit width to make it "fit" a desired physical implementation.

Their approach consists of adding auxiliary spins to the Ising model for problematic interactions or magnetic fields in such a way that the ground state (solution) of the transformed model is the same as that of the original model while also requiring a lower bit width. The technique is relatively simple and completely guaranteed to produce an equivalent Ising model with the same solution as the original. "Our strategy is the world's first to efficiently and theoretically address the bit-width reduction problem in the spin interactions and magnetic field coefficients in Ising models," remarks Professor Nozomu Togawa from Waseda University, Japan, who led the study.

The scientists also put their method to the test in several experiments, which further confirmed its validity. Prof. Togawa has high hopes, and he concludes by saying, "The approach developed in this study will widen the applicability of quantum annealers and make them much more attractive for people dealing with not only physical Ising models but all kinds of combinatorial optimization problems. Such problems are common in cryptography, logistics, and artificial intelligence, among many other fields."

In a recent study, researchers from Imperial College London developed a model to assess the effect of different measures used to curb the spread of the coronavirus. However, the model had fundamental shortcomings and cannot be used to draw the published conclusions, claim Swedish researchers from Lund University, and other institutions, in the journal Nature.

The results from Imperial indicated that it was almost exclusively the complete societal lockdown that suppressed the wave of infections in Europe during spring. 

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The study estimated the effects of different measures such as social distancing, self-isolating, closing schools, banning public events, and the lockdown itself.

"As the measures were introduced at roughly the same time over a few weeks in March, the mortality data used simply does not contain enough information to differentiate their individual effects. We have demonstrated this by conducting a mathematical analysis. Using this as a basis, we then ran simulations using Imperial College's original code to illustrate how the model's sensitivity leads to unreliable results," explains Kristian Soltesz, associate professor in automatic control at Lund University and first author of the article. 

The group's interest in the Imperial College model was roused by the fact that it explained almost all of the reduction in transmission during the spring via lockdowns in ten of the eleven countries modeled. The exception was Sweden, which never introduced a lockdown.

"In Sweden, the model offered an entirely different measure as an explanation for the reduction - a measure that appeared almost ineffective in other countries. It seemed almost too good to be true that an effective lockdown was introduced in every country except one, while another measure appeared to be unusually effective in this country", notes Soltesz.  {module INSIDE STORY}

Soltesz is careful to point out that it is entirely plausible that individual measures are affected, but that the model could not be used to determine how effective they were.

"The various interventions do not appear to work in isolation from one another, but are often dependent upon each other. A change in behavior as a result of one intervention influences the effect of other interventions. How much and in what way is harder to know, and requires different skills and collaboration", says Anna Jöud, associate professor in epidemiology at Lund University and co-author of the study.

Analyses of models from Imperial College and others highlight the importance of epidemiological models being reviewed, according to the authors.

"There is a major focus in the debate on sources of data and their reliability, but an almost total lack of systematic review of the sensitivity of different models in terms of parameters and data. This is just as important, especially when governments across the globe are using dynamic models as a basis for decisions", Soltesz and Jöud point out.

The first step is to carry out a correct analysis of the model's sensitivities. If they pose too great a problem then more reliable data is needed, often combined with a less complex model structure.

"With a lot at stake, it is wise to be humble when faced with fundamental limitations. Dynamic models are usable as long as they take into account the uncertainty of the assumptions on which they are based and the data they are led by. If this is not the case, the results are on a par with assumptions or guesses", concludes Soltesz.