# Quantum Optimization Isn't All It's Cracked

**Quantum Optimization Isn't All It's Cracked Up To Be**

We spend a lot of time solving optimization problems intuitively because they are ubiquitous. We run into them when we're trying to figure out the quickest route between after-work errands, when we're looking for a restaurant that can accommodate a variety of tastes and dietary needs, and when we're trying to figure out which grocery store queue will move the fastest. We utilise classical computers to execute algorithms that seek out the best feasible answers for more complicated optimization problems, but these issues quickly become extremely difficult and computationally expensive. Quantum computers, according to researchers, might solve these problems faster and with more accuracy.

These theoretical advancements have sparked a lot of interest in the quantum world throughout the years. This is partly due to the prevalence of optimization challenges. They can be used by quantitative professionals to depict a wide range of essential and extremely relevant real-world problems in industries such as financial planning, supply chain management, civil engineering, and countless others. It's not a big stretch to believe that a quantum optimization breakthrough could revolutionise the globe by slashing millions of miles from the supply chain, combating climate change, or even removing traffic congestion. Experts are increasingly claiming that such claims are exaggerated, particularly when it comes to existing quantum optimization techniques, and that further study is needed to fully realise the field's potential.

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"While individuals outside the field have exaggerated optimization, researchers in the field have never had reason to expect that optimization was as likely to display exponential quantum advantage as, say, certain fields of quantum chemistry," IBM researcher Giacomo Nannicini said. This is because today's quantum algorithms only provide minor speedups over their classical counterparts. When it comes to particularly huge optimization issues and other specific circumstances, those slight increases can be important, but in many other cases, you'd hardly notice the difference.

On the other hand, if researchers can uncover an optimization problem for which a quantum computer can provide an exponential speedup over traditional methods, the field — and the world — might be irrevocably changed. When it comes to identifying real-world applications, these problems that quantum computers may address tenfold quicker than classical computers are the most significant. For black box optimization tasks, where we don't know anything about the dataset from which we're trying to discover an optimal solution, quantum computers don't appear to offer exponential speedups. However, in circumstances where you know a little more about the problem, there may be some exponential speedup. That's why it's crucial to continue optimising to discover how and where quantum computers may help,as well as to progress the science more broadly.

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However, in order to understand why this is the case for quantum optimization, we must first understand what optimization problems are, how classical computers deal with them, and how quantum computers might be able to improve on traditional optimization techniques.

Mathematical Optimization: Trying to Find the Best of All Worlds

Above is an example of a Sudoku puzzle and its answer (below).

By en:User:Cburnett, CC BY-SA 3.0 (https://commons.wikimedia.org/w/index.php?curid=57831971), https://commons.wikimedia.org/w/index.php? curid=57831971

Any issue in which the goal is to find the best of all possible worlds with respect to specific variables and restrictions — ie, the best, or "optimal," solution from a finite (or countably infinite) collection of alternative solutions — is an optimization problem. For example, the popular puzzle game Sudoku can be thought of as an optimization problem in which the player must fill a 9x9 grid with numeric digits in such a way that each row, column, and 3x3 sub-grid contains all digits 1– 9. Each vacant cell in an unsolved Sudoku puzzle symbolises a "decision variable" that the player must compute in order to solve the puzzle, while the game's rules indicate the limitations that the player must meet in order to find an optimal solution.

How does a computer deal with a situation like this? There are a variety of optimization methods available that model issues in various ways.

"A simple gradient descent technique is one example you could look at," said IBM researcher and optimization expert Srinivasan Arunachalam. "The most fundamental approach for optimization is gradient descent," he explained. It's also the workhorse of machine learning, but that's a different storey.

A gradient is simply a description of the direction and steepness of a line or surface in mathematics. Any optimization problem may be represented mathematically as a function, and any mathematical function can be graphically represented as a line or multi-dimensional surface, depending on how many variables the function contains. A one-variable function depicts a line, a two-variable function explains the geometry of a plane, and so on. Consider those raised relief topographic globes you may find in a classroom, the kind with bumps and ridges that allow you to feel the height of the Himalayas and the flatness of the Great Planes with your fingers. These are just three-dimensional graphs of height, a function with two variables: latitude and longitude, with elevation as the output.It can be difficult to visualise graphical representations of functions with three or more variables, but the same concepts apply.

This method of depicting an optimization issue frequently results in something that resembles a topographic map of a mountainous region, complete with hills and valleys. The lowest points in those valleys usually reflect the most ideal solutions in optimization issues, while the summits of the hills usually represent the least optimal options. It's called "gradient descent" for a reason. Working your way down to the map's lowest point is your ultimate goal. However, because there is frequently no way to acquire a bird's eye view of the complete function, this can be difficult. In fact, you can't see anything other than what's there in front of you. It's as though you're descending a mountain in the dead of night, making your way through a dense forest.

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A gradient function is shown graphically. Image courtesy of the Public Domain.

In traditional computing, the only way to go to the lowest point on the map — the best solution — is to look around, figure out which direction is up the local gradient and which is down, and then take a step in the direction that appears to be going downwards. "A standard gradient descent begins at a certain position, computes the gradient of the function at that moment, and then...determines where I should walk," Arunachalam explained. A "gradient update rule" governs this decision , as it dictates how the algorithm progresses with each new computation.

Every step you take in gradient descent necessitates a new round of computation, and you can never be certain that you're getting to the lowest point on the map. You could be on your way to a small valley distant from the true lowest point, sometimes known as the "global minimum." This is one of the reasons why gradient descent is so difficult and computationally expensive on traditional systems. Each step's computations are resource-intensive on their own, but when viewed as a whole, they become overpowering. Complex optimization issues, on the other hand, may take millions of steps to get a global minimum.

That's a lot of math to do. Let's take a look at how quantum computers could speed up the process.