A Beautiful Computation

When I first started my studies in science, I didn't fully understand the close and intricate connection between mathematics and physics. My doubts came from both the strict rules and their interpretations. Even when I agreed with certain principles, I hesitated to translate my understanding into mathematical formulas. Over the years, I began to see the parallels between different branches of mathematics. For instance, linear equations could be visualized as straight lines, and geometrical intersections as systems of equations. Mastering this connection is a useful tool, like solving complex equations using graphical methods. Later, I realized the convenience of expressing physical concepts through formulas.

I would like to illustrate the tight relationship between mathematics and physics through the work and ideas of a great physicist and mathematician, Richard Feynman. Simplicity is the essence of mathematical beauty, and Feynman had much to say about it. In the 1980s, he introduced an idea that laid the foundation for quantum computation as we know it today.

Let's consider a problem to understand this better: a set of atoms can be described by equations. We might want to solve these equations to predict some features of the material, but we may not be able to solve them, even with the help of a powerful computer. Sometimes, we may even want to solve the equations independently of the atoms themselves! Feynman's brilliant idea was to tackle the tricky computation for complex systems using a fundamental property of matter: the principle of minimum energy.

When you launch an object into the air, it falls back down because it is attracted by gravity. This phenomenon has a deeper meaning: the object tends to lose energy, moving towards the ground to minimize its gravitational energy. This is a basic principle of mechanics. Quantum systems in our cutting-edge computers also strive to reach the lowest energy state. Both mechanical and quantum systems can be described by a function, which specialists call the Hamiltonian, accounting for the energy of the system. Feynman's stroke of genius was to let nature itself search for the minimum of this function. By the parallelism previously introduced, we can map our Hamiltonian to equivalent problems within its complexity class. Hence, the Hamiltonian solution is the same for those problems that we cannot solve with a supercomputer. We solve these hard problems by following the evolution of a quantum state to its ground energy state.

An example of such a beautiful computation can be easily reproduced with soap bubbles, two plexiglass planes, and nails. Let's scatter some points on a plane and connect them with segments. What is the arrangement of segments that sums up to the shortest path? When building a railroad infrastructure, this is a practical issue. Well, this problem cannot be solved easily by calculus or computers, but nature can solve it in a matter of seconds! Pour soap bubbles between the plexiglass planes, and they will arrange themselves among the nails by the shortest air-water interface, creating our perfect street map.

This example shows how nature can provide elegant solutions to complex problems, illustrating the profound connection between mathematics and the physical world.