Have you always thought mathematics is dull and complicated? You are certainly not alone. But there is a lot of beauty hidden in it and in the way it describes our world.

Theoretical physics is all about using maths to describe nature. As the universe we live in is vast and filled with myriads of phenomena — starting with the universe itself expanding due to dark energy, galaxies held together thanks to dark matter, new stars being born and dying, planets and asteroids orbiting these start and colliding with one another; through processes happening on and inside those planets including the miracle of life; down to the perplexing world of molecules, atoms, and subatomic particles — so the mathematical language in which these processes are described uses a lot of tools, often very complex. And yet, there is a surprising level of similarity between different systems.

For a theoretical physicist, there is no difference between an oscillating pendulum, a vibrating string, and a propagating beam of light. Heat transfer and particle diffusion are equivalent because they obey the same mathematical law. According to quantum field theorists, every type of particle (be it a proton, an electron, or a photon) can be seen as a harmonic oscillator and there is almost no qualitative difference between them.

Some physicists are trying to take this idea one step further and find a single physical theory encompassing all physics as we know it. Thus, the Grand Unified Theory was developed which unifies three of the four fundamental interactions in nature — electromagnetic, weak, and strong. Including the fourth one — gravitational — is a feat that has not yet been achieved. Some even doubt that such a Theory of Everything will ever be formulated.

Many theoretical physicists (like me, for example) do not pursue such noble quests but focus on smaller, albeit not less meaningful tasks: How does X work? Can it be used for something worthwhile? What is the best way to do it? These are not important questions on the global scale (compared to questions such as ‘How did the universe come to be?’) but the more important for technological progress. Such development is ultimately the domain of experimental physicists and engineers but finding ways of using new bits of physics in ways humanity can benefit from is a part of theoretical physicist’s work.

Such a process can be illustrated on a problem that is occupying many a scientific mind: building the quantum computer. It will, of course, be experimental physicists who will build the first functioning prototype (assuming we ever develop one) but theoretical physicists examine how such a device should be built. Should we use atoms as the information carriers? Photons? Something more exotic? Those are some of the questions quantum information theorists are trying to answer.

There is a lot of mathematical beauty in solving such tasks, too. After all, theoretical physicist sees a quantum computer as a large register of quantum bits on which an arbitrary operation can be performed, which can be stored for a long time in a quantum memory, and which can be sent to another quantum computer via a quantum internet channel. The need for investigating various platforms comes from the experimental realisation — each potential platform has its own unique advantages and disadvantages that have to be carefully weighed when finding the optimal architecture for a successful quantum computer.

All that said, there are many more surprises hidden in quantum information theory. One often finds other, unexpected connections between the weirdest parts of the theory. And finding them is always one of the biggest delights working with theoretical physics can bring.

All this mathematical beauty can, actually, also be useful. If two different systems behave in a similar way, we can use one to simulate dynamics of the other. This is more and more often used in quantum physics where complicated systems (especially those that cannot be observed directly in a laboratory) can be simulated using much simpler systems. This way, we can relatively easily learn a lot about the elaborate system which cannot be simulated on classical computers. (Due to their nature, it is possible to simulate only small quantum systems on classical computers.)

The field of quantum simulations (i.e., using simple quantum systems to simulate evolution of more complicated systems) is still in its infancy. But it will probably not take long before we can simulate systems that too difficult to solve for regular computers. We can then expect better understanding of many physical and chemical processes such as high-temperature superconductivity, quantum phase transitions, dynamics of chemical reactions, or photosynthesis. And all that thanks to the incredibly rich and intriguing structure of the mathematical language we use to describe our world.