Yet, there are fundamental obstacles to simulating physical systems as efficiently as we want, which can be formalized using the tools computational complexity. For instance, it is known that general spin glasses are NP-hard. Thus, unless we believe that P=NP, simulating such systems will take an exponential time (roughly). However, it should be noted that pretty much none of the relevant separations is proven: Even PSPACE, the class containing the simulation of any quantum system, is not known to differ from P (the class which is generally understood as "efficiently solvable").

So why am I claiming that a separation in computational complexity is nowhere as fundamental as the speed of light or the uncertainty relation? The latter form a "hard wall" for our attempts to increase speed or precision: There is a hard cutoff we cannot pass. The growth in computational complexity - be it time, space, or the energy needed to carry out the computation - is much more subjective. What should we use as a time cutoff? The time of a PhD thesis? The lifetime of a human? The age of the universe? Which you consider "fundamental" is rather subjective. The same is true for energy scales, or space: What exactly you obtain depends on what you consider a "reasonable" investment.

As a comparison, consider that there were no speed of light, but speeding up a mass would require an energy which would grow exponentially with the speed: This would not yield a "hard" speed limit independent of the situation considered - if would really depend on how much energy you are willing to invest. Of course, you could argue that all the energy available in the universe is such a hard cutoff, but this would make such a limit much more subjective, and depending on global properties of the universe, rather than the speed of light or the uncertainty relations, which are limitations known to hold in any scenario locally, without requiring us to take into account the whole universe altogether.

In principle, it has been shown that are undecidable questions about physical systems one can ask. Thus, one could argue that these are fundamental limitations to simulating physical systems. But then again, there is no experiment which could answers these undecidable questions, and thus, these are rather statements about the "expressive power" of physical theories than about the ability to simulate processes in nature.

And finally, in simulating quantum mechanical systems, one is obviously limited by the limitations of quantum theory, where the observer takes a special role. Simulating the entire universe quantum mechanically is thus something which, I would argue, is outside the scope of the theory.

Yet, there are fundamental obstacles to simulating physical systems as efficiently as we want, which can be formalized using the tools of computational complexity. For instance, it is known that simulating general spin glasses is NP-hard. Thus, unless we believe that P=NP, simulating such systems will take an exponential time (roughly). However, it should be noted that pretty much none of the relevant separations is proven: Even PSPACE, the class containing the simulation of any quantum system, is not known to differ from P (the class which is generally understood as "efficiently solvable").

So why am I claiming that a separation in computational complexity is nowhere as fundamental as the speed of light or the uncertainty principle? The latter form a "hard wall" for our attempts to increase speed or precision: There is a hard cutoff we cannot pass. The growth in computational complexity - be it time, space, or the energy needed to carry out the computation - is much more subjective. What should we use as a time cutoff? The time it takes to complete the PhD thesis? The lifetime of a human? The age of the universe? Which you consider "fundamental" is rather subjective. The same is true for energy scales, or space: What exactly you obtain depends on what you consider a "reasonable" investment.

As a comparison, consider that there were no speed of light, but speeding up a mass would require an energy which would grow exponentially with the speed: This would not yield a "hard" speed limit independent of the situation considered - if would really depend on how much energy you are willing to invest, and how large the mass you want to accelerate is. Of course, you could argue that all the energy available in the universe is such a hard cutoff, but this would make such a limit much more subjective, and depending on global properties of the universe. This is in stark contrast to the speed of light or the uncertainty relation, which are limitations known to hold in any scenario locally, without requiring us to take into account the whole universe altogether to make sense of them.

Then, it has been shown that there are undecidable questions about physical systems one can ask. Thus, one could argue that these are fundamental limitations to simulating physical systems. But then again, there is no experiment which could answers these undecidable questions, and thus, these are rather statements about the "expressive power" of physical theories than about the ability to simulate processes in nature.

And finally, in simulating quantum mechanical systems, one is obviously limited by the limitations of quantum theory, where the observer takes a special role. Simulating the entire universe quantum mechanically is thus something which, I would argue, is outside the scope of the theory. But again, this is a limitation of the theory, not a fundamental computational limitation.

Of course, simulating larger and larger systems is growing increasingly complex. This can be better or worse, depending e.g. whether the system is chaotic, or whether we want to describe classical or quantum mechanical systems.

However, there are two main points to be observed about this growth in complexity: Firstly, to some extent it relates to our current understanding of the system. For instance, quantum systems are hard to simulate, yet it turns out there are classes of systems we can solve exactly analytically. Similarly, as it turns out, the naive assumption that in order to simulate a quantum system, one has to store an exponentially large state vector is incorrect, as one can e.g. use Monte Carlo sampling to approximate the path integral path which allows for the simulation of systems whose state vector could never be stored in a computer. Thus, there's plenty of room for improvement.

Of course, simulating larger and larger systems is growing increasingly complex. This growth in complexity can be better or worse, depending e.g. whether the system is chaotic, or whether we want to describe classical or quantum mechanical systems.

However, there are two main points to be observed about this growth in complexity: Firstly, to some extent it relates to our current understanding of the system. For instance, quantum systems are hard to simulate, yet it turns out there are classes of systems we can solve exactly analytically. Similarly, as it turns out, the naive assumption that in order to simulate a quantum system, one has to store an exponentially large state vector is incorrect, as one can e.g. use Monte Carlo sampling to approximate the path integral term by term, which allows for the simulation of systems whose state vector could never be stored in a computer. Thus, there's plenty of room for improvement, and in particular there might well be plenty of room for improvement we are not aware of as of today.