Computing a Universe Simulation- Script (Patreon)

Physics seems to be telling us that it’s possible to simulate the entire universe on a computer smaller than the universe. If we go along with this crazy seeming notion, how powerful would that computer need to be, and how long would it take. Believe it or not, we can figure this out.

I’m not saying the universe is a simulation. I mean it might be – I’m just not saying it. And perhaps it doesn’t make any difference. Even if this is the prime, the original physical universe, rather than somewhere deep in a simulation nest, we can STILL think of our universe’s underlying mechanics as computation. Imagine a universe in which the most elementary components are stripped of all properties besides some binary notion of existence or non-existence. Like, if the tiniest chunks of spacetime, or chunks of quantum fields, or elements in the abstract space of quantum-mechanical states can either be full or empty. These elements interact with their neighbors by a simple set of rules, leading to oscillations, elementary particles, atoms, and ultimately to all of the emergent laws of physics, physical structure, and ultimately the universe.

I just described the cellular automaton hypothesis. In this picture the universe is a multidimensional version of Conway’s game of life. Such a universe could be reasonably thought of as a computation – cells stripped of all properties until they are indistinguishable from pure information, and together they form a sort of computer whose sole task is to compute its own evolution. This may not be how our reality works, but it’s an idea that many physicists take seriously. And even if we aren’t emergent patterns of a cellular automaton, we can think of any physical reality as a computation so long as its underlying mechanics are a rules-based evolution over time. That includes most formulations of quantum mechanics and proposals for theories of everything.

We’ll come back to the question: “Is the universe a computer?” and look at cellular automata and pan-computationalism, and in general the idea of digital physics and an informational universe. But today, let’s answer a simple question: if the universe is a computer, how good a computer is it? And an even more fun question: could you build a computer inside this universe to simulate this universe?  In answering this we’ll also answer the recent challenge question, and I encourage you to watch that episode before you watch this one.

The power of a computer can crudely be broken down into two things: how much information can it store, and how quickly can it perform computations on that information? The laws of physics prescribe fundamental limits on both. The first one – the memory capacity of the universe – is a topic we’ve looked at. It’s defined by the Bekenstein bound, which tells us that the maximum information that can be stored in a volume of space is proportional to the surface area of that volume. Specifically, it’s the number of tiny Planck areas you can fit over that surface area divided by 4. It was in studying black holes that Jacob Bekenstein realized that they must contain the maximum possible amount of hidden information – the maximum possible entropy. If you fill a region of the universe with information equal to its Bekenstein bound, it’ll immediately become a black hole.

We saw in our episode on the information content of the universe that the maximum information content – the Bekenstein bound - of the observable universe is around 10^120 bits, based on its surface area. At the same time, the actual information content in matter and radiation is probably more like 10^90 bits, roughly corresponding to the number of particles of matter and radiation. 

Bizarrely, the Bekenstein bound suggests we could hold all of the information in the observable universe within a storage device smaller than the observable universe. Which brings us to the first part of the challenge question: assuming you can build a computer that stores information at the Bekenstein bound – essentially your memory device is the event horizon of a black hole – how large would that black hole need to be to store all information about all particles in the universe? We’ll figure out the cases for just matter and for matter and radiation. 

So there are something like 10^80 hydrogen atoms in the universe. If we instead count all elementary particles with mass we might get to 10^81 particles of matter. But let’s just go with the nice round order of magnitude - 10^80 bits assuming 1 bit per particle. 

Again, the Bekenstein bound is the event horizon surface area in Planck areas, with an extra factor of a quarter that Stephen Hawking figured out. So our Bekenstein black hole hard drive needs roughly 4x10^80 Planck areas, corresponding to 40 billion square meters. That corresponds to a spherical black hole with a radius of around 100km. So, informationally speaking, you could store the entire observable universe of non-radiation particles on the surface of a black hole the size of Switzerland.

The radius I gave is the Schwarzschild radius - the radius of the event horizon of a non-rotating black hole. It’s directly proportional to its mass. The mass of a 100km-radius black hole would be 30 times that of our Sun. It’s a hard drive the size of a picturesque European country, with the mass of the heaviest stars in the universe, and with the storage capacity to register every atom in the universe. Not exactly portable, but you’ll never need to defrag again.

If you want to include photons, neutrinos, dark matter, etc. and not just atoms, you need to scale up the surface area by a factor of 10 billion and radius by a factor of 100,000. You’d need a black hole a few million times the mass of the sun and 10 million km in radius. That’s pretty close to the size of the supermassive black hole in the center of the Milky Way. Those are some big masses from our puny human perspective, but remember that we’re storing all of the information in the universe on just one of these black holes. It’s quite the zip file.  

Storing all of the information in the universe is one thing, but a real computer must compute. Every instant in time must progress to the next instant – every quantum state has to be processed into the following state. There’s also a fundamental limit to the speed with which quantum states can change. The Margolus–Levitin theorem tells us the maximum rate at which the quantum states of a system can shift into a completely independent quantum states – or orthogonal states in quantum jargon. That rate is proportional to the average energy of the system. That makes some intuitive sense: the more energy in the system, the quicker it’s quantum states can evolve.

This maximum rate of state changes is also the maximum speed of logical operations for any computer. For example, a simple quantum system would be a group of electrons with spins pointing either up or down, corresponding to a single bit of information each. Flipping the spin of an electron is a change to an orthogonal state, but it can also be thought of as a single operation. In this case the NOT operation. This sort of quantum spin array is exactly the system used in most quantum computers, and so the Margolus-Levitin theorem also gives the speed limit of operations in quantum computing. In fact, for any computing – but only quantum computers can expect to get close to this limit.

Using this theorem we can also figure out the computational capacity of the whole universe. This follows the paper by Seth Lloyd that I pointed you to in the challenge question. For the energy of the system we use mass of the observable universe -  around 10^52 kg, then apply good old E=mc^2 to get energy. We get that if every single particle in the universe were used to make a computation, it should process 5x10^102 logical operations per second. Not bad. That’s why the universe can dial all its graphics settings up so high. The universe has been around for 13.8 billion years, or 4x10^17 seconds. So it could have performed around 10^120 operations in that time. And that’s actually independent of the number of particles or “degrees of freedom” the universe is using to do the computation. The number is based on its energy content, and it has to spread that computational power over its particles.

On the other hand, our Switzerland-sized black hole computer is a little slower. Instead of using the mass of the universe to figure the computation speed we only have 30 solar masses. It can do 3x10^82 logical ops per second, a factor of 10^20 slower than the computational speed of the whole universe. Assuming the universe is computing is own evolution at the maximum speed, our black hole computer would take a factor of 10^20 longer, or 10^30 years, to simulate the universe to the present day. Protons will have started to decay by the time we simulate last Monday. Our supermassive black hole-scale computer does a bit better, taking only 10^25 years. I dunno, maybe if we turn off anti-aliasing?

Again, this assumes that we needed all of those 10^120 ops. We can instead estimate the computational history of the universe by assuming all entropy generated over the history of the universe comes from its internal computation. The Landauer limit gives the entropy cost for computation, and approximating very crudely it’s 1 bit of entropy per operation. Matter and radiation combined have an entropy of 10^90 bits, so maybe they processed 10^90 ops since the Big Bang. Our smaller black hole computer could do that in a year, and our big one in 5 minutes. Unfortunately you can only read out the simulation results in Hawking radiation as those black holes evaporate, which’ll take 10^70 years minimum. Hell of an input-output bottleneck. But hey, you could simulate 10^70 universes in that time.

OK, so, wildly different results depending on your assumptions. Either way, getting the results of the simulation will take forever, nearly literally. Some of you gave answer close to these, by various ingenious methods. We chose 6 correct-enough responses at random – listed below. If you see your name below you’re a lucky winner! Of a space time t-shirt! Shoot us an email at PBSspacetime@gmail.com with your name, address, US t-shirt size – small, medium, extra-extra-large, etc. and let us know which T you’d like.

For me the big takeaway from this exercise isn’t so much the numbers, which are enormous and hard to get our heads around. It’s first that the universe can be simulated within the universe. I suppose that has implications for the simulation hypothesis, but I remain dubious. It’s not clear that programming and then reading out from an event horizon computer are even possible. What blows my mind even more is that we can use our current incomplete understanding of physics – in particular very standard ideas on general relativity and quantum mechanics – to figure out the computational properties of our universe. This WILL be important as quantum computers develop. These insights may also lead to real paradigm shifts, perhaps ultimately revealing the fundamentally informational nature of space time.