## An intuitive approach

It is trivial to show from combinatorics that, classically, to represent the state of one atom, you must have more than one atom (in fact, many more than one atom).

The proof:
Let's assume your computer's memory works by storing bits in the spin state of an atom (the type of atom doesn't really matter). Atomic spins are quantized, and can either be "up" or "down," which is convenient for building a binary system, where we can say 0 is "up" and 1 is "down."

If you assume you require 32 bits to represent all the possible states of a single hydrogen atom, it will take 32 memory atoms just to represent this single hydrogen atom.

In reality, for all the possible properties an atom can have, you will need a *lot* more than 32 bits. The number of bits you *actually* need is dependent on the number of properties your atom can have (spin, momentum, charge, etc...), as well as the resolution you need (the dynamic range).

This implies that, classically, in order to represent a simulation of a room down to the atomic level, you need a room much, much larger (in mass) than the room you intend to simulate to contain all of your computing hardware.

Even if we look at it from a quantum point of view (i.e. a post-singularity society that has created working general quantum computers), you can trivially prove that there is a 1:1 correlation.

If your simulated hydrogen atom has 500 possible quantum states (a gross underestimate to be sure), and you can somehow store this in the quantum state of a real hydrogen atom, then you need *at least* one real atom for every simulated atom you want to compute, simply to store the information about its state.

## But what *do* we need then?

All of these intuitive concepts about what it takes to simulate the world with "exact precision" led to a more exact formulation known as the Berkenstein Bound.

Essentially what the Berkenstein bound says is that the amount of information you can place in a given amount of space is limited. Conversely, it also shows that the amount of information you need to represent any physical system at the quantum level is *directly related to its mass and volume.* It also shows that there is an upper limit on the amount of processing you can do with any given amount of mass and space.

The Berkenstein bound was almost immediately found to have a direct relation to black holes: Namely that if you attempt to exceed the Berkenstein bound (i.e. put more information in a given volume than it can support), your computer will collapse into a black hole!

Thinking back to our intuitive thought experiment before, this makes sense. To simulate your world you need bits. If you need atoms to represent bits, and you place too many atoms together in a given volume, of course they'd exceed the Schwarzchild radius and collapse into a black hole.

So what does the Berkenstein Bound say about your simulated room?

Well, as we've established, the amount of information you need to simulate a given space at its quantum level is directly related to the size of that space and the *amount of mass in it.*

Your question doesn't state anything about the mass in the room, but gives us its dimensions, which approximate a sphere of around 12 m^{3} (as an aside, rather than a cube, a sphere is the best configuration for your room as it minimizes surface area).

So, by the Berkenstein bound, your room requires approximately

**3.08 x 10**^{44} bits / kg

to *exactly* represent at the quantum level, and this is *just* the memory to store the states of all the atoms. It says nothing about *computing* the states of those atoms.