Why can't even the largest supercomputer simulate a small molecule exactly?
Some problems do not just get bigger as they grow: they explode. A good example comes from nature itself.
Take particles that each have two possible states, like the electrons of a molecule that can point up or down. Quantum physics says that a group of them is not simply in one arrangement: to describe it exactly you need one number for every possible arrangement at once. With n particles that is 2^n numbers.
Every particle you add doubles the numbers needed. How much memory do 50 particles take? 2^50 numbers × 16 bytes ≈ 18 million billion bytes, about 16 petabytes (counted in powers of 1,024): the memory of the largest supercomputers in the world. Ten more particles and no computer on Earth could hold them.
In 1981 the physicist Richard Feynman pointed out that nature has no trouble with this: a molecule simply is in its quantum state. So, he proposed, why not build a computer out of quantum parts, which holds such states naturally?
That is why simulating chemistry and materials is the most natural job for a quantum computer: designing catalysts, batteries, fertilisers or drugs needs exactly this kind of calculation, and today it is done with approximations.
Other problems are hard for different reasons. Splitting a very large number into its prime factors is easy to check but, as far as we know, extremely slow for ordinary computers. Much of the encryption that protects the internet relies on that difficulty.
n two-state particles have 2^n arrangements; an exact description needs one complex number for each, and a complex number takes 16 bytes (two 64-bit numbers).
n = 30: 2^30 × 16 bytes = 16 gibibytes, the memory of a laptop. Each extra 10 particles multiplies that by 2^10 = 1,024: n = 40 needs 16 tebibytes, n = 50 needs 16 pebibytes.
Describing quantum systems exactly takes a number of values that doubles with every particle; a computer made of quantum parts can handle such states directly.