How are quantum computers more powerful than classical computers?

Question

I feel the answer to this question is just out of reach - I "understand" the implication that a quantum computer uses all combinations of bits simultaneously compared to a classic computer, and that clearly gives a huge boost to processing times, however I'm struggling to quite grasp how.

I watched a TED talk on quantum computing in which the gentleman said that a 300 qubit QC would be more powerful than a regular PC with one bit of memory for every atom in the universe. They also mentioned that Google have created a QC with 53 qubits.

So to try and get my head around it, I decided to compare this to my PC.

Forgetting memory needed for anything other than storing combinations, if I have 16GB of RAM available, that's 2^{17'179'869'184} possible combinations, or ~57 million lots of 53 bits (with each one in one specific combination).

So for each cycle of combinations, my PC is able to hold ~57 million combinations of 53 bits, and if my PC is running at 4.1GHz, that's 7.0E+19, or in the region of 2⁶⁶ combinations per second.

Now I have a good PC, but it seems to be reasonably on a par with Google's quantum computer, which just seems wrong. I know that I've not used an exact science to calculate things but I can't follow where I've screwed up the maths?

The claim for the 300 qubit QC I also can't follow with the above logic.

As excel won't work with stupidly large numbers I tried to do this on paper, so bear with me:

There are 10E+80 atoms in the universe, so this number of bits would give 2^(10E+80), or 2⁸⁰⁰ combinations. An unholy number that no-one could possibly comprehend.

For the QC there are 300 qubits or ~2⁸ therefore one could store 2⁷⁹² combinations at any one time, which is already way above the 2³⁰⁰ combinations available to a 300 qubit QC..?

Answer 1

We can not talk about how quantum computers are much powerful than classical computers without mentioning other quantum phenomena like entanglement and interference. For a good comparison you can read this article.

Regarding you calculations, I think you did not get it right. Two classical bit can be either 00, 01, 10, or 11. Two qubits can have all the four possibilities simultaneously. Which means 2 qubits can carry 4 2-bits of information at the same time. That is 4 times what 2 bits can carry.

In general, $n$ qubits can carry $2^n$ times what $n$ bits can carry. $300$ qubits can carry $2^{300}$ $300$-bits binary strings. $2^{300}$ ~ 2e+90. That is more than the atoms in the observable universe.

There are 10E+80 atoms in the universe, so this number of bits would give 2(10E+80), or 2800 combinations. An unholy number that no-one could possibly comprehend.

Number of combinations does not matter, because at any moment a classical bit can be either one or zero. That means at any time 10E+80 atoms can carry 10E+80 bits.