The idea of quantum computers is very interesting, but also very complex as it is rooted in the fundamental ideas of quantum mechanics. The "classical" computers that we are all familiar with are based on binary bits. This means that each bit can be in one of two different states, 0 or 1. Another feature is that classical computers can only do calculations on one set of numbers at a time. Quantum computers on the other hand, are based on quantum bits, or "qubits." Whereas a binary bit can only be a 0 or a 1, a qubit can be a 0, a 1, or some combination, or superposition, of the two states.

Qubits are what would make quantum computers so powerful. To illustrate, let's imagine a system comprised of some number, n, different bits. In terms of binary bits, each bit would either be a 0 or a 1 and there would be 2ⁿ different ways that these could appear. For example, if n=2, there are 2² or 4 possibilities - you could have 0,0 or 0,1 or 1,0 or 1,1. You can see how one might have to do (up to) 4 different calculations (one for each possible combination) to cover all the possibilities. Now, if these bits are instead qubits, each bit exists as a superposition of the two possible states, meaning that all four of the possibilities can be considered at once. Hence we can effectively do an operation on all the combinations at once! It is this inherent quantum parallelism in the computing that would make a quantum computer so powerful. Furthermore, the advantages become more clear as you increase n. For n=100, there are 2¹⁰⁰ (~10³⁰) possibilities of 0s and 1s! I think I have read that a quantum computer with as few as 25 to 30 qubits would be comparable for some types of calculations to the most powerful supercomputers of today.

One other thing to think about is how to get the answer you want amidst all those possibilities? Earlier, we said that qubits exist as a superposition of possible states. Here, it might be useful to think of the superposition of states as a (probability) wave. If we think of each qubit as a wave, then we can imagine how those waves can interfere with each other (in the same way that waves on water interact with each other). Thus, the idea is that depending on what conditions are set on the qubits, the right answer will appear from constructive interference, and all the other answers will be eliminated by the destructive interference. Here, I should note that quantum computers are many years away from becoming a "practical" item as much research is currently focused on making and controlling qubits, and other problems such as de-coherence.

Furthermore, quantum computers may not be perfectly suited for every type of calculation. However, one calculation it would be extremely proficient at (leading to much of the funding for research in this field) would be executing algorithms to factor very large numbers. The reason that is so significant is because many encryption codes (inclusing RSA) depend on the difficulty of factoring very large numbers, and a quantum computer could somewhat easily crack these codes.