state 0 and 1 at the same time. If we keep adding qubits to the register we increase its storage capacity exponentially i.e. three qubits can store 8 different numbers at once, four qubits can store 16 different numbers at once, and so on; in general L qubits can store 2L numbers at once. Once the register is prepared in a superposition of different numbers we can perform operations on all of them. For example, if qubits are atoms then suitably tuned laser pulses affect atomic electronic states and evolve initial superpositions of encoded numbers into different superpositions. During such evolution each number in the superposition is affected and as the result we generate a massive parallel computation albeit in one piece of quantum hardware. This means that a quantum computer can in only one computational step perform the same mathematical operation on 2L different input numbers encoded in coherent superpositions of L qubits. In order to acomplish the same task any classical computer has to repeat the same computation 2L times or one has to use 2L different processors working in parallel. In other words a quantum computer offers an enormous gain in the use of computational resources such as time and memory. But this, after all, sounds as yet another purely technological progress. It looks like classical computers can do the same computations as quantum computers but simply need more time or more memory. The catch is that classical computers need exponentially more time or memory to match the power of quantum computers and this is really asking for too much because an exponential increase is really fast and we run out of available time or memory very quickly. Let us have a closer look at this issue. In order to solve a particular problem computers follow a precise set of instructions that can be mechanically applied to yield the solution to any given instance of the problem. A specification of this set of instructions is called an algo...
