(taken from wikipedia -- needs major editing and moving of pieces, perhaps complete deletion)
Bra-ket notation
Paul Dirac invented a powerful and intuitive mathematical notation to describe quantum states, known as bra-ket notation. For instance, one can refer to an |excited atom> or to 
for a spin-up particle, hiding the underlying complexity of the mathematical description, which is revealed when the state is projected onto a coordinate basis. For instance, the simple notation |1s> describes the first hydrogen atom bound state, but becomes a complicated function in terms of Laguerre polynomials and spherical harmonics when projected onto the basis of position vectors |r>. The resulting expression Ψ(r)=<r|1s>, which is known as the wavefunction, is a special representation of the quantum state, namely, its projection into position space. Other representations, such as projection into momentum space, are possible. The various representations are simply different expressions of a single physical quantum state.
Bits vs qubits
Consider first a classical computer that operates on a 3 bit register. At a given time, the state of the register is determined by a single string of 3 bits, such as "101". This is usually expressed by saying that the register contains a single string of 3 bits. A quantum computer, on the other hand, can be in a state which is a mixture of all the classically allowed states. The particular state is determined by 8 complex numbers. In quantum mechanics notation we would write:
where a, b, c, d, e, f, g, and h are complex. Let us consider a particular example:
| State | Amplitude | Probability |
|---|---|---|
| * | (α+i β) | (|α|2+|β|2) |
| 000 | a = 0.37 + i 0.04 | 0.14 |
| 001 | b = 0.35 + i 0.43 | 0.31 |
| 010 | c = 0.09 + i 0.31 | 0.10 |
| 011 | d = 0.30 + i 0.30 | 0.18 |
| 100 | e = 0.11 + i 0.18 | 0.04 |
| 101 | f = 0.40 + i 0.01 | 0.16 |
| 110 | g = 0.09 + i 0.12 | 0.02 |
| 111 | h = 0.15 + i 0.16 | 0.05 |
For an n qubit quantum register, this table would have had 2n rows; for n=300, this is roughly 1090, more rows than there are atoms in the known universe. Note that these values are not all independent, since the probability constraint must be met. The representation is also non-unique, since there is no way to physically distinguish between this quantum register and a similar one where all of the amplitudes have been multiplied by the same phase such as -1, i, or in general any number on the complex unit circle. One can show the dimension of the set of states of an n qubit register is 2n+1 − 2. See Bloch sphere.
The first column shows all classically allowed states for three bits. Whereas a classical computer can hold only one such pattern at a time, a quantum computer can be in a superposition state of all 8 patterns. The second column shows the "amplitude" for each of the 8 states. These 8 complex numbers are a snapshot of the register at a given time. In this sense, a 3-qubit quantum computer has far more memory than a 3-bit classical computer because it can simultaneously represent all possible states of the classical computer.
When the qubit is measured, it is projected onto one of the classically allowed states. The absolute value squared of the amplitude of each classical state gives the probability that the qubit will be measured in that state. Looking at the table, the third column gives the probability for measuring each possible register configuration. In this example, there is a 14% chance that the returned string will be "000", a 31% chance it will be "001", and so on. Each complex number (α+βi) is called an (complex valued) amplitude, and each probability (|α|2+|β|2) is the absolute square of the amplitude, because it equals |α+ βi|2. The probabilities must sum to 1.
