Qubits are made up of controlled particles and the means of control (e.g. devices that trap particles and switch them from one state to another).[8]

For example: Consider first a classical computer that operates on a three-bitregister. The state of the computer at any time is a probability distribution over the23 = 8 different three-bit strings 000, 001, 010, 011, 100, 101, 110, 111. If it is a deterministic computer, then it is in exactly one of these states with probability 1. However, if it is a probabilistic computer, then there is a possibility of it being in anyone of a number of different states. We can describe this probabilistic state by eight nonnegative numbers A,B,C,D,E,F,G,H(where A = probability computer is in state000, B = probability computer is in state001, etc.). There is a restriction that these probabilities sum to 1.

The state of a three-qubit quantum computer is similarly described by an eight-dimensional vector (a,b,c,d,e,f,g,h), called a ket. However, instead of adding to one, the sum of the squares of the coefficient magnitudes, | a | 2 + | b | 2 + … + | h | 2, must equal one. Moreover, the coefficients are complex numbers. Since the probability amplitudes of the states are represented with complex numbers, the phase between any two states is a meaningful parameter, which is a key difference between quantum computing and probabilistic classical computing.[9]

If you measure the three qubits, you will observe a three-bit string. The probability of measuring a given string is the squared magnitude of that string’s coefficient (i.e., the probability of measuring 000 = | a | 2, the probability of measuring 001 = | b | 2, etc..). Thus, measuring a quantum state described by complex coefficients (a,b,…,h) gives the classical probability distribution ( | a | 2, | b | 2,…, | h | 2) and we say that the quantum state “collapses” to a classical state as a result of making the measurement.

Note that an eight-dimensional vector can be specified in many different ways depending on what basis is chosen for the space. The basis of bit strings (e.g., 000, 001, …, 111) is known as the computational basis. Other possible bases are unit-length, orthogonal vectors and the eigenvectors of the Pauli-x operator. Ket notation is often used to make the choice of basis explicit. For example, the state (a,b,c,d,e,f,g,h) in the computational basis can be written as:

where, e.g.,
The computational basis for a single qubit (two dimensions) is and .

Using the eigenvectors of the Pauli-x operator, a single qubit is and .