Basic concepts in quantum computation

The first one can represent, for example, the number 3 (in binary) and the second one the number 7. In general three physical bits can be prepared in 2 = 8 different configurations that can represent, for example, the integers from 0 to 7. However, a register composed of three classical bits can store only one number at a given moment of time. Enter qubits and quantum registers: A qubit is a quantum system in which the Boolean states 0 and 1 are represented by a prescribed pair of normalised and mutually orthogonal quantum states labeled as {|0〉, |1〉} [1]. The two states form a ‘computational basis’ and any other (pure) state of the qubit can be written as a superposition α|0〉+β|1〉 for some α and β such that |α|2 + |β|2 = 1. A qubit is typically a microscopic system, such as an atom, a nuclear spin, or a polarised photon. A collection of n qubits is called a quantum register of size n. We shall assume that information is stored in the registers in binary form. For example, the number 6 is represented by a register in state |1〉 ⊗ |1〉 ⊗ |0〉. In more compact notation: |a〉 stands for the tensor product |an−1〉 ⊗ |an−2〉 . . . |a1〉 ⊗ |a0〉, where ai ∈ {0, 1}, and it represents a quantum register prepared with the value a = 2a0 + 2 a1 + . . . 2 n−1an−1. There are 2 states of this kind, representing all binary strings of length n or numbers from 0 to 2n−1, and they form a convenient computational basis. In the following a ∈ {0, 1}n (a is a binary string of length n) implies that | a〉 belongs to the computational basis. Thus a quantum register of size three can store individual numbers such as 3 or 7,