In quantum computing, a quantum register is a system comprising multiple qubits. It is the quantum analog of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register.
An size quantum register is a quantum system comprising qubits.
The Hilbert space, , in which the data stored in a quantum register is given by .
Quantum vs. Classical Register
First, there's a conceptual difference between the quantum and classical register. An size classical register refers to an array of flip flops. An size quantum register is merely a collection of qubits.
Moreover, while an size classical register is able to store a single value of the possibilities spanned by classical pure bits, a quantum register is able to store all possibilities spanned by quantum pure qubits in the same time.
For example, consider a 2-bit-wide register. A classical register is able to store only one of the possible values represented by 2 bits - accordingly.
If we consider 2 pure qubits in superpositions and , using the quantum register definition it follows that it is capable of storing all the possible values spanned by two qubits simultaneously.