Quantum register

A quantum register is a system comprising multiple qubits and is the quantum analog of the classical processor register.

Definition

An $n$ size quantum register is a quantum system comprising $n$ qubits.

The Hilbert space, ${\mathcal {H}}$, in which the data stored in a quantum register is${\mathcal {H}}={\mathcal {H_{n-1}}}\otimes {\mathcal {H_{n-2}}}\otimes \ldots \otimes {\mathcal {H_{0}}}$.

Quantum vs. Classical Register

First, there's a conceptual difference between the quantum and classical register. An $n$ size classical register refers to an array of $n$ flip flops. An $n$ size quantum register is merely a collection of $n$ qubits.

Moreover, while an $n$ size classical register is able to store a single value of the $2^{n}$ possibilities spanned by $n$ classical pure bits, a quantum register is able to store all $2^{n}$ 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 - $00,01,10,11\quad (0,1,2,3)$ accordingly.

If we consider 2 pure qubits in superpositions $|a_{0}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle +|1\rangle )$ and $|a_{1}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle -|1\rangle )$, using the quantum register definition $|a\rangle =|a_{0}\rangle \otimes |a_{1}\rangle ={\frac {1}{2}}(|00\rangle -|01\rangle +|10\rangle -|11\rangle )$ it follows that it is capable of storing all the possible values spanned by two qubits simultaneously.