Quantum register

Definition

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

The Hilbert space, H , in which the data stored in a quantum register is H = H n − 1 ⊗ H n − 2 ⊗ … ⊗ 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 ( 0 , 1 , 2 , 3 ) accordingly.

If we consider 2 pure qubits in superpositions | a 0 ⟩ = 1 2 ( | 0 ⟩ + | 1 ⟩ ) and | a 1 ⟩ = 1 2 ( | 0 ⟩ − | 1 ⟩ ) , using the quantum register definition | a ⟩ = | a 0 ⟩ ⊗ | a 1 ⟩ = 1 2 ( | 00 ⟩ − | 01 ⟩ + | 10 ⟩ − | 11 ⟩ ) it follows that it is capable of storing all the possible values spanned by two qubits simultaneously.