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In quantum computing, a quantum register
is a system comprising multiple qubits^[1]. It is the quantum analog of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register.

Definition

An n{displaystyle n} size quantum register is a quantum system comprising n{displaystyle n} qubits.

The Hilbert space, H{displaystyle {mathcal {H}}}, in which the data stored in a quantum register is given by H=Hn−1⊗Hn−2⊗…⊗H0{displaystyle {mathcal {H}}={mathcal {H_{n-1}}}otimes {mathcal {H_{n-2}}}otimes ldots otimes {mathcal {H_{0}}}}.^[2]

Quantum vs. Classical Register

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

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

If we consider 2 pure qubits in superpositions |a0⟩=12(|0⟩+|1⟩){displaystyle |a_{0}rangle ={frac {1}{sqrt {2}}}(|0rangle +|1rangle )} and |a1⟩=12(|0⟩−|1⟩){displaystyle |a_{1}rangle ={frac {1}{sqrt {2}}}(|0rangle -|1rangle )}, using the quantum register definition |a⟩=|a0⟩⊗|a1⟩=12(|00⟩−|01⟩+|10⟩−|11⟩){displaystyle |arangle =|a_{0}rangle otimes |a_{1}rangle ={frac {1}{2}}(|00rangle -|01rangle +|10rangle -|11rangle )} it follows that it is capable of storing all the possible values spanned by two qubits simultaneously.

References