It is usually assumed that the register consists of qubits. It is also generally assumed that registers are not density matrices, but that they are pure, although the definition of "register" can be extended to density matrices.

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

The Hilbert space, ${\displaystyle {\mathcal {H}}}$ , in which the data is stored in a quantum register is given by ${\displaystyle {\mathcal {H}}={\mathcal {H_{n-1}}}\otimes {\mathcal {H_{n-2}}}\otimes \ldots \otimes {\mathcal {H_{0}}}}$  where ${\displaystyle \otimes }$  is the tensor product.^[3]

The number of dimensions of the Hilbert spaces depends on what kind of quantum systems the register is composed of. Qubits are 2-dimensional complex spaces (${\displaystyle \mathbb {C} ^{2}}$ ), while qutrits are 3-dimensional complex spaces (${\displaystyle \mathbb {C} ^{3}}$ ), et.c. For a register composed of N number of d-dimensional (or d-level) quantum systems we have the Hilbert space ${\displaystyle {\mathcal {H}}=(\mathbb {C} ^{d})^{\otimes N}=\underbrace {\mathbb {C} ^{d}\otimes \mathbb {C} ^{d}\otimes \dots \otimes \mathbb {C} ^{d}} _{N{\text{ times}}}\cong \mathbb {C} ^{d^{N}}.}$ 

The registers quantum state can in the bra-ket notation be written ${\displaystyle |\psi \rangle =\sum _{k=0}^{d^{N}-1}a_{k}|k\rangle =a_{0}|0\rangle +a_{1}|1\rangle +\dots +a_{d^{N}-1}|d^{N}-1\rangle .}$  The values ${\displaystyle a_{k}}$  are probability amplitudes. Because of the Born rule and the 2nd axiom of probability theory, ${\displaystyle \sum _{k=0}^{d^{N}-1}|a_{k}|^{2}=1,}$  so the possible state space of the register is the surface of the unit sphere in ${\displaystyle \mathbb {C} ^{d^{N}}.}$ 

Examples:

• The quantum state vector of a 5-qubit register is a unit vector in ${\displaystyle \mathbb {C} ^{2^{5}}=\mathbb {C} ^{32}.}$ 
• A register of four qutrits similarly is a unit vector in ${\displaystyle \mathbb {C} ^{3^{4}}=\mathbb {C} ^{81}.}$ 

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

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

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

• Arora, Sanjeev; Barak, Boaz (2016). Computational Complexity: A Modern Approach. Cambridge University Press. pp. 201–236. ISBN 978-0-521-42426-4.