Quantum register

Definition

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 ${\displaystyle \mathcal{H}={\mathcal{H}}_{\mathcal{n}\mathcal{-}\mathcal{1}}\otimes {\mathcal{H}}_{\mathcal{n}\mathcal{-}\mathcal{2}}\otimes \dots \otimes {\mathcal{H}}_{\mathcal{0}}}$ where ${\displaystyle \otimes }$ is the tensor product.

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 (${\displaystyle {\mathbb{C}}^3}$), etc. For a register composed of N number of d-dimensional (or d-level) quantum systems we have the Hilbert space ${\displaystyle \mathcal{H}=({\mathbb{C}}^d{)}^{\otimes N}=\underset{N\text{ times}}{\underbrace{{\mathbb{C}}^d\otimes {\mathbb{C}}^d\otimes \cdots \otimes {\mathbb{C}}^d}}\cong {\mathbb{C}}^{d^N}.}$

The registers quantum state can in the bra-ket notation be written ${\displaystyle |\psi ⟩=\sum _{k=0}^{d^N-1}{a}_k|k⟩=a_0|0⟩+a_1|1⟩+\cdots +a_{d^N-1}|d^N-1⟩.}$ 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 ${\displaystyle {\mathbb{C}}^{3^4}={\mathbb{C}}^{81}.}$