Quantum Register

quantum-computing

Definition

Quantum Register

An $n$-qubit quantum register is an ordered collection of $n$ qubits whose pure quantum state is a unit vector in the tensor product state space

$${\mathcal{H}}_n=({\mathbb{C}}^2{)}^{\otimes n}\cong {\mathbb{C}}^{2^n}.$$

Its computational basis is indexed by bit strings $b\in \{0,1{\}}^n$:

$$\{|b⟩\mid b\in \{0,1{\}}^n\}.$$

A general pure state of the register has the form

$$|\psi ⟩=\sum _{b\in \{0,1{\}}^n}{\alpha }_b|b⟩,\sum _{b\in \{0,1{\}}^n}|{\alpha }_b{|}^2=1.$$

Quantum gates act on the register as unitary operators $U:{\mathcal{H}}_n\to {\mathcal{H}}_n$.

Example

Example

A 2-qubit register has state space ${\mathbb{C}}^4$ with computational basis $\{|00⟩,|01⟩,|10⟩,|11⟩\}$. A general state is

$$|\psi ⟩={\alpha }_{00}|00⟩+{\alpha }_{01}|01⟩+{\alpha }_{10}|10⟩+{\alpha }_{11}|11⟩$$

with ${\sum }_{i,j}|{\alpha }_{ij}{|}^2=1$.