Quantum Register

quantum-computing

Definition

Quantum Register

A quantum register is a physical system consisting of $n$ qubits that stores quantum information. Its state space is the $n$-fold tensor product of individual qubit spaces:

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

The register is initialised in a predefined start state, typically $|0{⟩}^{\otimes n}$, and its state evolves through the application of quantum gates during computation.

Property: An $n$-qubit register can exist in superposition of all $2^n$ computational basis states simultaneously, enabling quantum parallelism.

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