Quantum register

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

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

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 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 - ${\displaystyle 00,01,10,11\quad (0,1,2,3)}$ 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 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.