Quantum register

In quantum computing, a quantum register is a system comprising multiple qubits^[1]. It is the quantum analog of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register.

Definition

An $n$ size quantum register is a quantum system comprising $n$ qubits.

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

Quantum vs. Classical Register

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

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

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

References

↑ Ekert, Artur; Hayden, Patrick; Inamori, Hitoshi (2008). "Basic concepts in quantum computation". arXiv:quant-ph/0011013.
↑ Major, Günther W., V.N. Gheorghe, F.G. (2009). Charged particle traps II : applications. Berlin: Springer. p. 220. ISBN 978-3540922605.

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[1] Ekert, Artur; Hayden, Patrick; Inamori, Hitoshi (2008). "Basic concepts in quantum computation". arXiv:quant-ph/0011013.

[2] Major, Günther W., V.N. Gheorghe, F.G. (2009). Charged particle traps II : applications. Berlin: Springer. p. 220. ISBN 978-3540922605.