Quantum register

In quantum computing, a quantum register is a system comprising multiple qubits. 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 is stored in a quantum register is given by ${\mathcal {H}}={\mathcal {H_{n-1}}}\otimes {\mathcal {H_{n-2}}}\otimes \ldots \otimes {\mathcal {H_{0}}}$ where $\otimes$ is the tensor product.

The number of dimensions of the Hilbert spaces depend on what kind of quantum systems the register is composed of. Qubits are 2-dimensional complex spaces, while qutrits are 3-dimensional, et.c. For a register composed of n number of d-level quantum systems we have the Hilbert space ${\mathcal {H}}=(\mathbb {C} ^{d})^{\otimes n}=\underbrace {\mathbb {C} ^{d}\otimes \mathbb {C} ^{d}\otimes \dots \otimes \mathbb {C} ^{d}} _{n{\text{ times}}}.$

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 at 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.

Further reading

• Arora, Sanjeev; Barak, Boaz (2016). Computational Complexity: A Modern Approach. Cambridge University Press. pp. 201–236. ISBN 978-0-521-42426-4.