User:Omrika/sandbox/QIP/Quantum register

Editing Quantum Register
Quantum Register is a system comprised of multiple qubits and is the quantum analog of the classical processor register.

Definition
An $$n$$ size quantum register is a quantum system comprised of $$n$$ qubits.

The Hilbert space, $$\mathcal{H}$$, in which the data stored in a quantum register is$$\mathcal{H} = \mathcal{H_{n-1}}\otimes\mathcal{H_{n-2}}\otimes\ldots\otimes\mathcal{H_0}$$.

Quantum vs. Classical Register
First, there's a conecptual 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 $$2^n-1$$ possibilities spanned by $$n$$ classical pure bits, a quantum register is able to store all $$2^n-1$$ 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 comprised of 2 bits - $$ 00, 01, 10, 11\quad(0, 1, 2, 3)$$ accordingly.

If we consider 2 pure qubits in superposition $$|a_0\rangle=\frac{1}{\sqrt2}(|0\rangle + |1\rangle), |a_1\rangle=\frac{1}{\sqrt2}(|0\rangle - |1\rangle)$$, using the quantum register definition $$|a\rangle=|a_{0}\rangle\otimes|a_{1}\rangle = \frac{1}{2}(|00\rangle - |01\rangle + |10\rangle - |11\rangle)$$ follows that it is capable of storing all the possible values spanned by two qubits simultaneously.