Quantum parallelism

The quantum parallelism is the reason for the higher computing power of a quantum computer compared to a classic computer , since when a gate is used once on a state of qubits, manipulations are carried out in total , as opposed to just one manipulation in a classic computer. ${\displaystyle n}$ ${\displaystyle 2^{n}}$

A classic computer processes information by manipulating bits with gates. For example, the application of a non-gate to the bit sequence [010111] results in the bit sequence [101000].

Quantum computers make use of the superposition of quantum states. A qubit is in a state of superposition (before any measurement)

${\displaystyle |\psi \rangle =a\,|0\rangle +b\,|1\rangle }$

With

${\displaystyle |a|^{2}+|b|^{2}=1}$

Applying a gate operation (e.g. NOT) to this state then results

${\displaystyle |\psi '\rangle =a\,|1\rangle +b\,|0\rangle }$

With a single application of a gate operation, two manipulations have already been carried out. If you take the whole thing further and consider an ( entangled ) state of two qubits

${\displaystyle |\psi \rangle =a\,|00\rangle +b\,|10\rangle +c\,|01\rangle +d\,|11\rangle }$

so NOT gives the result

${\displaystyle |\psi '\rangle =a\,|11\rangle +b\,|01\rangle +c\,|10\rangle +d\,|00\rangle }$

So you can see that when using two qubits, a total of four manipulations were carried out by a single gate operation. In general, when using interlaced qubits, manipulations are carried out by means of a gate operation. ${\displaystyle n}$ ${\displaystyle 2^{n}}$