# Example of a quantum algorithm better than its classical counterpart which involves only $1$ qubit?

I was reading over the proof of the Deutsch-Jozsa algorithm, which in its simplest case, involves at least 2 qubits.

Is there an example of a quantum algorithm that is better than it's classical counterpart which only involves a single qubit?

If not, could you provide an explanation of why such an algorithm cannot exist?

Thank you very much. I have only recently started my journey into quantum computing.

• Hi! There have been similar questions asked. See, for example, here. – Mark S Jun 13 at 1:59

There aren't many examples! The main reason for advantages in quantum computers is the ability to constructively combine amplitudes - if you've only got 1 qubit, there aren't any amplitudes to combine!

The best use case I can think of is randomness. A quantum computer (implemented with arbitrary error) could theoretically be a near perfect source of entropy, whereas a classical computer requires some outside source to contribute randomness (see random.org for more stuff on randomness!)

Seriously, though, to take advantage of constructive interference, you'll need amplitudes over different bitstrings to constructively interfere. :)

Great question!

• alright, thank you! – Pranav Jain Jun 13 at 2:50

We can construct quantum verified delay functions ( QVDF ) and delay authentication ( QDA ) circuits using single quibit quantum circuits. Like quantum randomness generators ( QRN ), these delay functions can be used for auction and lottery systems. We can possibly construct quantum ring structures (QRS) as building blocks for quibit storage, sensing elements and oscillators. Quantum Ring Oscillators ( QRO ) are the circuits where the feedback qubit state alternates in binary basis states with a period proportional to the delay of the circuit elements. When the overall transfer function for the feed-forward stage in a QRS is equivalent to a Pauli-X rotation, it results in a QRO. Qubit storage is achieved through continuous regeneration of the qubit state rather than attempting to preserve the same qubit. Please find further details about single qubit Quantum Ring Structures and the applications in the following research report.

Single Qubit Quantum Ring Structures and Applications