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It is known that in a classical computer we can't generate a purely random number by deterministic process. I am taking a course in Quantum Computing, recently I learnt that using state we can generate a deterministic process which will produce random number $|0\rangle$ and $|1 \rangle $. But, I am not aware of the physical implementation and very excited to know about it. I have searched online and got this paper, but this needs knowledge of architecture inside QC I think.

Can anyone explain(from basic please) how we can develop a deterministic process to generate $|0\rangle$ and $|1 \rangle $ random-ly? Also explain please how that process is a deterministic one.

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We start with the following qbit state:

$|\psi\rangle = |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

Then, we apply the Hadamard gate to that qbit:

$H|\psi\rangle = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}\begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix}\frac{1}{\sqrt{2}} \\ \frac{-1}{\sqrt{2}} \end{bmatrix}$

The resulting state is also known as the $|-\rangle$ state.

We can measure this state in the $|0\rangle$, $|1\rangle$ basis (also called the computational basis) and it will collapse to $|0\rangle$ or $|1\rangle$ with the following probabilities:

$P[|0\rangle] = |\frac{1}{\sqrt{2}}|^2 = \frac 1 2$

$P[|1\rangle] = |\frac{-1}{\sqrt{2}}|^2 = \frac 1 2$

Assuming that quantum mechanics is indeed fundamentally probabilistic, this gives you a deterministic random number generator.

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  • $\begingroup$ Weird use of terminology, calling quantum indeterminism deterministic. $\endgroup$ – jknappen - Reinstate Monica Sep 10 '18 at 17:57
  • $\begingroup$ The process we follow to generate the random number is deterministic. I thought it best to answer the spirit of OP's question rather than delve into semantics. $\endgroup$ – ahelwer Sep 10 '18 at 18:02
  • $\begingroup$ Yes, from the terminology of automata theory or algorithms, this is perfectly deterministic. But the physical side isn't. $\endgroup$ – jknappen - Reinstate Monica Sep 10 '18 at 18:03
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We can physically realise Quantum Randomness using Quantum Random Number Generators which harness quantum randomness through various processes such as boson sampling. When you look at a physical implementation of Quantum Randomness from Quantum Circuits, Hadamard transforms on Qubits would be an interesting approach. In addition, we can try techniques such as Bernoulli Trials or Quantum Oracles for realising Quantum Randomness from deterministic quantum circuits. Random Sampling of deterministic circuit is a recommended approach in this regard. Please find an interesting research concept note in this area. http://www.hri.res.in/~iwqi12/iwqi12_talks/Poster_Akshata.pdf

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