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In the paper by Stamatopoulos et al., the authors say that it is possible to load a distribution on a three qubit state to obtain: In Qiskit finance this is performed using the uncertainty model function. My question is, how do you encode random numbers on qubit states?

Later is their paper they report that you can use controlled-Y rotations to load a random distribution, this way: but I don't know how they pick the angle for the controlled rotation in order to obtain the desired number.

Maybe you already know this but I just want to point out that it seems to me that the "Loading random distribution" part of the circuit is coming from the RealAmplitude function: https://qiskit.org/documentation/stubs/qiskit.circuit.library.RealAmplitudes.html Which only generates quantum states with real amplitudes. The question whether this can prepare an arbitrary quantum state with real amplitudes is a good question, see this other question here related to that: https://quantumcomputing.stackexchange.com/questions/14032/preparing-arbitrary-two-and-multi-qubit-states-with-real-amplitudes

– KAJ226 – 2020-11-02T18:26:27.153

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From this paper: Entanglement types for two-qubit states with real amplitudes there is a theorem on page 3 that said that:

If we consider the subsets of two qubits states $$RQ_2$$ given by $$\{ |w\rangle = w_1 |00 \rangle + w_2 |01\rangle + w_2|10 \rangle + w_4|11\rangle : w_i \in \mathbb{R} \}$$ then for any pair of states $$|\phi_1 \rangle$$ and $$|\phi_2 \rangle$$ in $$RQ_2$$ thre exists angleds $$\theta_0$$, $$\theta_1$$, $$\theta_2$$ and $$\theta_3$$ such that the circuit send $$|\phi_1\rangle$$ tp $$|\phi_2\rangle$$.

We can decompose the CZ gate into $$(I \otimes H) CNOT (I \otimes H)$$ and absorbed the Hadamard gate into the RY rotation as well to get to the formed in the paper you posted.

In term of determine the specific angles to obtain the right amplitude, I am not sure. But I guess you can use optimization techniques like VQE type technique to find the paprameters that can generates a state with amplitudes that is close to the one you have in mind.

Hopefully other people will provide more in depth answer :)