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I am confused about how the VQE is able to print out a decimal number for the ground state energy of a molecule.

For example, for $\text{LiH}$, the ground state energy I get for an interatomic distance of $1.5$ was $-7.88210$. What I know is that after a quantum algorithm, it measures the qubit state, and therefore, it would read the state $|0\rangle$ or $|1\rangle$ from a qubit.

So I would like to know how does by measuring a qubit state of $|1\rangle$ or $|0\rangle$ turns into a decimal number to represent a ground state energy?

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Each measurement bitstring corresponds to an eigenvalue of the observable you're measuring. The output of VQE is an estimate of the expectation of the observable given by the average of these eigenvalues over all shots.

For example, suppose your observable is $Y_0 X_1$. Then the eigenvalues corresponding to the four possible outcomes $[00, 01, 10, 11]$ are $[1, -1, -1, 1]$ respectively.

Suppose then that you run 10 shots, getting the results

$$ [11, 00, 00, 01, 10, 10, 11, 11, 00, 00] $$

Plugging in the eigenvalues and taking the average gives the expectation estimate $$ \langle Y_0 X_1 \rangle \approx \frac{1 + 1 + 1 - 1 - 1 - 1 + 1 + 1 + 1 + 1}{10} = 0.4 $$

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