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Given a Hermitian operator $H$, I can calculate the variance of the Hamiltonian $Var[H]$ as

$$ Var[H] = \langle H^2 \rangle -\langle H \rangle^2 $$

Now, $H$ can be decomposed as

$$ H = \sum_i \alpha_i P_i \hspace{1 cm} P_i \in \{ I,X,Y,Z\}^{\otimes n} $$

Hence by linearity we have $$\langle H \rangle = \sum_i \alpha_i \langle P_i \rangle $$

In the Variational Quantum Eigensolver (VQE), we want to minimize the above expression, that is, $\min \langle H \rangle_{|\psi (\theta) \rangle} $.

Note that if $|\psi (\theta)\rangle$ is indeed an eigenvector of $H$ then $Var[H] = 0 $. And so one can verify whether they have indeed reached/found an eigenstate by performing such calculation.

The question is how do we calculate $\langle H^2 \rangle$?

Yes, I can decomposed $H^2$ into sum of Pauli terms, that is

$$ H^2 = \sum_k \beta_k P_k \hspace{1 cm} P_i \in \{ I,X,Y,Z\}^{\otimes n} $$

and then using the optimized $|\psi (\theta) \rangle$ I have found through VQE to perform $$\langle H^2 \rangle = \langle \psi(\theta) | \sum_k \beta_k P_k | \psi(\theta) \rangle $$

This would be the same process as I did to calculate $\langle H \rangle$, applying the appropriate rotations to my circuit before measuring in the $Z$ basis... Then once I have this, $\langle H^2 \rangle$, I can put it together with the expectation value I have found through VQE to calculate $Var[H]$. This is the naive way of doing it I guess...

What is a better way to do this?

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  • $\begingroup$ Phase estimation? $\endgroup$ Commented Mar 20, 2021 at 10:17
  • $\begingroup$ @NorbertSchuch Thanks for reading through the question and the comment. Yes, that is one way to do it. I was thinking about something more resource friendly. I thought there might be some properties that of $H^2$ that I can take advantage of... for instance, if $H$ was a pauli string then $\langle H^2 \rangle = 1$. $\endgroup$
    – KAJ226
    Commented Mar 20, 2021 at 17:55
  • $\begingroup$ Well, that would be a rather boring Hamiltonian. Do you know something about the Hamiltonian, structure etc? $\endgroup$ Commented Mar 20, 2021 at 18:18
  • $\begingroup$ I am thinking about Hamiltonian that is chemistry related. In particular, the electronic structure problem Hamiltonian, which has the form of equation 1 on page 4 in the following paper ihttps://arxiv.org/pdf/1001.3855v3.pdf $\endgroup$
    – KAJ226
    Commented Mar 20, 2021 at 18:55
  • $\begingroup$ What I am doing right now is I took that Hamiltonian, map it into the form of qubit operators which is a linear combinations of Pauli string ($P_i$) as described in my question. Then I calculate the decomposition of $H^2$ using the linear combinations of Pauli strings of $H$. My program go through the terms to do the multiplication and simplification... at the end I will again have a linear combination of pauli strings. Since $H$ has polynomial number of Pauli strings, $H^2$ will too as well. Then I calculate $\langle H^2 \rangle$ as I have calculated $\langle H \rangle$. $\endgroup$
    – KAJ226
    Commented Mar 20, 2021 at 18:57

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