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I am reading through the following article https://arxiv.org/abs/1601.02036.

Eq. (22) describes one of the terms of the gradient of the log-likelihood cost function, which can be estimated using Boltzmann averaging.

Eq. (22) looks like this

$$\frac{Tr[e^{-H}\delta_\theta H]}{Tr[e^{-H}]},$$ and the article states that this term can be estimated by sampling from eq. (12).

Eq. (12) looks like this

$$ \rho = \frac{e^{-H}}{Tr[e^{-H}]}.$$

Can someone give me an explanation of how this is done on a quantum computer? I was thinking that the expectation of the truncated Taylor expansion term $e^{-H} \approx I - H$ could be evaluated $$ \langle v| I - H |v \rangle, $$ for every basis state $|v \rangle$ and this would make it possible to estimate the diagonal of $\rho$. However, this task would be exponential in the number of qubits so I suspect that this is not how to do it. Any ideas?

1reading section V in the paper, it seems like they are saying that these types of $\rho$ can be generated using e.g. annealer devices, but it's not that easy a task – glS – 2020-11-13T17:11:14.457