Include tomography for Gibbs States (Maximum Entropy Principle)

[sebgrijalva]

There is increasing interest in the task of learning quantum Gibbs states. These are parametrized by some numbers $\lambda$ and formed from a set of operators ${E_i}$:

$$ \sigma(\lambda) = \frac{\exp(-\beta \sum_i λ_i E_i) }{ Z(\lambda)} $$

$$ Z(\lambda) = \textrm{tr}( \exp(-\beta \sum_i λ_i E_i) ) $$

Such quantum Gibbs states arise for example as thermal states from a given Hamiltonian, or as states produced by shallow quantum circuits. Learning such state has important implications for the field of Hamiltonian learning (which is connected to QML and Quantum Process Learning). One of the most exciting approaches [1] has been to mix the previously established results using the "Maximum Entropy Principle"[2], with tools of optimal control and Classical Shadows.

[1] https://arxiv.org/abs/2107.03333
[2] See Suplementary Material A of [1]