Paper ID: 2402.05552

Learning quantum Hamiltonians at any temperature in polynomial time with Chebyshev and bit complexity

Ales Wodecki, Jakub Marecek

We consider the problem of learning local quantum Hamiltonians given copies of their Gibbs state at a known inverse temperature, following Haah et al. [2108.04842] and Bakshi et al. [arXiv:2310.02243]. Our main technical contribution is a new flat polynomial approximation of the exponential function based on the Chebyshev expansion, which enables the formulation of learning quantum Hamiltonians as a polynomial optimization problem. This, in turn, can benefit from the use of moment/SOS relaxations, whose polynomial bit complexity requires careful analysis [O'Donnell, ITCS 2017]. Finally, we show that learning a $k$-local Hamiltonian, whose dual interaction graph is of bounded degree, runs in polynomial time under mild assumptions.

Submitted: Feb 8, 2024