Learning quantum Hamiltonians at any temperature in polynomial time with Chebyshev and bit complexity
Ales Wodecki, Jakub Marecek
TL;DR
Given copies of the Gibbs state $\rho = \frac{e^{-\beta H}}{\mathrm{Tr}(e^{-\beta H})}$ at known inverse temperature $\beta$, the paper tackles learning a $k$-local quantum Hamiltonian with bounded-degree dual interaction graphs. It introduces a flat exponential approximation based on Chebyshev expansions, via the product construction $Q_{k,l}$, to enable polynomial optimization and moment/SOS relaxations with controlled dimension and bit complexity. The authors prove that under mild assumptions the learning problem can be solved in polynomial time with polynomial-bit complexity, leveraging explicit bounding conditions for SOS. This work advances scalable quantum Hamiltonian identification from Gibbs-state data and has implications for quantum control and open-system modeling.
Abstract
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.