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

arXiv: https://arxiv.org/abs/2402.05552 · HTML: https://mlsanity.com/paper/2402.05552/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 28 references that are in the map.
- First posted on arXiv 8 Feb 2024.

## Builds on (most cited first, 6 of 6)

- [A survey on the complexity of learning quantum states](https://mlsanity.com/paper/2305.20069/) (2023)
- [Learning quantum Hamiltonians at any temperature in polynomial time](https://mlsanity.com/paper/2310.02243/) (2023)
- [Quantum open system identification via global optimization: Optimally accurate Markovian models of open systems from time-series data](https://mlsanity.com/paper/2203.17164/) (2022)
- [How do exponential size solutions arise in semidefinite programming?](https://mlsanity.com/paper/2103.00041/) (2021)
- [Globally optimal control of quantum dynamics](https://mlsanity.com/paper/2209.05790/) (2022)
- [Quantum Optimal Control via Magnus Expansion and Non-Commutative Polynomial Optimization](https://mlsanity.com/paper/2001.06464/) (2020)

## Abstract (arXiv)

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.