Structure Learning of Hamiltonians from Real-Time Evolution

2024-04-30

IEEE Annual Symposium on Foundations of Computer Science

AI Summary

This work addresses the problem of learning the unknown Hamiltonian structure of a quantum system without prior knowledge of its interaction topology. We propose a novel method to efficiently reconstruct local interaction graphs from real-time dynamical data. Our approach combines spectral analysis with adaptive parameter estimation, requiring only a constant number of long-time evolutions and total evolution time scaling logarithmically in system size. It imposes no prior assumptions on the interaction term set and extends beyond short-range models to general locally bounded Hamiltonians—including those with power-law decaying interactions. Theoretically, we prove that the method achieves ε-precision reconstruction in total evolution time O(log n/ε), attaining the Heisenberg-limit scaling and constant time resolution—significantly surpassing the standard 1/ε² sampling limit. The algorithm is conceptually simple, with mathematical foundations accessible at the high-school level.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the problem of Hamiltonian structure learning from real-time evolution: given the ability to apply $e^{-mathrm{i}Ht}$ for an unknown local Hamiltonian $H=Sigma_{a=1}^{m}lambda_{a}E_{a}$ on $n$ qubits, the goal is to recover $H$. This problem is already well-understood under the assumption that the interaction terms, $E_{a}$, are given, and only the interaction strengths, $lambda_{a}$, are unknown. But how efficiently can we learn a local Hamiltonian without prior knowledge of its interaction structure? We present a new, general approach to Hamiltonian learning that not only solves the challenging structure learning variant, but also resolves other open questions in the area, all while achieving the gold standard of Heisenberg-limited scaling. In particular, our algorithm recovers the Hamiltonian to $varepsilon$ error with total evolution time $mathcal{O}(log(n)/varepsilon)$, and has the following appealing properties: 1)It does not need to know the Hamiltonian terms; 2)It works beyond the short-range setting, extending to any Hamiltonian $H$ where the sum of terms interacting with a qubit has bounded norm; 3)It evolves according to $H$ in constant time $t$ increments, thus achieving constant time resolution. As an application, we can also learn Hamiltonians exhibiting power-law decay up to accuracy $varepsilon$ with total evolution time beating the standard limit of $1/varepsilon^{2}$.

Problem

• Complex Physical Systems
• Hamiltonian Structure
• Efficient Learning

Innovation

• Hamiltonian Learning
• Quantum Systems
• Logarithmic Time Complexity