Structure learning of Hamiltonians from real-time evolution (2405.00082v2)

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 $H = \sum_{a = 1}^m \lambda_a E_a$ 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 $O(\log (n)/\varepsilon)$, 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 $1/\varepsilon^2$.

• The paper introduces an algorithm that learns quantum Hamiltonian dynamics without requiring prior knowledge of interaction terms.
• It achieves Heisenberg-limited scaling where recovery error inversely correlates with observation time across generalized bounded interactions.
• This breakthrough advances quantum sensing and computation by enabling resource-efficient and scalable characterization of quantum systems.

Unraveling the Heisenberg-Limited Learning of Hamiltonian Structure

Introduction to Hamiltonian Learning

Hamiltonian learning sits at the intersection of quantum physics and machine learning. It's a process that aims to deduce the Hamiltonian of a quantum system based on observations of its dynamics. This is particularly challenging when the Hamiltonian includes interactions that aren't known a priori, a scenario we refer to as "Hamiltonian structure learning".

Hamiltonian Structure Learning

Hamiltonians are pivotal in quantum mechanics; they describe the total energy of a system and dictate its evolution. Conventional Hamiltonian learning techniques assume known interaction terms, focusing solely on determining their strengths. However, the real challenge emerges when these terms aren't known - a quandary called Hamiltonian structure learning.

Gold Standard: Heisenberg-Limited Scaling

A landmark goal in Hamiltonian learning is achieving Heisenberg-limited scaling; this means the error in the Hamiltonian recovery decreases inversely with the time of observation. This scaling is optimal and signifies a profound understanding of the quantum system's dynamics.

Bridging Quantum Sensing and Machine Learning

Hamiltonian learning bridges the theory-heavy world of quantum sensing with practical machine learning algorithms. By interpreting quantum systems through a machine learning lens, researchers can leverage classical data analysis techniques to unravel quantum mechanical properties.

Main Results: A Leap in Hamiltonian Learning

The paper presents an innovative algorithm capable of learning quantum Hamiltonian dynamics without prior knowledge of interaction terms. This algorithm achieves Heisenberg-limited scaling and introduces robustness to several conditions:

• Unknown Interaction Terms: It effectively guesses the interaction structure, significantly widening the application scope.
• Generalized Settings: Unlike previous methods confined to short-range or low-dimensional settings, this algorithm extends to any Hamiltonians with bounded local norm interactions, regardless of the range or spatial dimension.
• Optimal Resource Utilization: It requires a minimal quantum evolution time and maintains constant resolution, marking a technical advancement over prior works that either required fine resolution or suffered from increased time complexity.

Implications and Future Directions

This breakthrough paves the way for practical and scalable benchmarks for quantum computers. It allows for the characterization and verification of quantum devices in more arbitrary and possibly noisy settings without extensive recalibration for each new system tested.

Speculative Outlook

Looking ahead, this algorithm could potentially accommodate even more generalized forms of Hamiltonians, such as those with dynamic or time-dependent interactions. Furthermore, the convergence of techniques from classical machine learning, such as structure learning in graphical models, with quantum system analysis hints at a rich seam of research that might yield even more versatile and powerful quantum learning algorithms.

Concluding Thoughts

In conclusion, the development of a Hamiltonian learning algorithm with Heisenberg-limited scaling and the ability to operate without known interaction terms is a significant stride in quantum computation and sensing. As quantum technology continues to evolve, the tools and techniques for understanding and harnessing its capabilities must also advance, and this research marks a substantial step forward in that direction.