Heisenberg-limited Hamiltonian learning without short-time control

TL;DR

This paper demonstrates that Heisenberg-limited Hamiltonian learning is possible without short-time control, using a new framework that emulates continuous control with minimum evolution times, thus overcoming experimental challenges.

Contribution

It introduces a method to achieve Heisenberg-limited Hamiltonian learning without requiring arbitrarily short control pulses, addressing a key experimental limitation.

Findings

Achieves Heisenberg-limited scaling with minimum evolution time T

Reduces learning to sparse pure-state tomography

Shows polynomial tradeoff for many-body systems

Abstract

Characterizing quantum systems by learning their underlying Hamiltonians is a central task in quantum information science. While recent algorithmic advances have achieved near-optimal efficiency in this task, they critically rely on accessing arbitrarily short-time dynamics. This reliance poses severe experimental challenges due to finite control bandwidth and transient pulse errors. In this work, we demonstrate that Heisenberg-limited Hamiltonian learning can be achieved without short-time control. We introduce a framework in which every query to the unknown dynamics has duration at least a prescribed minimum time $T$, and show that this restriction does not preclude Heisenberg-limited scaling. The key ingredient is a method for emulating the continuous quantum control required by iterative learning algorithms using only such lower-bounded evolution times. This reduces the learning task to sparse pure-state tomography. Notably, for logarithmically sparse Hamiltonians, our algorithm achieves the information-theoretically optimal $1/\varepsilon$ scaling in total evolution time for any arbitrary constant minimum evolution time $T$. For many-body (polynomially sparse) systems, we uncover a rigorous quantitative tradeoff, showing that the minimum required evolution time can be significantly relaxed from the standard limit at a polynomial cost in total evolution time. Our results affirmatively resolve a prominent open problem in the field and reveal that high-bandwidth, ultra-short pulses are not fundamentally necessary for optimal quantum learning.