# Heisenberg-limited Hamiltonian learning without short-time control

arXiv: https://arxiv.org/abs/2604.27838 · HTML: https://mlsanity.com/paper/2604.27838/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Apr 2026.

## Built on by (most cited first, 3 of 3)

- [Provable learning separation for predicting time-evolution of quantum many-body systems](https://mlsanity.com/paper/2607.06472/) (2026)
- [Characterizing Arbitrary Lindbladian Dynamics with a Few Pauli Measurements](https://mlsanity.com/paper/2607.23044/) (2026)
- [Hamiltonian locality testing and certification do not achieve the Heisenberg limit](https://mlsanity.com/paper/2610.03205/) (2026)

## Abstract (arXiv)

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.