Heisenberg-limited Hamiltonian learning without short-time control

2026-04-30

AI Summary

This work proposes a Hamiltonian learning framework that achieves Heisenberg-limited precision without requiring ultrafast control pulses, under the experimental constraint that all evolution operations must last at least a minimum time \( T \). By integrating continuous control simulation with sparse pure-state tomography, the method exclusively employs evolution durations no shorter than \( T \). The study establishes, for the first time, that information-theoretically optimal scaling of total evolution time—specifically \( 1/\varepsilon \) to achieve estimation error \( \varepsilon \)—remains attainable under this constraint: logarithmically sparse Hamiltonians reach the Heisenberg limit, while polynomially sparse many-body systems incur only polynomial overhead due to the minimal-time restriction. These results demonstrate that ultra-high-bandwidth, sub-\( T \) pulses are not essential for optimal quantum learning, resolving a key open question in the field.

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.

Problem

• Hamiltonian learning
• Heisenberg-limited
• short-time control
• quantum dynamics
• evolution time

Innovation

• Heisenberg-limited learning
• Hamiltonian learning
• short-time control
• quantum control emulation
• sparse tomography