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Structure learning of Hamiltonians from real-time evolution

Ainesh Bakshi, Allen Liu, Ankur Moitra, Ewin Tang

TL;DR

The paper tackles Hamiltonian structure learning from real-time evolution for an unknown $\mathfrak{K}$-local Hamiltonian on $n$ qubits, aiming to recover all coefficients without prior knowledge of interaction terms. It introduces a recursive, bootstrapped framework that achieves Heisenberg-limited scaling, with total evolution time $t_{total}=O(\mathfrak{r}\log n/\varepsilon)$ and constant time resolution $t_{min}=\Theta(1/\mathfrak{r})$, while handling long-range and power-law decaying interactions via bounded local norms. A key innovation is a constant-time Trotterization bound enabling discrete control to replace continuous quantum control, together with a structure-learning layer that uses Goldreich–Levin–style queries on Pauli spectra to identify large coefficients efficiently. The method is applicable to both bounded-range and non-local Hamiltonians, including power-law decays that beat the standard $1/\varepsilon^2$ scaling, and it yields a fixed-parameter tractable runtime in many realistic settings. Overall, the work provides a general, scalable framework for quantum-device characterization that bridges quantum metrology, Hamiltonian simulation, and classical structure-learning paradigms.

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 $H = \sum_{a = 1}^m λ_a E_a$ 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, $λ_a$, 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 $1/\varepsilon^2$.

Structure learning of Hamiltonians from real-time evolution

TL;DR

The paper tackles Hamiltonian structure learning from real-time evolution for an unknown -local Hamiltonian on qubits, aiming to recover all coefficients without prior knowledge of interaction terms. It introduces a recursive, bootstrapped framework that achieves Heisenberg-limited scaling, with total evolution time and constant time resolution , while handling long-range and power-law decaying interactions via bounded local norms. A key innovation is a constant-time Trotterization bound enabling discrete control to replace continuous quantum control, together with a structure-learning layer that uses Goldreich–Levin–style queries on Pauli spectra to identify large coefficients efficiently. The method is applicable to both bounded-range and non-local Hamiltonians, including power-law decays that beat the standard scaling, and it yields a fixed-parameter tractable runtime in many realistic settings. Overall, the work provides a general, scalable framework for quantum-device characterization that bridges quantum metrology, Hamiltonian simulation, and classical structure-learning paradigms.

Abstract

We study the problem of Hamiltonian structure learning from real-time evolution: given the ability to apply for an unknown local Hamiltonian on qubits, the goal is to recover . This problem is already well-understood under the assumption that the interaction terms, , are given, and only the interaction strengths, , 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 error with total evolution time , 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 where the sum of terms interacting with a qubit has bounded norm; (3) it evolves according to in constant time increments, thus achieving constant time resolution. As an application, we can also learn Hamiltonians exhibiting power-law decay up to accuracy with total evolution time beating the standard limit of .
Paper Structure (28 sections, 21 theorems, 123 equations, 2 figures, 6 algorithms)

This paper contains 28 sections, 21 theorems, 123 equations, 2 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Results
  3. Technical overview
  4. A recursive framework for Hamiltonian learning with Heisenberg scaling.
  5. Improving an estimate with continuous quantum control.
  6. Implementing the algorithm without continuous quantum control.
  7. Understanding constant-time Trotterization.
  8. Performing Hamiltonian learning with weaker locality guarantees.
  9. Time-efficient structure learning.
  10. Related work
  11. Hamiltonian learning.
  12. Classical learning problems.
  13. Estimating the Pauli spectrum.
  14. Discussion
  15. Background
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $H = \sum_{a=1}^m \lambda_a E_a$ be an unknown $n$-qubit, $\mathfrak{K}$-local Hamiltonian. Suppose $\mathfrak{K} = \bigO{1}$ and $\lonorm{H} = \bigO{1}$. Then given $0< \varepsilon <1$, there exists a quantum algorithm $\mathcal{A}$ that outputs a set of estimated terms, $\{(P, \widehat{\lambda

Figures (2)

  • Figure 1: The basic subroutine of our main algorithm. All of the circuits we consider are of this form. Here, $H_0$ is a known Hamiltonian, and $U$ and $V$ are layers of single-qubit Clifford gates.
  • Figure 2: A comparison of the quantum resources required by Hamiltonian learning algorithms: its total evolution time, time resolution, number of experiments used, and whether it can perform structure learning, respectively. For this comparison, we suppose we have a constant-local $n$-qubit Hamiltonian on a constant-dimensional lattice, and we ignore $\log(n)$ and $\log\log$ dependences.

Theorems & Definitions (61)

  • Theorem 1.1: Learning a local Hamiltonian from real-time evolution, \ref{['thm:main']}
  • Remark 1.2: On $\varepsilon$ dependence
  • Remark 1.3: Other properties
  • Remark 1.4: On gate complexity
  • Corollary 1.5: Informal version of \ref{['cor:power-law']}
  • Remark 1.6
  • Lemma 1.8: Informal version of \ref{['lem:main']}
  • Lemma 1.10: Informal version of \ref{['lem:gl-general']}
  • Definition 2.1: Pauli matrices
  • Definition 2.2: Subset order on Pauli matrices
  • ...and 51 more