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Learning $k$-body Hamiltonians via compressed sensing

Muzhou Ma, Steven T. Flammia, John Preskill, Yu Tong

TL;DR

The paper tackles learning a general $k$-body Hamiltonian with $M$ Pauli terms in a setting free of locality assumptions, using a compression-based approach to identify the sparse set of coefficients. It builds a novel protocol that reshapes the Hamiltonian into a commuting effective form via random Pauli insertions, then recovers the sparse coefficient vector through weight-$k$ Hadamard compressed sensing and robust frequency estimation, all with nonadaptive, SPAM-robust experiments that require only single-qubit control. The main contributions are a nearly Heisenberg-limited total evolution time scaling $T = \widetilde{\mathcal{O}}\left(\frac{M^{1/2+1/p}}{\epsilon}\right)$ for $1\le p\le 2$, a polynomial-time classical post-processing via $\ell^1$-minimization (and its SDP formulation), and a matching lower bound up to log factors, plus explicit robustness to modeling and SPAM errors. The results hold for applications including sparse Sachdev-Ye models and power-law interacting Hamiltonians, offering scalable Hamiltonian learning without locality constraints and with strong practical implications for quantum simulation and control.

Abstract

We study the problem of learning a $k$-body Hamiltonian with $M$ unknown Pauli terms that are not necessarily geometrically local. We propose a protocol that learns the Hamiltonian to precision $ε$ with total evolution time ${\mathcal{O}}(M^{1/2+1/p}/ε)$ up to logarithmic factors, where the error is quantified by the $\ell^p$-distance between Pauli coefficients. Our learning protocol uses only single-qubit control operations and a GHZ state initial state, is non-adaptive, is robust against SPAM errors, and performs well even if $M$ and $k$ are not precisely known in advance or if the Hamiltonian is not exactly $M$-sparse. Methods from the classical theory of compressed sensing are used for efficiently identifying the $M$ terms in the Hamiltonian from among all possible $k$-body Pauli operators. We also provide a lower bound on the total evolution time needed in this learning task, and we discuss the operational interpretations of the $\ell^1$ and $\ell^2$ error metrics. In contrast to most previous works, our learning protocol requires neither geometric locality nor any other relaxed locality conditions.

Learning $k$-body Hamiltonians via compressed sensing

TL;DR

The paper tackles learning a general -body Hamiltonian with Pauli terms in a setting free of locality assumptions, using a compression-based approach to identify the sparse set of coefficients. It builds a novel protocol that reshapes the Hamiltonian into a commuting effective form via random Pauli insertions, then recovers the sparse coefficient vector through weight- Hadamard compressed sensing and robust frequency estimation, all with nonadaptive, SPAM-robust experiments that require only single-qubit control. The main contributions are a nearly Heisenberg-limited total evolution time scaling for , a polynomial-time classical post-processing via -minimization (and its SDP formulation), and a matching lower bound up to log factors, plus explicit robustness to modeling and SPAM errors. The results hold for applications including sparse Sachdev-Ye models and power-law interacting Hamiltonians, offering scalable Hamiltonian learning without locality constraints and with strong practical implications for quantum simulation and control.

Abstract

We study the problem of learning a -body Hamiltonian with unknown Pauli terms that are not necessarily geometrically local. We propose a protocol that learns the Hamiltonian to precision with total evolution time up to logarithmic factors, where the error is quantified by the -distance between Pauli coefficients. Our learning protocol uses only single-qubit control operations and a GHZ state initial state, is non-adaptive, is robust against SPAM errors, and performs well even if and are not precisely known in advance or if the Hamiltonian is not exactly -sparse. Methods from the classical theory of compressed sensing are used for efficiently identifying the terms in the Hamiltonian from among all possible -body Pauli operators. We also provide a lower bound on the total evolution time needed in this learning task, and we discuss the operational interpretations of the and error metrics. In contrast to most previous works, our learning protocol requires neither geometric locality nor any other relaxed locality conditions.

Paper Structure

This paper contains 34 sections, 25 theorems, 205 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main results.
  3. Techniques.
  4. Prior work.
  5. Applications.
  6. Overview
  7. Background.
  8. The Hamiltonian learning protocol.
  9. Lower bounds.
  10. Notations
  11. Background
  12. Hamiltonian reshaping
  13. Compressed sensing
  14. Robust frequency estimation
  15. The Hamiltonian learning protocol
  16. ...and 19 more sections

Key Result

Theorem 1

There exists a learning protocol that uses $N=\widetilde{\mathcal{O}}(M)$ independent non-adaptive experiments to learn, with probability at least $1-\delta$, an estimate $\widehat{\boldsymbol{\mu}}$ of all coefficients with weight $\le k$ that has $\ell^p$ error ($1\leq p\leq 2$) at most $\epsilon$ The protocol is robust to a constant amount of state preparation and measurement (SPAM) noise.

Figures (1)

  • Figure 1: The learning protocol. Quantum processes are illustrated in blue blocks, and classical processes are illustrated in orange blocks. State preparation and measurement (SPAM) errors are taken into account in the experiments.

Theorems & Definitions (50)

  • Theorem : Informal version of Theorem \ref{['thm:ham_learning_upper_bound']}
  • Theorem : Informal version of Theorem \ref{['thm:lower_bound']}
  • Corollary 1: The sparse Sachdev-Ye model
  • Corollary 2: The power law interaction Hamiltonians
  • Theorem : Learning a $k$-body Hamiltonian containing $M$ terms
  • Lemma 1
  • proof
  • Theorem 3
  • Definition 1: Weight-$k$ Hadamard matrix $\mathbf{H}^{(k)}$
  • Theorem 4: Compressed sensing with the weight-$k$ Hadamard matrix
  • ...and 40 more