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Quantum Probe Tomography

Sitan Chen, Jordan Cotler, Hsin-Yuan Huang

TL;DR

This work introduces quantum probe tomography, a framework for learning unknown quantum many-body Hamiltonians using only a small local probe. It develops a rigorous identifiability analysis by marrying algebraic geometry with smoothed analysis, showing that generic Hamiltonians are identifiable up to simple symmetries under constrained access. Building on these insights, the authors design ProbeLearn, an end-to-end algorithm that, from single-site Gibbs-state probes, recovers Hamiltonian parameters to accuracy $\varepsilon$ with query complexity polynomial in $1/\varepsilon$ and polylogarithmic classical post-processing, and they instantiate the results for translation- and rotation-invariant nearest-neighbor Hamiltonians on $1$D, $2$D, and $3$D square lattices. This demonstrates robust Hamiltonian learning under severely constrained experimental access and provides a concrete, scalable path toward characterizing realistic quantum materials and devices where full control is impractical. The work also lays a versatile algebraic-geometric toolkit with potential applications to broader quantum learning problems under restricted control, offering a foundation for future extensions to long-range, disordered, and many-body bosonic/fermionic systems.

Abstract

Characterizing quantum many-body systems is a fundamental problem across physics, chemistry, and materials science. While significant progress has been made, many existing Hamiltonian learning protocols demand digital quantum control over the entire system, creating a disconnect from many real-world settings that provide access only through small, local probes. Motivated by this, we introduce and formalize the problem of quantum probe tomography, where one seeks to learn the parameters of a many-body Hamiltonian using a single local probe access to a small subsystem of a many-body thermal state undergoing time evolution. We address the identifiability problem of determining which Hamiltonians can be distinguished from probe data through a new combination of tools from algebraic geometry and smoothed analysis. Using this approach, we prove that generic Hamiltonians in various physically natural families are identifiable up to simple, unavoidable structural symmetries. Building on these insights, we design the first efficient end-to-end algorithm for probe tomography that learns Hamiltonian parameters to accuracy $\varepsilon$, with query complexity scaling polynomially in $1/\varepsilon$ and classical post-processing time scaling polylogarithmically in $1/\varepsilon$. In particular, we demonstrate that translation- and rotation-invariant nearest-neighbor Hamiltonians on square lattices in one, two, and three dimensions can be efficiently reconstructed from single-site probes of the Gibbs state, up to inversion symmetry about the probed site. Our results demonstrate that robust Hamiltonian learning remains achievable even under severely constrained experimental access.

Quantum Probe Tomography

TL;DR

This work introduces quantum probe tomography, a framework for learning unknown quantum many-body Hamiltonians using only a small local probe. It develops a rigorous identifiability analysis by marrying algebraic geometry with smoothed analysis, showing that generic Hamiltonians are identifiable up to simple symmetries under constrained access. Building on these insights, the authors design ProbeLearn, an end-to-end algorithm that, from single-site Gibbs-state probes, recovers Hamiltonian parameters to accuracy with query complexity polynomial in and polylogarithmic classical post-processing, and they instantiate the results for translation- and rotation-invariant nearest-neighbor Hamiltonians on D, D, and D square lattices. This demonstrates robust Hamiltonian learning under severely constrained experimental access and provides a concrete, scalable path toward characterizing realistic quantum materials and devices where full control is impractical. The work also lays a versatile algebraic-geometric toolkit with potential applications to broader quantum learning problems under restricted control, offering a foundation for future extensions to long-range, disordered, and many-body bosonic/fermionic systems.

Abstract

Characterizing quantum many-body systems is a fundamental problem across physics, chemistry, and materials science. While significant progress has been made, many existing Hamiltonian learning protocols demand digital quantum control over the entire system, creating a disconnect from many real-world settings that provide access only through small, local probes. Motivated by this, we introduce and formalize the problem of quantum probe tomography, where one seeks to learn the parameters of a many-body Hamiltonian using a single local probe access to a small subsystem of a many-body thermal state undergoing time evolution. We address the identifiability problem of determining which Hamiltonians can be distinguished from probe data through a new combination of tools from algebraic geometry and smoothed analysis. Using this approach, we prove that generic Hamiltonians in various physically natural families are identifiable up to simple, unavoidable structural symmetries. Building on these insights, we design the first efficient end-to-end algorithm for probe tomography that learns Hamiltonian parameters to accuracy , with query complexity scaling polynomially in and classical post-processing time scaling polylogarithmically in . In particular, we demonstrate that translation- and rotation-invariant nearest-neighbor Hamiltonians on square lattices in one, two, and three dimensions can be efficiently reconstructed from single-site probes of the Gibbs state, up to inversion symmetry about the probed site. Our results demonstrate that robust Hamiltonian learning remains achievable even under severely constrained experimental access.

Paper Structure

This paper contains 37 sections, 32 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Defining Quantum Probe Tomography
  4. Computational Model
  5. Physical Rationale
  6. Quantum Instruments for Probe Operations
  7. Constrained Access
  8. Interleaved Dynamics
  9. Probing Translation-Invariant Hamiltonians
  10. Identifiability Problem
  11. End-to-End Learning Guarantee
  12. Discussion
  13. Roadmap for Appendices.
  14. Related Work
  15. Preliminaries
  16. ...and 22 more sections

Key Result

Theorem 2.1

Let $P(\mathbf{x}) \approx \tilde{\mathbf{c}}$ denote the approximate polynomial system obtained from estimating certain derivatives of $A_{\mu,C}(t;\beta)$. If $H$ is a generic member (in the smoothed analysis sense) of a Hamiltonian family for which $P(\mathbf{x})$ satisfies certain efficiently ce

Figures (1)

  • Figure 1: Cartoon depiction of quantum probe tomography. The goal is to learn about a quantum many-body system using a small quantum probe. (a) An illustration of an experimental platform for learning a quantum many-body system (depicted as a two-dimensional array of spins) using a small quantum probe (depicted as a diamond). The probe can only observe and affect a small region of the much larger quantum many-body system. (b) In the model of computation, we consider an experiment to consist of three types of steps: 1. the many-body system thermalizes to an inverse temperature $\beta$ resulting in the thermal state $\frac{e^{-\beta H}}{\mathsf{tr}(e^{-\beta H})}$, 2. the quantum probe performs a quantum channel, a quantum measurement, or a combination of both (quantum instrument) to a small region of the spins, 3. the system evolves under the native Hamiltonian $H$.

Theorems & Definitions (67)

  • Theorem 2.1: Informal, see Theorem \ref{['thm:main_general']}
  • Theorem 2.2
  • Definition B.1
  • Example B.2: Translation-invariant nearest-neighbor Hamiltonians in 1D
  • Definition B.3: Smoothed analysis
  • Lemma B.4: Hadamard formula
  • Theorem B.5: Theorem 3.1 from haah2024learning
  • Lemma B.6: Carbery-Wright
  • Lemma B.7
  • proof
  • ...and 57 more