Quantum Probe Tomography

2025-10-09

AI Summary

This work addresses the problem of learning Hamiltonian parameters of a many-body quantum system from thermal-state time evolution, using only local probes—reflecting realistic experimental constraints where global control or measurement is infeasible. We propose “quantum probe tomography,” a novel framework that integrates algebraic geometry with smooth analysis to rigorously establish identifiability: generic many-body Hamiltonians are uniquely reconstructible from local probe data. Methodologically, we design the first end-to-end efficient learning algorithm, achieving polynomial query complexity—$ ext{poly}(1/varepsilon)$—and polylogarithmic classical post-processing time—$ ext{polylog}(1/varepsilon)$. We validate the approach on translation- and rotation-invariant nearest-neighbor lattice models in one, two, and three dimensions, demonstrating $varepsilon$-accurate Hamiltonian reconstruction. Our key contributions are: (i) the first rigorous proof of local identifiability for generic many-body Hamiltonians under thermal dynamics; and (ii) the first locally constrained Hamiltonian learning scheme with provable polynomial-time guarantees.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Data transparency and provenance

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.

Problem

• Learning many-body Hamiltonian parameters via local probes
• Addressing identifiability of Hamiltonians from probe data
• Developing efficient algorithms for quantum probe tomography

Innovation

• Quantum probe tomography learns Hamiltonians from local probes
• Algebraic geometry and smoothed analysis address identifiability problem
• Efficient algorithm reconstructs Hamiltonians with polynomial query complexity