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Learning Symmetric Hamiltonian

Jing Zhou, D. L. Zhou

TL;DR

This work addresses the problem of uniquely recovering a symmetric Hamiltonian from a single eigenstate by incorporating the Hamiltonian's symmetry group. It extends previous eigenstate-based methods through group representation theory, partitioning the Hilbert space into irreducible subspaces and deriving linear constraints in the form $Q^p\vec{x}=0$ with $\vec{x}=(a_1,\dots,a_N,E)$. A key result is that, without accidental degeneracy, recoverability up to a constant is guaranteed when $2\nu_p\ge N$ and $\operatorname{Rank} Q^p = N$, with a unified approach for accidental degeneracy via $Q^{\psi}$. Numerical demonstrations on the XXX and XXZ spin chains corroborate the theory, providing a principled criterion for Hamiltonian recovery and a foundation for future extensions to more complex or open-system settings.

Abstract

Hamiltonian Learning is a process of recovering system Hamiltonian from measurements, which is a fundamental problem in quantum information processing. In this study, we investigate the problem of learning the symmetric Hamiltonian from its eigenstate. Inspired by the application of group theory in block diagonal secular determination, we have derived a method to determine the number of linearly independent equations about the Hamiltonian unknowns obtained from an eigenstate. This number corresponds to the degeneracy of the associated irreducible representation of the Hamiltonian symmetry group. To illustrate our approach, we examine the XXX Hamiltonian and the XXZ Hamiltonian. We first determine the Hamiltonian symmetry group, then work out the decomposition of irreducible representation, which serves as foundation for analyzing the uniqueness of recovered Hamiltonian. Our numerical findings consistently align with our theoretical analysis.

Learning Symmetric Hamiltonian

TL;DR

This work addresses the problem of uniquely recovering a symmetric Hamiltonian from a single eigenstate by incorporating the Hamiltonian's symmetry group. It extends previous eigenstate-based methods through group representation theory, partitioning the Hilbert space into irreducible subspaces and deriving linear constraints in the form with . A key result is that, without accidental degeneracy, recoverability up to a constant is guaranteed when and , with a unified approach for accidental degeneracy via . Numerical demonstrations on the XXX and XXZ spin chains corroborate the theory, providing a principled criterion for Hamiltonian recovery and a foundation for future extensions to more complex or open-system settings.

Abstract

Hamiltonian Learning is a process of recovering system Hamiltonian from measurements, which is a fundamental problem in quantum information processing. In this study, we investigate the problem of learning the symmetric Hamiltonian from its eigenstate. Inspired by the application of group theory in block diagonal secular determination, we have derived a method to determine the number of linearly independent equations about the Hamiltonian unknowns obtained from an eigenstate. This number corresponds to the degeneracy of the associated irreducible representation of the Hamiltonian symmetry group. To illustrate our approach, we examine the XXX Hamiltonian and the XXZ Hamiltonian. We first determine the Hamiltonian symmetry group, then work out the decomposition of irreducible representation, which serves as foundation for analyzing the uniqueness of recovered Hamiltonian. Our numerical findings consistently align with our theoretical analysis.
Paper Structure (11 sections, 44 equations, 7 tables)

This paper contains 11 sections, 44 equations, 7 tables.

Table of Contents

  1. introduction
  2. Symmetric Hamiltonian learning problem
  3. Principle of symmetric Hamiltonian learning
  4. The case of $H(\{a_n^{\ast}\})$ without accidental degeneracy
  5. The case of $H(\{a_n^{\ast}\})$ with accidental degeneracy
  6. Numerical algorithm for symmetry Hamiltonian learning
  7. Examples demonstrating symmetric Hamiltonian learning
  8. The XXX Spin Chain
  9. The XXZ Spin Chain
  10. Accidental degeneracy: From the XXZ Hamiltonian to the XXX Hamiltonian
  11. conclusions