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Improved algorithms for learning quantum Hamiltonians, via flat polynomials

Shyam Narayanan

TL;DR

The paper presents a Hamiltonian-learning algorithm that reconstructs a low-interaction quantum Hamiltonian from copies of its Gibbs state at any temperature, achieving a singly exponential dependence on the inverse temperature $\beta$ rather than a doubly exponential one. The key advance is a novel flat exponential polynomial $P$ of degree $O(\beta^2 \log(1/\varepsilon))$ that closely approximates $e^{-x}$ on a broad interval, with truncation reducing the effective degree to $O(\beta \log(1/\varepsilon))$ to optimize dependence on $\varepsilon$. This polynomial is integrated into a Sum-of-Squares framework to certify required nonnegativity constraints, allowing the final algorithm to invoke Bakshi–Liu–Moitra–Tang results with improved complexity: $\mathfrak{n} = O\big(m^6 (1/\varepsilon)^{O(\beta^2)} + \log m /(\beta^2 \varepsilon^2)\big)$ samples and a corresponding runtime, both polynomial in $m$ and sub-exponential in $\beta$. By combining a carefully constructed flat-approximation polynomial with robust SOS proofs, the work closes an open question about achieving polynomial-time performance at fixed low temperatures, offering practical gains for quantum Hamiltonian learning in lattice-like systems. The results have potential impact on quantum simulation, material science, and quantum information processing where characterizing many-body Hamiltonians from thermal states is essential.

Abstract

We give an improved algorithm for learning a quantum Hamiltonian given copies of its Gibbs state, that can succeed at any temperature. Specifically, we improve over the work of Bakshi, Liu, Moitra, and Tang [BLMT24], by reducing the sample complexity and runtime dependence to singly exponential in the inverse-temperature parameter, as opposed to doubly exponential. Our main technical contribution is a new flat polynomial approximation to the exponential function, with significantly lower degree than the flat polynomial approximation used in [BLMT24].

Improved algorithms for learning quantum Hamiltonians, via flat polynomials

TL;DR

The paper presents a Hamiltonian-learning algorithm that reconstructs a low-interaction quantum Hamiltonian from copies of its Gibbs state at any temperature, achieving a singly exponential dependence on the inverse temperature rather than a doubly exponential one. The key advance is a novel flat exponential polynomial of degree that closely approximates on a broad interval, with truncation reducing the effective degree to to optimize dependence on . This polynomial is integrated into a Sum-of-Squares framework to certify required nonnegativity constraints, allowing the final algorithm to invoke Bakshi–Liu–Moitra–Tang results with improved complexity: samples and a corresponding runtime, both polynomial in and sub-exponential in . By combining a carefully constructed flat-approximation polynomial with robust SOS proofs, the work closes an open question about achieving polynomial-time performance at fixed low temperatures, offering practical gains for quantum Hamiltonian learning in lattice-like systems. The results have potential impact on quantum simulation, material science, and quantum information processing where characterizing many-body Hamiltonians from thermal states is essential.

Abstract

We give an improved algorithm for learning a quantum Hamiltonian given copies of its Gibbs state, that can succeed at any temperature. Specifically, we improve over the work of Bakshi, Liu, Moitra, and Tang [BLMT24], by reducing the sample complexity and runtime dependence to singly exponential in the inverse-temperature parameter, as opposed to doubly exponential. Our main technical contribution is a new flat polynomial approximation to the exponential function, with significantly lower degree than the flat polynomial approximation used in [BLMT24].
Paper Structure (22 sections, 40 theorems, 58 equations)

This paper contains 22 sections, 40 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Prior work:
  3. Our work.
  4. Problem Statement
  5. Main Theorem
  6. Technical Overview
  7. Flat exponential approximation.
  8. Prior approach.
  9. Our approach.
  10. Reducing the dependence on $\varepsilon$.
  11. Remainder of the algorithm
  12. Preliminaries
  13. Chebyshev Polynomials
  14. Truncation of Polynomials
  15. Sum-of-Squares
  16. ...and 7 more sections

Key Result

Theorem 1.6

Let $H = \sum_{a=1}^m \lambda_a E_a \in \mathbb C^{N \times N}$ be a low-interaction Hamiltonian on $n$ qubits (i.e., the locality $\mathfrak{K}$ and maximum degree $\mathfrak{d}$ of $\mathfrak{G}(H)$ are bounded by some fixed constant). Given knowledge of $E_a$, $\varepsilon \in (0, 1)$, and $\beta copies of the Gibbs state and runtime The big $O$ notation may hide dependencies on $\mathfrak{K}$

Theorems & Definitions (74)

  • Definition 1.2: Local Term
  • Definition 1.3: Hamiltonian
  • Definition 1.4: Low-interaction Hamiltonian haah2022optimal
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4: Markov Brothers' Inequality
  • Proposition 2.5
  • ...and 64 more