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Simple algorithms to test and learn local Hamiltonians

Francisco Escudero Gutiérrez

TL;DR

The paper addresses testing and learning $n$-qubit $k$-local Hamiltonians from queries to their time-evolution operators, measured with the $2$-norm of the Pauli spectrum. It introduces a tolerant locality tester with complexity $O\bigl(1/((\varepsilon_2-\varepsilon_1))^8\bigr)$ queries and a learning algorithm with complexity $\exp\bigl(O(k^2+k\log(1/\varepsilon))\bigr)$, both leveraging Pauli-spectrum sampling and the Choi-state formalism. A key contribution is achieving these results without explicit $n$-dependence in learning and providing a simpler tester applicable to general Pauli-structure properties, improving prior work. The methods combine short-time evolutions, Bell-basis measurements, the Montanaro–Osborne SWAP test, and the non-commutative Bohnenblust–Hille inequality to bound large and small Pauli coefficients, enabling efficient verification and reconstruction of local Hamiltonians under the $2$-norm.

Abstract

We consider the problems of testing and learning an $n$-qubit $k$-local Hamiltonian from queries to its evolution operator with respect the 2-norm of the Pauli spectrum, or equivalently, the normalized Frobenius norm. For testing whether a Hamiltonian is $ε_1$-close to $k$-local or $ε_2$-far from $k$-local, we show that $O(1/(ε_2-ε_1)^{8})$ queries suffice. This solves two questions posed in a recent work by Bluhm, Caro and Oufkir. For learning up to error $ε$, we show that $\exp(O(k^2+k\log(1/ε)))$ queries suffice. Our proofs are simple, concise and based on Pauli-analytic techniques.

Simple algorithms to test and learn local Hamiltonians

TL;DR

The paper addresses testing and learning -qubit -local Hamiltonians from queries to their time-evolution operators, measured with the -norm of the Pauli spectrum. It introduces a tolerant locality tester with complexity queries and a learning algorithm with complexity , both leveraging Pauli-spectrum sampling and the Choi-state formalism. A key contribution is achieving these results without explicit -dependence in learning and providing a simpler tester applicable to general Pauli-structure properties, improving prior work. The methods combine short-time evolutions, Bell-basis measurements, the Montanaro–Osborne SWAP test, and the non-commutative Bohnenblust–Hille inequality to bound large and small Pauli coefficients, enabling efficient verification and reconstruction of local Hamiltonians under the -norm.

Abstract

We consider the problems of testing and learning an -qubit -local Hamiltonian from queries to its evolution operator with respect the 2-norm of the Pauli spectrum, or equivalently, the normalized Frobenius norm. For testing whether a Hamiltonian is -close to -local or -far from -local, we show that queries suffice. This solves two questions posed in a recent work by Bluhm, Caro and Oufkir. For learning up to error , we show that queries suffice. Our proofs are simple, concise and based on Pauli-analytic techniques.
Paper Structure (5 sections, 3 theorems, 7 equations)

This paper contains 5 sections, 3 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Testing locality
  4. Learning local Hamiltonians
  5. Acknowledgements.

Key Result

Theorem 3

There is an algorithm that solves the locality testing problem (prob:localitytesting) by making $O(1/(\varepsilon_2-\varepsilon_1)^8\cdot\log(1/\delta))$ queries to the evolution operator and with $O(1/(\varepsilon_2-\varepsilon_1)^7\cdot\log(1/\delta))$ total evolution time.

Theorems & Definitions (5)

  • Theorem 3: Locality testing
  • Theorem 4: Local Hamiltonian learning
  • Claim 5
  • Remark 6
  • Theorem 7: Non-Commutative Bohnenblust-Hille inequality