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Analysis of sum-of-squares relaxations for the quantum rotor model

Sujit Rao

TL;DR

A family of Hamiltonians (called the quantum rotor model in condensed matter literature or lattice $O(k)$ vector model in quantum field theory) with infinite-dimensional local Hilbert space is considered and it is shown that a degree-2 ncSoS relaxation approximates the ground state energy better than any product state.

Abstract

The noncommutative sum-of-squares (ncSoS) hierarchy was introduced by Navascués-Pironio-Acín as a sequence of semidefinite programming relaxations for approximating values of noncommutative polynomial optimization problems, which were originally intended to generalize quantum values of nonlocal games. Recent work has started to analyze the hierarchy for approximating ground energies of local Hamiltonians, initially through rounding algorithms which output product states for degree-2 ncSoS applied to Quantum Max-Cut. Some rounding methods are known which output entangled states, but they use degree-4 ncSoS. Based on this, Hwang-Neeman-Parekh-Thompson-Wright conjectured that degree-2 ncSoS cannot beat product state approximations for Quantum Max-Cut and gave a partial proof relying on a conjectural generalization of Borrell's inequality. In this work we consider a family of Hamiltonians (called the quantum rotor model in condensed matter literature or lattice $O(k)$ vector model in quantum field theory) with infinite-dimensional local Hilbert space $L^{2}(S^{k - 1})$, and show that a degree-2 ncSoS relaxation approximates the ground state energy better than any product state.

Analysis of sum-of-squares relaxations for the quantum rotor model

TL;DR

A family of Hamiltonians (called the quantum rotor model in condensed matter literature or lattice vector model in quantum field theory) with infinite-dimensional local Hilbert space is considered and it is shown that a degree-2 ncSoS relaxation approximates the ground state energy better than any product state.

Abstract

The noncommutative sum-of-squares (ncSoS) hierarchy was introduced by Navascués-Pironio-Acín as a sequence of semidefinite programming relaxations for approximating values of noncommutative polynomial optimization problems, which were originally intended to generalize quantum values of nonlocal games. Recent work has started to analyze the hierarchy for approximating ground energies of local Hamiltonians, initially through rounding algorithms which output product states for degree-2 ncSoS applied to Quantum Max-Cut. Some rounding methods are known which output entangled states, but they use degree-4 ncSoS. Based on this, Hwang-Neeman-Parekh-Thompson-Wright conjectured that degree-2 ncSoS cannot beat product state approximations for Quantum Max-Cut and gave a partial proof relying on a conjectural generalization of Borrell's inequality. In this work we consider a family of Hamiltonians (called the quantum rotor model in condensed matter literature or lattice vector model in quantum field theory) with infinite-dimensional local Hilbert space , and show that a degree-2 ncSoS relaxation approximates the ground state energy better than any product state.
Paper Structure (43 sections, 19 theorems, 107 equations)

This paper contains 43 sections, 19 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Classical polynomial optimization
  3. Noncommutative polynomial optimization
  4. Results
  5. Motivation for continuous-variable Hamiltonians
  6. Prior work on complexity of infinite-dimensional or continuous-variable Hamiltonians
  7. Classical motivation for continuous-variable Hamiltonians
  8. The quantum rotor model
  9. Main result
  10. Open questions
  11. Technical overview
  12. Problem
  13. Approach
  14. Approach to writing down a ncSoS relaxation
  15. Approach to rounding the SDP solution
  16. ...and 28 more sections

Key Result

Theorem 1

For $k$ sufficiently large, a level-1 ncSoS relaxation achieves an approximation ratio better than the best possible product state ratio for $H_{G,n,a,b}$.

Theorems & Definitions (35)

  • Theorem 1: informal
  • Theorem 2: Briet_Oliveira_Vallentin_2014
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • ...and 25 more