General conditions for universality of Quantum Hamiltonians

TL;DR

This paper classifies the simulation capabilities of quantum Hamiltonians using complexity theory, providing necessary and sufficient conditions for universality and simplifying previous proofs.

Contribution

It introduces a complete classification of universal quantum Hamiltonians based on their complexity classes, advancing the understanding of Hamiltonian universality.

Findings

Derived necessary and sufficient complexity conditions for universality.

Provided simplified proofs of existing universality results.

Connected Hamiltonian universality to complexity theory classes.

Abstract

Recent work has demonstrated the existence of universal Hamiltonians - simple spin lattice models that can simulate any other quantum many body system to any desired level of accuracy. Until now proofs of universality have relied on explicit constructions, tailored to each specific family of universal Hamiltonians. In this work we go beyond this approach, and completely classify the simulation ability of quantum Hamiltonians by their complexity classes. We do this by deriving necessary and sufficient complexity theoretic conditions characterising universal quantum Hamiltonians. Although the result concerns the theory of analogue Hamiltonian simulation - a promising application of near-term quantum technology - the proof relies on abstract complexity theoretic concepts and the theory of quantum computation. As well as providing simplified proofs of previous Hamiltonian universality results, and offering a route to new universal constructions, the results in this paper give insight into the origins of universality. For example, finally explaining the previously noted coincidences between families of universal Hamiltonian and classes of Hamiltonians appearing in complexity theory.