# Universal Quantum Hamiltonians

## TL;DR

This paper proves that simple spin-lattice models can replicate the physics of any quantum many-body system, establishing a universal framework for quantum simulation and advancing the understanding of quantum phases.

## Contribution

It introduces a rigorous definition of quantum system replication and demonstrates that certain simple models are universally capable of simulating all quantum many-body physics.

## Key findings

- Heisenberg and XY models are universal on 2D lattices.
- Complete classification of two-qubit interactions regarding universality.
- Supports the feasibility of error correction-free quantum simulation.

## Abstract

Quantum many-body systems exhibit an extremely diverse range of phases and physical phenomena. Here, we prove that the entire physics of any other quantum many-body system is replicated in certain simple, "universal" spin-lattice models. We first characterise precisely what it means for one quantum many-body system to replicate the entire physics of another. We then show that certain very simple spin-lattice models are universal in this very strong sense. Examples include the Heisenberg and XY models on a 2D square lattice (with non-uniform coupling strengths). We go on to fully classify all two-qubit interactions, determining which are universal and which can only simulate more restricted classes of models. Our results put the practical field of analogue Hamiltonian simulation on a rigorous footing and take a significant step towards justifying why error correction may not be required for this application of quantum information technology.