# Effective descriptions of bosonic systems can be considered complete

## TL;DR

This paper proves that simplified models of bosonic systems accurately capture their true physics, enabling reliable simulations and quantum computing applications.

## Contribution

The authors rigorously show that finite-dimensional approximations and polynomial Hamiltonians fully capture bosonic system behavior.

## Key findings

- Any physical bosonic unitary evolution can be approximated by a finite-dimensional evolution.
- Finite-dimensional evolutions can be generated exactly by polynomial bosonic Hamiltonians.

## Abstract

Bosonic statistics give rise to remarkable phenomena, from the Hong–Ou–Mandel effect to Bose–Einstein condensation, with applications spanning fundamental science to quantum technologies. Modeling bosonic systems relies heavily on effective descriptions: typically, truncating their infinite-dimensional state space or restricting their dynamics to a simple class of Hamiltonians, such as polynomials of canonical operators. However, many natural bosonic Hamiltonians do not belong to these simple classes, and some quantum effects harnessed by bosonic computers inherently require infinite-dimensional spaces. Can we trust that results obtained with such simplifying assumptions capture real effects? We solve this outstanding problem, showing that these effective descriptions do correctly capture the physics of bosonic systems. Our technical contributions are twofold: firstly, we prove that any physical bosonic unitary evolution can be accurately approximated by a finite-dimensional unitary evolution; secondly, we show that any finite-dimensional unitary evolution can be generated exactly by a bosonic Hamiltonian that is a polynomial of canonical operators. Beyond their fundamental significance, our results have implications for classical and quantum simulations of bosonic systems, provide universal methods for engineering bosonic quantum states and Hamiltonians, show that polynomial Hamiltonians generate universal gate sets for quantum computing over bosonic modes, and lead to a bosonic Solovay–Kitaev theorem.

Bosonic systems live in an infinite-dimensional space, and in order to be able to describe them one usually reduces it to an effective finite dimension or constrain the dynamics in some way, but it is not fully understood whether this captures all the physics at play. Here, the authors fill this gap showing that such representations can successfully and rigorously approximate bosonic physics.