Bootstrapping the stationary state of bosonic open quantum systems

TL;DR

This paper introduces a hierarchy-based method to accurately compute expectation values in the stationary states of bosonic open quantum systems, improving bounds with increasing occupation number, suitable for simulating complex quantum states.

Contribution

It presents a novel hierarchy of semi-definite relaxations that provide rigorous bounds for stationary state expectations, robust to degeneracies, and scalable with occupation number.

Findings

Bounds become tighter with higher occupation numbers

Method is robust to stationary state degeneracies

Applicable to simulating dissipatively stabilized cat qubits

Abstract

We propose a method to compute expectation values of observables in the stationary state of a (Markovian) bosonic open quantum system. Using a hierarchy of semi-definite relaxations, we obtain finer and finer upper and lower bounds to any expectation value of interest. The bounds are rigorous, robust to stationary state degeneracies, and numerically improve as the occupation number increases on the examples we considered. This makes it adapted to the simulation of stationary states of bosonic qubits and in particular dissipatively stabilized cat qubits.