The universe as a nonlinear quantum simulation: Large $n$ limit of the central spin model

We investigate models of nonlinear qubit evolution based on mappings to an $n$-qubit central spin model (CSM) in the large $n$ limit, where mean field theory is exact. Extending a theorem of Erd\"os and Schlein, we establish that the CSM is rigorously dual to a nonlinear qubit when $n \rightarrow \infty$. The duality supports a type of nonlinear quantum computation in systems, such as a condensate, where a large number of ancilla couple symmetrically to a "central" qubit. It also enables a gate-model implementation of nonlinear quantum simulation with a rigorous error bound. Two variants of the model, with and without coupling between ancilla, map to effective models with different nonlinearity and symmetry. The duality discussed here might also be interesting from a quantum foundations perspective. There has long been interest in whether quantum mechanics might possess some type of small, unobserved nonlinearity. If not, what is the principle prohibiting it? The duality implies that there is not a sharp distinction between universes evolving according to linear and nonlinear quantum mechanics: A one-qubit "universe" prepared in a pure state $| \varphi \rangle $ at the time of the big bang and symmetrically coupled to ancilla prepared in the same state, would appear to evolve nonlinearly for any finite time $t>0$ as long as there are exponentially many ancilla $n \gg {\rm exp}(O(t))$.