Universe as a nonlinear quantum simulation: Large- n limit of the central-spin model

Abstract

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 [J. Stat. Phys. 134, 859 (2009)], we establish that the CSM is rigorously dual to a nonlinear qubit when $n\ensuremath{\rightarrow}\ensuremath{\infty}$. The duality supports a type of nonlinear quantum computation in systems, such as a condensate, where a large number of ancillas 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 ancillas, map to effective models with different nonlinearity and symmetry. Without coupling the CSM simulates initial-condition nonlinearity, where the Hamiltonian is a linear combination of $\mathrm{tr}({\ensuremath{\rho}}_{0}{\ensuremath{\sigma}}^{x}){\ensuremath{\sigma}}^{x}, \mathrm{tr}({\ensuremath{\rho}}_{0}{\ensuremath{\sigma}}^{y}){\ensuremath{\sigma}}^{y}$, and $\mathrm{tr}({\ensuremath{\rho}}_{0}{\ensuremath{\sigma}}^{z}){\ensuremath{\sigma}}^{z}$, where ${\ensuremath{\sigma}}^{x}, {\ensuremath{\sigma}}^{y}$, and ${\ensuremath{\sigma}}^{z}$ are Pauli matrices and ${\ensuremath{\rho}}_{0}$ is the initial density matrix. With symmetric ancillas coupling it simulates linear combinations of $\mathrm{tr}(\ensuremath{\rho}{\ensuremath{\sigma}}^{x}){\ensuremath{\sigma}}^{x}, \mathrm{tr}(\ensuremath{\rho}{\ensuremath{\sigma}}^{y}){\ensuremath{\sigma}}^{y}$, and $\mathrm{tr}(\ensuremath{\rho}{\ensuremath{\sigma}}^{z}){\ensuremath{\sigma}}^{z}$, where $\ensuremath{\rho}$ is the current state. This case can simulate qubit torsion, which has been shown by Abrams and Lloyd [Phys. Rev. Lett. 81, 3992 (1998)] to enable an exponential speedup for state discrimination in an idealized setting. The duality discussed here might also be interesting from a quantum foundation 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 $|\ensuremath{\varphi}\ensuremath{\rangle}$ at the time of the big bang and symmetrically coupled to ancillas prepared in the same state would appear to evolve nonlinearly for any finite time $t>0$ as long as there are exponentially many ancillas $n\ensuremath{\gg}\mathrm{exp}[O(t)]$.