Quantum Simulation of Hyperbolic Space with Circuit Quantum Electrodynamics: From Graphs to Geometry

We give a quantum field theoretic perspective on recent breakthrough experiments in circuit quantum electrodynamics, where hyperbolic lattices are realized with superconducting resonators and photons are tricked into believing that space is hyperbolic. We show how these finite lattice geometries can be mapped onto quantum field theories in continuous negatively curved space. We use this as a computational tool to quantitatively reproduce ground state energy, spectral gap, and correlation functions of the noninteracting lattice system by means of analytic formulas on the Poincare disk, and show how conformal symmetry emerges for large lattices. I will discuss how interaction effects can be induced by coupling qubits to the hyperbolic lattice. This sets the stage for studying interactions and disorder on hyperbolic graphs, and to resolve fundamental open problems at the interface of interacting many-body systems, quantum field theory in curved space, and quantum gravity using tabletop experiments.