  We explore the hyperbolic band theory under a magnetic field for the first
time. Our theory is a general extension of the conventional band theory defined
on a Euclidean lattice into the band theory on a general hyperbolic
lattice/Riemann surface. Our methods and results can be experimentally
confirmed by circuit quantum electrodynamics (cQED), which enables us to create
novel materials in a hyperbolic space. To investigate the band structures, we
directly construct the hyperbolic magnetic Bloch states and find that they form
Dirac cones on a coordinate neighborhood, by which they can be regarded as a
global quantum gravity solution detectable in a laboratory. Besides this is the
first explicit example of a massless Dirac state on a higher genus surface.
Moreover we show that the energy spectrum exhibits an unusual fractal
structure, when plotted as a function of a magnetic flux.
